how long is math phd

Ph.D. Program

Degree requirements.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

• Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics
• Pass the six-hour written Preliminary Examination covering calculus, real analysis, complex analysis, linear algebra, and abstract algebra; students must pass the prelim before the start of their second year in the program (within three semesters of starting the program)
• Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted within two years of entering the program
• Complete a seminar, giving a talk of at least one-hour duration
• Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee
• Meet the University residence requirement of two years or four semesters

Detailed Regulations

The detailed regulations of the Ph.D. program are the following:

Course Requirements

During the first year of the Ph.D. program, the student must enroll in at least 4 courses. At least 2 of these must be graduate courses offered by the Department of Mathematics. Exceptions can be granted by the Vice-Chair for Graduate Studies.

Preliminary Examination

The Preliminary Examination consists of 6 hours (total) of written work given over a two-day period (3 hours/day). Exam questions are given in calculus, real analysis, complex analysis, linear algebra, and abstract algebra. The Preliminary Examination is offered twice a year during the first week of the fall and spring semesters.

Qualifying Examination

To arrange the Qualifying Examination, a student must first settle on an area of concentration, and a prospective Dissertation Advisor (Dissertation Chair), someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective advisor, the student forms an examination committee of 4 members.  All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The QE chair and Dissertation Chair cannot be the same person; therefore, t he Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee . The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval, it is to be circulated to all faculty members of the appropriate research sections. The Qualifying Examination must cover material falling in at least 3 subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be reviewed online or in 910 Evans Hall. The student must attempt the Qualifying Examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program. For a student to pass the Qualifying Examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student.

Department of Mathematics

Mathematics phd program.

The Ph.D. program in the Department of Mathematics provides students with in-depth knowledge and rigorous training in all the subject areas of mathematics. A core feature is the first-year program, which helps bring students to the forefront of modern mathematics. Students work closely with faculty and each other and participate fully in both research and student-run seminars.

Questions? Email [email protected]

• The firm deadline for applications for Autumn 2024, is December 5, 2024.
• The (general and advanced) GRE tests are no longer accepted. Please do not submit these scores.

Overview of the PhD Program

For specific information on the Applied Mathematics PhD program, see the navigation links to the right. 

What follows on this page is an overview of all Ph.D. programs at the School; additional information and guidance can be found on the  Graduate Policies  pages. 

General Ph.D. Requirements

• 10 semester-long graduate courses, including at least 8 disciplinary.   At least 5 of the 10 should be graduate-level SEAS "technical" courses (or FAS graduate-level technical courses taught by SEAS faculty), not including seminar/reading/project courses.  Undergraduate-level courses cannot be used.  For details on course requirements, see the school's overall PhD course requirements  and the individual program pages linked therein.
• Program Plan (i.e., the set of courses to be used towards the degree) approval by the  Committee on Higher Degrees  (CHD).
• Minimum full-time academic residency of two years .
• Serve as a Teaching Fellow (TF) in one semester of the second year.
• Oral Qualifying Examination Preparation in the major field is evaluated in an oral examination by a qualifying committee. The examination has the dual purpose of verifying the adequacy of the student's preparation for undertaking research in a chosen field and of assessing the student's ability to synthesize knowledge already acquired. For details on arranging your Qualifying Exam, see the exam policies and the individual program pages linked therein.
• Committee Meetings : PhD students' research committees meet according to the guidelines in each area's "Committee Meetings" listing.  For details see the "G3+ Committee Meetings" section of the Policies of the CHD  and the individual program pages linked therein.
• Final Oral Examination (Defense) This public examination devoted to the field of the dissertation is conducted by the student's research committee. It includes, but is not restricted to, a defense of the dissertation itself.  For details of arranging your final oral exam see the  Ph.D. Timeline  page.
• Dissertation Upon successful completion of the qualifying examination, a committee chaired by the research supervisor is constituted to oversee the dissertation research. The dissertation must, in the judgment of the research committee, meet the standards of significant and original research.

Optional additions to the Ph.D. program

Harvard PhD students may choose to pursue these additional aspects:

• a Secondary Field (which is similar to a "minor" subject area).  SEAS offers PhD Secondary Field programs in  Data Science and in  Computational Science and Engineering .   GSAS  lists  secondary fields offered by other programs.
• a Master of Science (S.M.) degree conferred  en route to the Ph.D in one of several of SEAS's subject areas.  For details see here .
• a Teaching Certificate awarded by the Derek Bok Center for Teaching and Learning .

SEAS PhD students may apply to participate in the  Health Sciences and Technology graduate program  with Harvard Medical School and MIT.  Please check with the HST program for details on eligibility (e.g., only students in their G1 year may apply) and the application process.

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Learn more about Harvard’s Math community and our statement on diversity and inclusion.

The Harvard Griffin GSAS Office of Equity, Diversity, Inclusion & Belonging offers student affinity groups for graduate students and many other resources.

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What are the deadlines for applications? December 1st is the priority deadline for applicants;  we strongly encourage applicants to submit their applications by the priority deadline.  Please see the admissions page for the  final deadline for remaining applicants; a pplicants who apply by the final deadline will still have their files formally reviewed by the admissions committee for full consideration, but not necessarily on a priority basis. 

When are applications reviewed? The admissions committee will begin reviewing applications as early as December 1st, but everyone who applies by the deadline receives full consideration.  The admissions page shows when applicants can expect a response.  We kindly request that you not email us to ask about the admissions decision unless you have not heard from us and February 15th has passed.

How can I get feedback on my application? PhD applicants who identify as members of underrepresented or marginalized groups may receive feedback from the volunteers in the  PAR program . 

Can I apply for any quarter? We only offer autumn admission.

How can I request an application fee waiver?  Step 1: See if you qualify for a UW Grad School application fee  waiver  and follow the instructions provided if so.  Step 2: If ineligible for the above waiver, the department offers a limited number of waivers. Waivers may be granted to applicants who identify as members of underrepresented or marginalized groups. Waivers may also be granted on a need basis. If requesting a waiver for these reasons, please  email  [email protected]  to explain why you are seeking a waiver  (please include your citizenship).

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• Doing a PhD in Mathematics
• Doing a PhD

What Does a PhD in Maths Involve?

Maths is a vast subject, both in breadth and in depth. As such, there’s a significant number of different areas you can research as a math student. These areas usually fall into one of three categories: pure mathematics, applied mathematics or statistics. Some examples of topics you can research are:

• Number theory
• Numerical analysis
• String theory
• Random matrix theory
• Graph theory
• Quantum mechanics
• Statistical forecasting
• Matroid theory
• Control theory

Besides this, because maths focuses on addressing interdisciplinary real-world problems, you may work and collaborate with other STEM researchers. For example, your research topic may relate to:

• Biomechanics and transport processes
• Evidence-based medicine
• Fluid dynamics
• Financial mathematics
• Machine learning
• Theoretical and Computational Optimisation

What you do day-to-day will largely depend on your specific research topic. However, you’ll likely:

• Continually read literature – This will be to help develop your knowledge and identify current gaps in the overall body of knowledge surrounding your research topic.
• Undertake research specific to your topic – This can include defining ideas, proving theorems and identifying relationships between models.
• Collect and analyse data – This could comprise developing computational models, running simulations and interpreting forecasts etc.
• Liaise with others – This could take many forms. For example, you may work shoulder-to-shoulder with individuals from different disciplines supporting your research, e.g. Computer scientists for machine learning-based projects. Alternatively, you may need frequent input from those who supplied the data for your research, e.g. Financial institutions or biological research colleagues.
• Attend a wide range of lectures, seminars and events.

Browse PhD Opportunities in Mathematics

Application of artificial intelligence to multiphysics problems in materials design, study of the human-vehicle interactions by a high-end dynamic driving simulator, physical layer algorithm design in 6g non-terrestrial communications, machine learning for autonomous robot exploration, detecting subtle but clinically significant cognitive change in an ageing population, how long does it take to get a phd in maths.

The average programme duration for a mathematics PhD in the UK is 3 to 4 years for a full-time studying. Although not all universities offer part-time maths PhD programmes, those that do have a typical programme duration of 5 to 7 years.

Again, although the exact arrangement will depend on the university, most maths doctorates will require you to first register for an MPhil . At the end of your first year, your supervisor will assess your progress to decide whether you should be registered for a PhD.

Additional Learning Modules

Some Mathematics departments will require you to enrol on to taught modules as part of your programme. These are to help improve your knowledge and understanding of broader subjects within your field, for example, Fourier Analysis, Differential Geometry and Riemann Surfaces. Even if taught modules aren’t compulsory in several universities, your supervisor will still encourage you to attend them for your development.

Most UK universities will also have access to specialised mathematical training courses. The most common of these include Pure Mathematics courses hosted by Mathematics Access Grid Conferencing ( MAGIC ) and London Taught Course Centre ( LTCC ) and Statistics courses hosted by Academy for PhD Training in Statistics ( APTS ).

What Are the Typical Entry Requirements for A PhD in Maths?

In the UK, the typical entry requirements for a Maths PhD is an upper second-class (2:1) Master’s degree (or international equivalent) in Mathematics or Statistics [1] .

However, there is some variation on this. From writing, the lowest entry requirement is an upper second-class (2:1) Bachelor’s degree in any math-related subject. The highest entry requirement is a first-class (1st) honours Master’s degree in a Mathematics or Statistics degree only.

It’s worth noting if you’re applying to a position which comes with funding provided directly by the Department, the entry requirements will usually be on the higher side because of their competitiveness.

In terms of English Language requirements, most mathematics departments require at least an overall IELTS (International English Language Testing System) score of 6.5, with no less than 6.0 in each individual subtest.

Tips to Consider when Making Your Application

When applying to any mathematics PhD, you’ll be expected to have a good understanding of both your subject field and the specific research topic you are applying to. To help show this, it’s advisable that you demonstrate recent engagement in your research topic. This could be by describing the significance of a research paper you recently read and outlining which parts interested you the most, and why. Additionally, you can discuss a recent mathematics event you attended and suggest ways in how what you learnt might apply to your research topic.

As with most STEM PhDs, most maths PhD professors prefer you to discuss your application with them directly before putting in a formal application. The benefits of this is two folds. First, you’ll get more information on what their department has to offer. Second, the supervisor can better discover your interest in the project and gauge whether you’d be a suitable candidate. Therefore, we encourage you to contact potential supervisors for positions you’re interested in before making any formal applications.

How Much Does a Maths PhD Typically Cost?

The typical tuition fee for a PhD in Maths in the UK is £4,407 per year for UK/EU students and £20,230 per year for international students. This, alongside the range in tuition fees you can expect, is summarised below:

Note: The above tuition fees are based on 12 UK Universities [1]  for 2020/21 Mathematic PhD positions. The typical fee has been taken as the median value.

In addition to the above, it’s not unheard of for research students to be charged a bench fee. In case you’re unfamiliar with a bench fee, it’s an annual fee additional to your tuition, which covers the cost of specialist equipment or resources associated with your research. This can include the upkeep of supercomputers you may use, training in specialist analysis software, or travelling to conferences. The exact fee will depend on your specific research topic; however, it should be minimal for most mathematic projects.

What Specific Funding Opportunities Are There for A PhD in Mathematics?

Alongside the usual funding opportunities available to all PhD Research students such as doctoral loans, departmental scholarships, there are a few other sources of funding available to math PhD students. Examples of these include:

You can find more information on these funding sources here: DiscoverPhDs funding guide .

What Specific Skills Do You Gain from Doing a PhD in Mathematics?

A doctorate in Mathematics not only demonstrates your commitment to continuous learning, but it also provides you with highly marketable skills. Besides subject-specific skills, you’ll also gain many transferable skills which will prove useful in almost all industries. A sample of these skills is listed below.

• Logical ability to consider and analyse complex issues,
• Commitment and persistence towards reaching research goals,
• Outstanding verbal and written skills,
• Strong attention to detail,
• The ability to liaise with others from unique disciple backgrounds and work as part of a team
• Holistic deduction and reasoning skills,
• Forming and explaining mathematical and logical solutions to a wide range of real-world problems,
• Exceptional numeracy skills.

What Jobs Can You Get with A Maths PhD?

One of the greatest benefits maths PostDocs will have is the ability to pursue a wide range of career paths. This is because all sciences are built on core principles which, to varying extents, are supported by the core principles of mathematics. As a result, it’s not uncommon to ask students what path they intend to follow after completing their degree and receive entirely different answers. Although not extensive by any means, the most common career paths Math PostDocs take are listed below:

• Academia – Many individuals teach undergraduate students at the university they studied at or ones they gained ties to during their research. This path is usually the preferred among students who want to continue focusing on mathematical theories and concepts as part of their career.
• Postdoctoral Researcher – Others continue researching with their University or with an independent organisation. This can be a popular path because of the opportunities it provides in collaborative working, supervising others, undertaking research and attending conferences etc.
• Finance – Because of their deepened analytical skills, it’s no surprise that many PostDocs choose a career in finance. This involves working for some of the most significant players in the financial district in prime locations including London, Frankfurt and Hong Kong. Specific job titles can include Actuarial, Investment Analyst or Risk Modeller.
• Computer Programming – Some students whose research involves computational mathematics launch their career as a computer programmer. Due to their background, they’ll typically work on specialised projects which require high levels of understanding on the problem at hand. For example, they may work with physicists and biomedical engineers to develop a software package that supports their more complex research.
• Data Analyst – Those who enjoy number crunching and developing complex models often go into data analytics. This can involve various niches such as forecasting or optimisation, across various fields such as marketing and weather.

What Are Some of The Typical Employers Who Hire Maths PostDocs?

As mentioned above, there’s a high demand for skilled mathematicians and statisticians across a broad range of sectors. Some typical employers are:

• Education – All UK and international universities
• Governments – STFC and Department for Transport
• Healthcare & Pharmaceuticals – NHS, GSK, Pfizer
• Finance & Banking – e.g. Barclays Capital, PwC and J. P. Morgan
• Computing – IBM, Microsoft and Facebook
• Engineering – Boeing, Shell and Dyson

The above is only a small selection of employers. In reality, mathematic PostDocs can work in almost any industry, assuming the role is numerical-based or data-driven.

How Much Can You Earn with A PhD in Maths?

As a mathematics PhD PostDoc, your earning potential will mostly depend on your chosen career path. Due to the wide range of options, it’s impossible to provide an arbitrary value for the typical salary you can expect.

However, if you pursue one of the below paths or enter their respective industry, you can roughly expect to earn [3] :

Academic Lecturer

• Approximately £30,000 – £35,000 starting salary
• Approximately £40,000 with a few years experience
• Approximately £45,000 – £55,000 with 10 years experience
• Approximately £60,000 and over with significant experience and a leadership role. Certain academic positions can earn over £80,000 depending on the management duties.

Actuary or Finance

• Approximately £35,000 starting salary
• Approximately £45,000 – £55,000 with a few years experience
• Approximately £70,000 and over with 10 years experience
• Approximately £180,000 and above with significant experience and a leadership role.

Aerospace or Mechanical Engineering

• Approximately £28,000 starting salary
• Approximately £35,000 – £40,000 with a few years experience
• Approximately £60,000 and over with 10 years experience

Data Analyst

• Approximately £45,000 – £50,000 with a few years experience
• Approximately £90,000 and above with significant experience and a leadership role.

Again, we stress that the above are indicative values only. Actual salaries will depend on the specific organisation and position and responsibilities of the individual.

Facts and Statistics About Maths PhD Holders

The below chart provides useful insight into the destination of Math PostDocs after completing their PhD. The most popular career paths from other of highest to lowest is education, information and communication, finance and scientific research, manufacturing and government.

Note: The above chart is based on ‘UK Higher Education Leavers’ data [2] between 2012/13 and 2016/17 and contains a data size of 200 PostDocs. The data was obtained from the Higher Education Statistics Agency ( HESA ).

Which Noteworthy People Hold a PhD in Maths?

Alan turing.

Alan Turing was a British Mathematician, WW2 code-breaker and arguably the father of computer science. Alongside his lengthy list of achievements, Turning achieved a PhD in Mathematics at Princeton University, New Jersey. His thesis titled ‘Systems of Logic Based on Ordinals’ focused on the concepts of ordinal logic and relative computing; you can read it online here . To this day, Turning pioneering works continues to play a fundamental role in shaping the development of artificial intelligence (AI).

Ruth Lawrence

Ruth Lawrence is a famous British–Israeli Mathematician well known within the academic community. Lawrence earned her PhD in Mathematics from Oxford University at the young age of 17! Her work focused on algebraic topology and knot theory; you can read her interesting collection of research papers here . Among her many contributions to Maths, her most notable include the representation of the braid groups, more formally known as Lawrence–Krammer representations.

Emmy Noether

Emmy Noether was a German mathematician who received her PhD from the University of Erlangen, Germany. Her research has significantly contributed to both abstract algebra and theoretical physics. Additionally, she proved a groundbreaking theorem important to Albert Einstein’s general theory of relativity. In doing so, her theorem, Noether’s theorem , is regarded as one of the most influential developments in physics.

Other Useful Resources

Institute of Mathematics and its Applications (IMA) – IMA is the UK’s professional body for mathematicians. It contains a wide range of useful information, from the benefits of further education in Maths to details on grants and upcoming events.

Maths Careers – Math Careers is a site associated with IMA that provides a wide range of advice to mathematicians of all ages. It has a section dedicated to undergraduates and graduates and contains a handful of information about progressing into research.

Resources for Graduate Students – Produced by Dr Mak Tomford, this webpage contains an extensive collection of detailed advice for Mathematic PhD students. Although the site uses US terminology in places, don’t let that put you off as this resource will prove incredibly helpful in both applying to and undertaking your PhD.

Student Interviews – Still wondering whether a PhD is for you? If so, our collection of PhD interviews would be a great place to get an insider perspective. We’ve interviewed a wide range of PhD students across the UK to find out what doing a PhD is like, how it’s helped them and what advice they have for other prospective students who may be thinking of applying to one. You can read our insightful collection of interviews here .

[1] Universities used to determine the typical (median) and range of entry requirements and tuition fees for 2020/21 Mathematics PhD positions.

• http://www.lse.ac.uk/study-at-lse/Graduate/Degree-programmes-2020/MPhilPhD-Mathematics
• https://www.ox.ac.uk/admissions/graduate/courses/dphil-mathematics?wssl=1
• https://www.graduate.study.cam.ac.uk/courses/directory/mapmpdpms
• https://www.ucl.ac.uk/prospective-students/graduate/research-degrees/mathematics-mphil-phd
• http://www.bristol.ac.uk/study/postgraduate/2020/sci/phd-mathematics/
• https://www.surrey.ac.uk/postgraduate/mathematics-phd
• https://www.maths.ed.ac.uk/school-of-mathematics/studying-here/pgr/phd-application
• https://www.lancaster.ac.uk/study/postgraduate/postgraduate-courses/mathematics-phd/
• https://www.sussex.ac.uk/study/phd/degrees/mathematics-phd
• https://www.manchester.ac.uk/study/postgraduate-research/programmes/list/05325/phd-pure-mathematics/
• https://warwick.ac.uk/study/postgraduate/research/courses-2020/mathematicsphd/
• https://www.exeter.ac.uk/pg-research/degrees/mathematics/

[2] Higher Education Leavers Statistics: UK, 2016/17 – Outcomes by subject studied – https://www.hesa.ac.uk/news/28-06-2018/sfr250-higher-education-leaver-statistics-subjects

[3] Typical salaries have been extracted from a combination of the below resources. It should be noted that although every effort has been made to keep the reported salaries as relevant to Math PostDocs as possible (i.e. filtering for positions which specify a PhD qualification as one of their requirements/preferences), small inaccuracies may exist due to data availability.

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How Long Does It Take to Get a Ph.D. Degree?

Earning a Ph.D. from a U.S. grad school typically requires nearly six years, federal statistics show.

How Long It Takes to Get a Ph.D. Degree

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A Ph.D. is most appropriate for someone who is a "lifelong learner."

Students who have excelled within a specific academic discipline and who have a strong interest in that field may choose to pursue a Ph.D. degree. However, Ph.D. degree-holders urge prospective students to think carefully about whether they truly want or need a doctoral degree, since Ph.D. programs last for multiple years.

According to the Survey of Earned Doctorates, a census of recent research doctorate recipients who earned their degree from U.S. institutions, the median amount of time it took individuals who received their doctorates in 2017 to complete their program was 5.8 years. However, there are many types of programs that typically take longer than six years to complete, such as humanities and arts doctorates, where the median time for individuals to earn their degree was 7.1 years, according to the survey.

Some Ph.D. candidates begin doctoral programs after they have already obtained master's degrees, which means the time spent in grad school is a combination of the time spent pursuing a master's and the years invested in a doctorate. In order to receive a Ph.D. degree, a student must produce and successfully defend an original academic dissertation, which must be approved by a dissertation committtee. Writing and defending a dissertation is so difficult that many Ph.D. students drop out of their Ph.D. programs having done most of the work necessary for degree without completing the dissertation component. These Ph.D. program dropouts often use the phrase " all but dissertation " or the abbreviation "ABD" on their resumes.

According to a comprehensive study of Ph.D. completion rates published by The Council of Graduate Schools in 2008, only 56.6% of people who begin Ph.D. programs earn Ph.D. degrees.

Ian Curtis, a founding partner with H&C Education, an educational and admissions consulting firm, who is pursuing a Ph.D. degree in French at Yale University , says there are several steps involved in the process of obtaining a Ph.D. Students typically need to fulfill course requirements and pass comprehensive exams, Curtis warns. "Once these obligations have been completed, how long it takes you to write your dissertation depends on who you are, how you work, what field you're in and what other responsibilities you have in life," he wrote in an email. Though some Ph.D. students can write a dissertation in a single year, that is rare, and the dissertation writing process may last for several years, Curtis says.

Curtis adds that the level of support a Ph.D. student receives from an academic advisor or faculty mentor can be a key factor in determining the length of time it takes to complete a Ph.D. program. "Before you decide to enroll at a specific program, you’ll want to meet your future advisor," Curtis advises. "Also, reach out to his or her current and former students to get a sense of what he or she is like to work with."

Curtis also notes that if there is a gap between the amount of time it takes to complete a Ph.D. and the amount of time a student's funding lasts, this can slow down the Ph.D. completion process. "Keep in mind that if you run out of funding at some point during your doctorate, you will need to find paid work, and this will leave you even less time to focus on writing your dissertation," he says. "If one of the programs you’re looking at has a record of significantly longer – or shorter – times to competition, this is good information to take into consideration."

He adds that prospective Ph.D. students who already have master's degrees in the field they intend to focus their Ph.D. on should investigate whether the courses they took in their master's program would count toward the requirements of a Ph.D. program. "You’ll want to discuss your particular situation with your program to see whether this will be possible, and how many credits you are likely to receive as the result of your master’s work," he says.

How to Write M.D.-Ph.D. Application Essays

Ilana Kowarski May 15, 2018

Emmanuel C. Nwaodua, who has a Ph.D. degree in geology, says some Ph.D. programs require candidates to publish a paper in a first-rate, peer-reviewed academic journal. "This could extend your stay by a couple of years," he warns.

Pierre Huguet, the CEO and co-founder of H&C Education, says prospective Ph.D. students should be aware that a Ph.D. is designed to prepare a person for a career as a scholar. "Most of the jobs available to Ph.D. students upon graduation are academic in nature and directly related to their fields of study: professor, researcher, etc.," Huguet wrote in an email. "The truth is that more specialization can mean fewer job opportunities. Before starting a Ph.D., students should be sure that they want to pursue a career in academia, or in research. If not, they should make time during the Ph.D. to show recruiters that they’ve traveled beyond their labs and libraries to gain some professional hands-on experience."

Jack Appleman, a business writing instructor, published author and Ph.D. candidate focusing on organizational communication with the University at Albany—SUNY , says Ph.D. programs require a level of commitment and focus that goes beyond what is necessary for a typical corporate job. A program with flexible course requirements that allow a student to customize his or her curriculum based on academic interests and personal obligations is ideal, he says.

Joan Kee, a professor at the University of Michigan with the university's history of art department, says that the length of time required for a Ph.D. varies widely depending on what subject the Ph.D. focuses on. "Ph.D. program length is very discipline and even field-specific; for example, you can and are expected to finish a Ph.D, in economics in under five years, but that would be impossible in art history (or most of the humanities)," she wrote in an email.

Kee adds that humanities Ph.D. programs often require someone to learn a foreign language, and "fields like anthropology and art history require extensive field research." Kee says funding for a humanities Ph.D. program typically only lasts five years, even though it is uncommon for someone to obtain a Ph.D. degree in a humanities field within that time frame. "Because of this, many if not most Ph.D. students must work to make ends meet, thus further prolonging the time of completion," she says.

Jean Marie Carey, who earned her Ph.D. degree in art history and German from the University of Otago in New Zealand, encourages prospective Ph.D. students to check whether their potential Ph.D. program has published a timeline of how long it takes a Ph.D. student to complete their program. She says it is also prudent to speak with Ph.D. graduates of the school and ask about their experience.

Online Doctoral Programs: What to Expect

Ronald Wellman March 23, 2018

Kristin Redington Bennett, the founder of the Illumii educational consulting firm in North Carolina, encourages Ph.D. hopefuls to think carefully about whether they want to become a scholar. Bennett, who has a Ph.D. in curriculum and assessment and who previously worked as an assistant professor at Wake Forest University , says a Ph.D. is most appropriate for someone who is a "lifelong learner." She says someone contemplating a Ph.D. should ask themselves the following questions "Are you a very curious person... and are you persistent?"

Bennett urges prospective Ph.D. students to visit the campuses of their target graduate programs since a Ph.D. program takes so much time that it is important to find a school that feels comfortable. She adds that aspiring Ph.D. students who prefer a collaborative learning environment should be wary of graduate programs that have a cut-throat and competitive atmosphere, since such students may not thrive in that type of setting.

Alumni of Ph.D. programs note that the process of obtaining a Ph.D. is arduous, regardless of the type of Ph.D. program. "A Ph.D. is a long commitment of your time, energy and financial resources, so it'll be easier on you if you are passionate about research," says Grace Lee, who has a Ph.D. in neuroscience and is the founder and CEO of Mastery Insights, an education and career coaching company, and the host of the Career Revisionist podcast.

"A Ph.D. isn't about rehashing years of knowledge that is already out there, but rather it is about your ability to generate new knowledge. Your intellectual masterpiece (which is your dissertation) takes a lot of time, intellectual creativity and innovation to put together, so you have to be truly passionate about that," Lee says.

Curtis says a prospective Ph.D. student's enthusiasm for academic work, teaching and research are the key criteria they should use to decide whether to obtain a Ph.D. degree. "While the time it takes to complete a doctorate is an understandable concern for many, my personal belief is that time is not the most important factor to consider," he says. "Good Ph.D. programs provide their students with generous stipends, health care and sometimes even subsidized housing."

Erin Skelly, a graduate admissions counselor at the IvyWise admissions consulting firm, says when a Ph.D. students struggles to complete his or her Ph.D. degree, it may have more to do with the student's academic interests or personal circumstances than his or her program.

"The time to complete a Ph.D. can depend on a number of variables, but the specific discipline or school would only account for a year or two's difference," she wrote in an email. "When a student takes significantly longer to complete a Ph.D. (degree), it's usually related to the student's coursework and research – they need to take additional coursework to complete their comprehensive exams; they change the focus of their program or dissertation, requiring extra coursework or research; or their research doesn't yield the results they hoped for, and they need to generate a new theory and conduct more research."

Skelly warns that the average completion time of a Ph.D. program may be misleading in some cases, if the average is skewed based on one or two outliers. She suggests that instead of focusing on the duration of a particular Ph.D. program, prospective students should investigate the program's attritition and graduation rates.

"It is worthwhile to look at the program requirements and the school's proposed timeline for completion, and meet current students to get their input on how realistic these expectations for completion are," Skelly says. "That can give you an honest idea of how long it will really take to complete the program."

Searching for a grad school? Access our complete rankings of Best Graduate Schools.

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Department of Mathematics

Graduate program.

Application deadline is December 15th, 2023.

Test requirements:

GRE Subject Test:         GRE Subject Math Test scores are OPTIONAL.

GRE General Test:      GRE General Test scores are OPTIONAL.  

TOEFL or IELTS:      Scores are REQUIRED (the link below contains answers to common questions on these exams including who has to take them).

Standardized Test Questions:       Yale Graduate School of Arts & Sciences

Fee waiver:  if you wish to apply to waive the application fee (105$) please apply for the waiver here:  Application Fees & Fee Waivers | Yale Graduate School of Arts & Sciences . We recommend to do this as early as possible and, at least, several days before the deadline of January 2, 2023. Please note that the department has no control over the waivers. 

Program in Applied Mathematics . Note that there is a separate program in Applied Mathematics. You cannot apply for both programs. Follow  Welcome | Applied Mathematics Program (yale.edu)  for the general information about that program  and  https://applied.math.yale.edu/graduate-program-0  for the information about admissions, requirements, etc.

Welcome to the Yale graduate program in Mathematics.

The transition from mathematics student to working mathematician depends on ability, hard work and independence, but also on community. Yale’s graduate program provides an excellent environment for this, and we are proud of the talented students who come here and the leading faculty with whom they learn the profession.

In their first two years, students focus on building their general knowledge and passing the qualifying exams , but are also encouraged to use the time to think about their areas of interest, work together to explore them, and begin making connections with faculty advisors. There are few formal requirements and this flexibility allows students to develop independence, formulating and following their own goals.

Mathematics, while requiring intense individual focus, also thrives on collaborative work. Students form study groups and seminars together, and also benefit from our excellent cohort of Gibbs Assistant Professors and other Postdoctoral Fellows, who are a source of fresh mathematical perspectives and camaraderie.

Research, and the contribution of new ideas and results to the body of mathematical knowledge, naturally form the main focus of the next few years, and typically students complete their PhD by the end of the 5th (sometimes 6th) year. During this time they also get to know the faculty better, and continue building intellectual and personal connections, horizontally across the discipline and through time to our shared intellectual history and tradition.

Teaching is an important component of our profession, and the department provides support and training to graduate students. Teaching assignments proceed from individual coaching to classroom teaching, with careful mentoring provided by our dedicated team of lecturers.  The Lang Lunch Seminar, in the second year, provides in-depth training to graduate students before they begin to lecture.

Director of Graduate Studies : Van Vu .

Inquiries concerning the graduate program in mathematics should be sent to Van Vu .

Registrar of Graduate Studies: TBA 

Some useful links:

• The mathematics department page in the Graduate School catalog.
• Graduate school homepage for general information.
• Admissions information from the graduate school.
• Mathematics Graduate Program Advising Guidelines

PhD Qualifying Exams

The requirements for the PhD program in Mathematics have changed for students who enter the program starting in Autumn 2023 and later. 

Requirements for the Qualifying Exams

Students who entered the program prior to autumn 2023.

To qualify for the Ph.D. in Mathematics, students must pass two examinations: one in algebra and one in real analysis. 

Students who entered the program in Autumn 2023 or later

To qualify for the Ph.D. in Mathematics, students must choose and pass examinations in two of the following four areas: 

• real analysis
• geometry and topology
• applied mathematics

The exams each consist of two parts. Students are given three hours for each part.

Topics Covered on the Exams:

• Algebra Syllabus
• Real Analysis Syllabus
• Geometry and Topology Syllabus
• Applied Mathematics Syllabus

Check out some Past and Practice Qualifying Exams to assist your studying.

Because some students have already taken graduate courses as undergraduates, incoming graduate students are allowed to take either or both of the exams in the autumn. If they pass either or both of the exams, they thereby fulfill the requirement in those subjects. However, they are in no way penalized for failing either of the exams.

Students must pass both qualifying exams by the autumn of their second year. Ordinarily first-year students take courses in algebra and real analysis throughout the year to prepare them for the exams. The exams are then taken at the beginning of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Students who started in Autumn 2023 and later

Students must choose and pass two out of the four qualifying exams by the autumn of their second year. Students take courses in algebra, real analysis, geometry and topology, and applied math in the autumn and winter quarters of their first year to prepare them for the exams. The exams are taken during the first week of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Exam Schedule

Unless otherwise noted, the exams will be held each year according to the following schedule:

Autumn Quarter:  The exams are held during the week prior to the first week of the quarter. Spring Quarter:  The exams are held during the first week of the quarter.

The exams are held over two three-hour blocks. The morning block is 9:30am-12:30pm and the afternoon block is 2:00-5:00pm.

For the start date of the current or future years’ quarters please see the  Academic Calendar

Upcoming Exam Dates

Spring 2024.

The exams will be held on the following dates:

Monday, April 1st

Analysis Exam, Room 384H

Wednesday, April 3rd

Algebra Exam, Room 384I

Thursday, April 4th 

Geometry & Topology Exam, Room 384I

Friday, April 5th

Applied Math Exam, Room 384I

The department offers programs covering a broad range of topics leading to the Doctor of Philosophy and the Doctor of Science degrees (the student chooses which to receive; they are functionally equivalent). Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 120 Ph.D. students, about 2/3 are in Pure Mathematics, 1/3 in Applied Mathematics.

The two programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, mathematics of data, and the theory of computation. In addition, many mathematically-oriented courses are offered by other departments. Students in Applied Mathematics are especially encouraged to take courses in engineering and scientific subjects related to their research.

All students pursue research under the supervision of the faculty , and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.

Degree Requirements

Degree requirements consist of:

• Oral qualifying exam
• Classroom teaching
• Original thesis and defense

Prospective students are invited to consult the graduate career timeline for more information, and to read about the application procedure .

Graduate Co-Chairs

Graduate Student Issues, math graduate admissions

Jonathan Kelner , Davesh Maulik , and Zhiwei Yun

How Long Does It Take to Get a Ph.D. in Mathematics?

Education Requirements for Information Technology

If you love mathematics and have a passion to teach, earning a Ph.D. in mathematics may be the ideal next step. A Ph.D. in mathematics is a gateway to a career in teaching or research at the university level. While someone with a master's degree may be able to get jobs teaching at a community college, a Ph.D. will make candidates more competitive for those jobs, as well. The amount of time it takes to complete a Ph.D. in mathematics depends on a number of variables, including the program, the research project and the individual pace of student work. However, most programs are completed in five to six years in addition to the four years that most students take to complete an undergraduate degree.

The coursework for a Ph.D. program in mathematics focuses on higher-order mathematics and mathematical theory. Many programs also require that students take a language class and demonstrate some proficiency in that language. The number of credit hours required varies by program, but typically, coursework takes two to three years to complete. Some of the required credit hours can usually be applied to dissertation work or time spent as a teaching assistant.

Completing the Dissertation

Completing the dissertation will take up the bulk of the time dedicated to finishing your Ph.D. A dissertation must include original research and is typically a book-length manuscript. Students who have a good idea of the research they would like to conduct before the program begins will usually be able to finish their work faster and graduate sooner. However, many students explore ideas through their coursework and their discussions with professors before deciding on a dissertation topic.

Passing Exams

The final step to completing a Ph.D. program in mathematics is completing qualifying exams. These include both written and oral exams. In some programs, you may have to take several written exams, often at the end of each year of coursework. Oral exams usually involving defending the dissertation and oral exams that test your understanding of the subject matter. Often, other requirements include: a minimum GPA, the presentation of research or work as a teaching assistant.

Job Outlook

The Bureau of Labor Statistics predicts that jobs for mathematicians will grow about as fast as average for other industries by 2020. The agency predicts a growth of 500 jobs, or an increase of about 16 percent. The median pay for mathematicians was $84,760 in 2018. The bureau also notes that mathematicians earning only a master's degree have job opportunities in government agencies and private organizations.

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Maria Magher has been working as a professional writer since 2001. She has worked as an ESL teacher, a freshman composition teacher and an education reporter, writing for regional newspapers and online publications. She has written about parenting for Pampers and other websites. She has a Master's degree in English and creative writing.

University of Notre Dame

Department of Mathematics

College of Science

• Home ›

The PhD Program

The primary graduate program is the PhD program in Mathematics. All students admitted to this program receive financial support, with a full fellowship in the first year. Current students are pursuing exciting projects in a variety of areas and PhD work in mathematics prepares students for a variety of interesting careers, including but not limited to academic work. The department has active research groups in Algebra and Algebraic Geometry, Analysis and Partial Differential Equations, Discrete Math, Operations Research, and Probability, Logic, Mathematical Physics, and Topology.  See the research groups web page for more details. For a detailed description of the PhD program and requirements, see the  Regulations for Doctoral Students in Mathematics .

See this link for more information about joint mathematics/philosophy graduate degrees at Notre Dame.

Masters degrees  are available for Mathematics PhD candidates who have taken 30 credit hours and passed their candidacy examinations. We offer a Master of Science in Interdisciplinary Mathematics (MSIM) to students enrolled in other PhD programs at Notre Dame.  For more details talk to the Mathematics Director of Graduate Studies or see this link . We do not admit students directly into any masters program.

Interdisciplinary study is possible, but it requires extra dedication. One has to be first admitted to a PhD granting department at Notre Dame. A next step is the above mentioned MSIM. We do not admit student directly into an interdisciplinary program. 

More info about graduate school at Notre Dame can be found at  http://graduateschool.nd.edu

Contact Info

Dr. Samuel Evens , Director of Graduate Studies Email:  [email protected]

Dr. Samuel Evens , Director of Graduate Admissions Email: [email protected]

Tori Spalding, Graduate Coordinator Email: [email protected] Phone:  574-631-6008  Fax:  574-631-6579

Latest News

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April 23, 2024

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April 15, 2024

Mathematics

77 Massachusetts Avenue Building 2-110 Cambridge MA, 02139

617-253-2416 [email protected]

Website: Mathematics

Application Opens: September 14

Deadline: December 15 at 11:59 PM Eastern Time

Fee: $75.00

Terms of Enrollment

Interdisciplinary programs.

• Computational Science and Engineering
• Interdisciplinary Doctoral Program in Statistics (IDPS)

Standardized Tests

Graduate Record Examination (GRE)

• General test is optional
• Subject test is optional

International English Language Testing System (IELTS)

• Minimum score required: 6
• Electronic scores send to: MIT Graduate Admissions
• Preferably taken this year

Test of English as a Foreign Language (TOEFL)

• Institute code: 3514
• Department code: 72

Please note: Applicants should submit the IELTS or TOEFL, not both exams. Waivers of TOEFL/IELTS may be available.

Areas of Research

• Algebra/Algebraic Geometry
• Algebraic Topology
• Analysis/PDE
• Combinatorics
• General Applied Mathematics
• General Pure Mathematics
• Geometry/Geometric Topology
• Lie Groups/Algebraic Groups/Representation Theory
• Mathematical/Computational Biology
• Number Theory
• Physical Applied Mathematics
• Probability
• Scientific Computation/Numerical Analysis
• Theoretical Computer Science

Financial Support

Financial support is guaranteed for five years to students making satisfactory progress. Students receive full tuition, a stipend, and individual health coverage. Support is in the form of:

• fellowships
• teaching assistantships
• research assistantships

All students are supported on full fellowships in their first year, so have no teaching duties.

Teaching assistants either teach recitation sections or grade papers for more advanced courses. Both jobs are paid at the same rate, and both require about 10-12 hours per week. Teaching is an important part of the graduate education in Mathematics and all students are expected to teach at least one semester and preferably more, whether supported through other means or not.

Students are encouraged to seek outside fellowships and grants. We do not require students to begin fellowships in their first year in our graduate program (but the granting agency may have such a requirement).

If an outside fellowship partially covers a student’s full tuition and stipend, then the department will supplement the remainder, either through a teaching assistantship, a research assistantship, or a departmental fellowship, in order to bring all students up to the same level of support.

Application Requirements

• Online application
• Statement of objectives
• Three letters of recommendation
• Transcripts (required upon admission)
• English proficiency exam scores

Special Instructions

The program admits students starting in the Fall term of each year only. Admission is to the PhD program only; there is no Masters program. There is no separate application for financial support; all admitted students are offered support.

Submitting GRE scores is entirely optional: We will accept scores if submitted (and are most interested in the Math Subject test result, if any) but it will not hurt your application if not included.

Applicants should not send published papers or theses. The only paper documents needed are your transcripts (which will be requested only if admitted).

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RIT graduate pursues Ph.D. across time zones

Nastaran Nagshineh, center, defended her Ph.D. thesis at RIT in April. Faculty from RIT’s Rochester and Dubai campuses served on her thesis committee and include, from left to right, Kathleen Lamkin-Kennard, Steven Weinstein, Nathaniel Barlow, and David Kofke (a professor at the University at Buffalo). Mohamed Samaha participated remotely and appears on the video screen behind the group and alongside Nagshineh’s picture.

Nastaran Nagshineh is one of the first Ph.D. candidates to bridge RIT’s Rochester and Dubai campuses. Her accomplishment creates a path for future students at the university’s international campuses.

Nagshineh completed her Ph.D. in mathematical modeling while working full time as a mathematics lecturer at RIT Dubai in the United Arab Emirates, teaching as many as five classes a semester. She described her Ph.D. journey as “an exercise in perseverance” due to competing demands and long days. Rochester is eight hours behind Dubai, and the time difference meant many late-night classes and meetings.

“I saw this collaboration as an opportunity, rather than as a challenge, because my primary adviser, Dr. Steven Weinstein (RIT professor of chemical engineering), and my co-adviser, Dr. Mohamed Samaha (RIT Dubai associate professor of mechanical engineering), both have the same area of research interest,” she said. “They both worked toward my success.”

Nagshineh is one of 67 RIT Ph.D. students who defended their thesis this academic year and who will earn their doctorate. RIT awarded 63 Ph.D. degrees in 2023.

In 2020-2021, RIT’s Graduate School met and surpassed the university’s goal of conferring 50 Ph.D. degrees during an academic year. That number will continue to grow as students cycle through the seven new Ph.D. programs that RIT has added since 2017, said Diane Slusarski , dean of RIT’s Graduate School.

Meeting these goals puts RIT on a path toward achieving an “R1,” or research-intensive designation, from the Carnegie Classification of Institutions of Higher Learning. RIT is currently ranked as an R2 institution . Many factors go into changing a university’s status, including research investment and maintaining a three-year average of 70 Ph.D. degrees awarded per year, according to Slusarski.

“We have met the goals of the strategic plan, and now we look forward to contributing to the research innovation in the future,” Slusarski said. “We want to help the new programs thrive and win national research awards.”

RIT’s emphasis on high-level research is seen in Nagshineh’s Ph.D. work. She applies mathematical modeling to the field of fluid dynamics. Her research has been published in top-tier journals and has gained notice, said Weinstein, her thesis adviser.

Weinstein describes Nagshineh’s accomplishments as “a testament to a fantastic work ethic and commitment” and is inspirational to younger students at Rochester and Dubai.

“The collaboration between RIT Dubai/Rochester has continued,” he said. “Another paper was submitted a few weeks ago with Mohamed Samaha and Nate Barlow (RIT associate professor in the School of Mathematics and Statistics) as co-authors, as well as Cade Reinberger, a younger Ph.D. student in my research group.”

Mathematical modeling is one of RIT’s newer Ph.D. degree programs, and Nagshineh is among its earliest graduates. The program has doubled in size since it began accepting students in 2017, Slusarski said. This past fall, the mathematical modeling program had 35 students, with two graduating this year.

Altogether, RIT has 13 Ph.D. degree programs currently enrolling 438 students, with computing and information sciences accounting for the largest with 117 students. RIT’s other Ph.D. programs include astrophysical sciences and technology , biomedical and chemical engineering , business administration , color science , electrical and computer engineering, imaging science , mechanical and industrial engineering , microsystems engineering , and sustainability .

New programs in cognitive science and physics will launch in the fall.

The growth in RIT graduate education—with more than 3,000 master’s and doctoral students—reflects a demographic change in the student population, Slusarski said. “We have a higher percentage of women in the graduate programs than we have for RIT undergraduate programs.”

RIT’s graduate programs enroll 42 percent women, according to Christie Leone , assistant dean for the Graduate School.

Nagshineh, who also holds an MS in electrical engineering from RIT Dubai, welcomes her role as a mentor to other women students on both campuses.

“As a young woman in an Arabic country, the power of women is often underestimated and undervalued, and I hope to serve as a role model to female students, especially those that question their path,” Nagshineh said.

She plans to continue in her career as a professor and a researcher. “I would like to pursue a research program where I can advise my own students and teach them more deeply.”

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Long Multiplication Calculator

How do you multiply decimals, how to do long multiplication, long multiplication with decimals, how to multiply big numbers multiplication algorithm in practice, how (and when) to use the long multiplication calculator.

Welcome to the long multiplication calculator – a great tool that helps you solve multiplication yourself. If you ever asked yourself: How do you multiply decimals? or How to multiply big numbers? , here is the right place to find the answer.

Knowledge of the basic multiplication algorithm makes it possible to deal with more complex problems like multiplying fractions or matrices. Furthermore, if you learn how to do long multiplication and also practice with long division calculator , it will make math exercises with operations on numbers as simple as never before!

If you'd like to try an alternative approach to multiplying large numbers, check out Omni's partial products calculator !

Let's begin from the basics – multiplication is a compact way of writing the addition of repeating numbers. If we want to solve a problem like 6×2 , it would be the same as if we have to add 2 six times, 2+2+2+2+2+2 . In that meaning, multiplication is extremely useful, especially for big numbers.

Mathematicians call the first number in the multiplication the multiplier and the second one the multiplicand . The result of the multiplication is the product .

💡 Multiplication is commutative . It means that we can change the multiplier and multiplicand places , and the outcome wouldn't change.

From the beginning of our education, we all learn how to multiply numbers from one to ten – most pupils' nightmare, isn't it? But, actually, it's all you need to know about how to multiply big numbers or how long multiplication with decimals works!

So, how do you multiply decimals? In short, forget about the decimal dot and perform the multiplication with whole numbers. Next, add up the decimal places in the multiplier and the multiplicand . The sum is the number of decimal places in the product. We describe the whole process in detail in the dedicated section below .

We can describe the long multiplication algorithm in a few steps:

• Set both numbers one under the other and align them to the right so that the first significant digits are the first ones from the right.

🙋 It's a good practice to set the longer number as the multiplier and the shorter one as the multiplicand. It doesn't change the product, but it does reduce the number of steps .

Start multiplying the multiplier by the first digit (from the right) of the multiplicand, digit by digit. Anytime you finish with a number larger than 9, write down the digit of ones and carry the digit of tens to the next step (e.g. 7×5=35 , so write 5 and carry 3 ).

Repeat for the rest of the digits of the multiplier. Whenever you carry a number, add it to the product (e.g. 1×5 and 3 carried from the previous step gives us 8 ).

When you finish with the first digit of the multiplicand, you obtain a first intermediate product .

Repeat the same procedure for the rest of the digits from the multiplicand, each time starting one place further from the right (as you multiply by tens, hundreds, and so on). You can also write trailing zeros if you like.

🙋 If you face any 0 digits in the multiplicand, you can skip the step , as the product of zero and any other number is always zero.

When you're done with all of the intermediate products, add them up .

The result is your final product. Now you know how to do long multiplication!

Let's step up to the next level and find out how to multiply decimals using the long multiplication method. As for an example, let's multiply 4.37 by 8.5 . It turns out that we can treat it as a 3- and 2-digit multiplication problem. To get the answer, we can follow these steps:

Count the number of decimal digits in both numbers . There are two decimals in the first one and a single decimal in the second one.

The sum of decimal digits in the multiplier and the multiplicand is three ( 2+1 ). We will have three decimals in the product as well.

At this point, we can forget about decimal dots and perform the multiplication 437 × 85 .

The product of 437 and 5 is 2185 .

The product of 437 and 8 is 3496 . Remember to start writing it one place further from the right. Alternatively, you can add one 0 at the end, so the product becomes 34960 , and both numbers are right-aligned then.

Evaluate the sum of these two intermediate products. 2185 + 34960 = 37145 .

Finally, apply the decimal dot in the product . We know there should be three decimal digits , so our outcome equals 37.145 .

If you have any doubts, get lost at some point, or simply want to check the answer, you can always use our long multiplication calculator!

The good thing about long multiplication is that it doesn't make a problem significantly more difficult for larger numbers. What matters is the length of the numbers rather than the values themselves. Moreover, it may be even simpler to multiply big numbers if any of them (or both) ends with several trailing zeros. Why?

We can simply skip trailing zeros for multiplication , as any intermediate products would equal zero. We can add trailing zeros from both the multiplier and the multiplicand and write them next to the product . The procedure is very similar to the one with decimals.

Let's apply the long multiplication algorithm for two big numbers, let's say 34000 and 2870 :

Count the number of trailing zeros in both cases . There are three of them in the first number and a single zero in the second one.

Now, our new values are 34 and 287 , respectively. Note that in this case, the first number is shorter than the second one (in contrast to the initial numbers). We can swap them and work out the multiplication 287 times 34 .

The first intermediate product is 1148 , and the second one is 861 (remember about shifting this number one digit to the left). Summing them up, we receive 9758 .

That's the right time to apply missing trailing zeros to the product . We have four of them in total.

The final result of long multiplication is 97,580,000 . We can even write it as 9.758×10⁷ using scientific notation.

Have you ever tried to add or subtract fractions? If so, you're probably familiar with the concept of finding the least common denominator. Simply speaking, it's all about finding the least common multiple of two (or multiple) numbers. The process involves a certain skill in multiplication. This situation is where our long multiplication calculator becomes handy, especially for fractions containing decimals or big numbers.

The good thing is that our tool is pretty straightforward to use. As you already know how to do long multiplication with decimals by hand from previous sections, let's see how to do the same using the long multiplication calculator:

Type the first number as the multiplier, e.g., 0.00367 .

Input the second number as the multiplicand, e.g., 449300 .

And that's all! As a result, you receive an answer, which is 1648.931 . Additionally, you also get explanations and hints on how to deal with multiplying big numbers and decimals.

If you like this long multiplication calculator, make sure to check our distributive property calculator as well!

Lowest common denominator

One's complement.

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Students completing their degree requirements during any summer 2024 session must apply to graduate by Saturday, June 16, in MyPNW on the Graduation tab. This application is required whether or not you plan to participate in a Commencement ceremony. Information from the application is used to order your diploma.

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1250 BELLFLOWER BOULEVARD LONG BEACH, CALIFORNIA 90840 562.985.4111

Counseling Programs Information Session: Virtual 06/18/2024

Event Date and Time:    Tuesday, June 18, 2024 - 2:00PM - 3:00PM

Virtual:  

This information session will be hosted via Zoom Webinar.  Reservations for these information sessions are required, please RSVP .  

Overview:  

The Graduate Studies Office hosts Counseling Programs Information Sessions to provide an overview of the Counseling Programs: Counseling Psychology, School Counseling (K-12), and Student Development in Higher Education. The application process, degree requirements, and career objectives of the program will also be discussed.  

Reservations for these information sessions are required, please  RSVP .  

If you would like to request accommodations, please contact the Bob Murphy Access Center at least 72 hours in advance before the information session at 562.985.5401 or  [email protected]