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Number Systems

Number systems are systems in mathematics that are used to express numbers in various forms and are understood by computers. A number is a mathematical value used for counting and measuring objects, and for performing arithmetic calculations. Numbers have various categories like natural numbers, whole numbers, rational and irrational numbers, and so on. Similarly, there are various types of number systems that have different properties, like the binary number system, the octal number system, the decimal number system, and the hexadecimal number system.

In this article, we will explore different types of number systems that we use such as the binary number system, the octal number system, the decimal number system, and the hexadecimal number system. We will learn the conversions between these number systems and solve examples for a better understanding of the concept.

What are Number Systems?

A number system is a system representing numbers. It is also called the system of numeration and it defines a set of values to represent a quantity. These numbers are used as digits and the most common ones are 0 and 1, that are used to represent binary numbers. Digits from 0 to 9 are used to represent other types of number systems.

Number Systems Definition

A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner. The value of any digit in a number can be determined by a digit, its position in the number, and the base of the number system. The numbers are represented in a unique manner and allow us to operate arithmetic operations like addition, subtraction, and division.

Types of Number Systems

There are different types of number systems in which the four main types are as follows.

• Binary number system (Base - 2)
• Octal number system (Base - 8)
• Decimal number system (Base - 10)
• Hexadecimal number system (Base - 16)

We will study each of these systems one by one in detail after going through the following number system chart.

Number System Chart

Given below is a chart of the main four types of number system that we use to represent numbers.

Binary Number System

The binary number system uses only two digits: 0 and 1. The numbers in this system have a base of 2. Digits 0 and 1 are called bits and 8 bits together make a byte. The data in computers is stored in terms of bits and bytes. The binary number system does not deal with other numbers such as 2,3,4,5 and so on. For example: 10001 2 , 111101 2 , 1010101 2 are some examples of numbers in the binary number system.

Octal Number System

The octal number system uses eight digits: 0,1,2,3,4,5,6 and 7 with the base of 8. The advantage of this system is that it has lesser digits when compared to several other systems, hence, there would be fewer computational errors. Digits like 8 and 9 are not included in the octal number system. Just like the binary, the octal number system is used in minicomputers but with digits from 0 to 7. For example, 35 8 , 23 8 , and 141 8 are some examples of numbers in the octal number system.

Decimal Number System

The decimal number system uses ten digits: 0,1,2,3,4,5,6,7,8 and 9 with the base number as 10. The decimal number system is the system that we generally use to represent numbers in real life. If any number is represented without a base, it means that its base is 10. For example, 723 10 , 32 10 , and 4257 10 are some examples of numbers in the decimal number system.

Hexadecimal Number System

The hexadecimal number system uses sixteen digits/alphabets: 0,1,2,3,4,5,6,7,8,9 and A,B,C,D,E,F with the base number as 16. Here, A-F of the hexadecimal system means the numbers 10-15 of the decimal number system respectively. This system is used in computers to reduce the large-sized strings of the binary system. For example, 7B3 16 , 6F 16 , and 4B2A 16 are some examples of numbers in the hexadecimal number system.

Conversion of Number Systems

A number can be converted from one number system to another number system using number system formulas. Like binary numbers can be converted to octal numbers and vice versa, octal numbers can be converted to decimal numbers and vice versa, and so on. Let us see the steps required in converting number systems.

Steps for Conversion of Binary to Decimal Number System

To convert a number from the binary to the decimal system, we use the following steps.

• Step 1: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.
• Step 2: The exponents should start with 0 and increase by 1 every time we move from right to left.
• Step 3: Simplify each of the above products and add them.

Let us understand the steps with the help of the following example in which we need to convert a number from binary to decimal number system.

Example: Convert 100111 2 into the decimal system.

Step 1: Identify the base of the given number. Here, the base of 100111 2 is 2.

Step 2: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base. The exponents should start with 0 and increase by 1 every time as we move from right to left. Since the base is 2 here, we multiply the digits of the given number by 2 0 , 2 1 , 2 2 , and so on from right to left.

Step 3: We just simplify each of the above products and add them.

Here, the sum is the equivalent number in the decimal number system of the given number. Or, we can use the following steps to make this process simplified.

100111 = (1 × 2 5 ) + (0 × 2 4 ) + (0 × 2 3 ) + (1 × 2 2 ) + (1 × 2 1 ) + (1 × 2 0 )

= (1 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (1 × 1)

= 32 + 0 + 0 + 4 + 2 + 1

Thus, 100111 2 = 39 10 .

Conversion of Decimal Number System to Binary / Octal / Hexadecimal Number System

To convert a number from the decimal number system to a binary/octal/hexadecimal number system, we use the following steps. The steps are shown on how to convert a number from the decimal system to the octal system.

Example: Convert 4320 10 into the octal system.

Step 1: Identify the base of the required number. Since we have to convert the given number into the octal system, the base of the required number is 8.

Step 2: Divide the given number by the base of the required number and note down the quotient and the remainder in the quotient-remainder form. Repeat this process (dividing the quotient again by the base) until we get the quotient less than the base.

Step 3: The given number in the octal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

Therefore, 4320 10 = 10340 8

Conversion from One Number System to Another Number System

To convert a number from one of the binary/octal/hexadecimal systems to one of the other systems, we first convert it into the decimal system, and then we convert it to the required systems by using the above-mentioned processes.

Example: Convert 1010111100 2 to the hexadecimal system.

Step 1: Convert this number to the decimal number system as explained in the above process.

Thus, 1010111100 2 = 700 10 → (1)

Step 2: Convert the above number (which is in the decimal system), into the required number system (hexadecimal).

Here, we have to convert 700 10 into the hexadecimal system using the above-mentioned process. It should be noted that in the hexadecimal system, the numbers 11 and 12 are written as B and C respectively.

Thus, 700 10 = 2BC 16 → (2)

From the equations (1) and (2), 1010111100 2 = 2BC 16

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Number Systems Examples

Example 1: Convert 300 10 into the binary number system with base 2.

Solution: 300 10 is in the decimal system. We divide 300 by 2 and note down the quotient and the remainder. We will repeat this process for every quotient until we get a quotient that is less than 2.

The equivalent number in the binary system is obtained by reading all the remainders and just the last quotient from bottom to top as shown above.

Thus, 300 10 = 100101100 2

Example 2: Convert 5BC 16 into the decimal system.

Solution: 5BC 16 is in the hexadecimal system. We know that B = 11 and C = 12 in the hexadecimal system. So we get the equivalent number in the decimal system using the following process:

Thus, 5BC 16 = 1468 10

Example 3: Convert 144 8 into the hexadecimal system.

Solution: The base of 144 8 is 8. First, we will convert this number into the decimal system as follows:

Thus, 144 8 = 100 10 → (1). Now we will convert this into the hexadecimal system as follows:

Thus, 100 10 = 64 16 → (2)

From the equations (1) and (2), we can conclude that: 144 8 = 64 16

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Practice Questions on Number Systems

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FAQs on Number Systems

What are number systems with examples.

A number system is a system of writing or expressing numbers. In mathematics, numbers are represented in a given set by using digits or symbols in a certain manner. Every number has a unique representation of its own and numbers can be represented in the arithmetic and algebraic structure as well. There are different types of number systems that have different properties, like the binary number system, the octal number system, the decimal number system, and the hexadecimal number system. Some examples of numbers in different number systems are 10010 2 , 234 8 , 428 10 , and 4BA 16 .

What are the Different Types of Number Systems?

There are four main types of number systems:

What are the Conversion Rules of Number Systems?

To convert a number from binary/octal/hexadecimal system to a decimal number system, we use the following steps:

• Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.
• The exponents should start with 0 and increase by 1 every time we move from right to left.
• Simplify each of the above products and add them.

To convert a number from decimal system to binary/octal/hexadecimal system, we use the following steps:

• Divide the given number by the base of the required number and note down the quotient and the remainder in the “quotient-remainder” form.
• Repeat this process (dividing the quotient again by the base) until we get the quotient less than the base.
• The given number in the decimal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

To convert a number from one of the binary/octal/hexadecimal systems to one of the other systems:

• We first convert it into the decimal system.
• Then we convert it to the required system.

What are the Uses of Each Number System?

There are different purposes of each number system, such as:

• The binary number system is used to store the data in computers.
• The advantage of the octal number system is that it has fewer digits when compared to several other systems, hence, there would be fewer computational errors.
• The decimal number system is the system that we use in daily life.
• The hexadecimal number system is used in computers to reduce the large-sized strings of the binary system.

What is the Importance of Number Systems?

Number systems help in representing the numbers in a small symbol set. Binary numbers are mostly used in computers that use digits like 0 and 1 for calculating simple problems. The number systems also help in converting one number system to another.

How are Number Systems Classified?

The number systems can be classified mainly into two categories: Positional and Non-positional number systems. For positional number systems, each digit is associated with a weight and its examples are binary, octal, decimal, etc. In non-positional number systems, the digit values are independent of their positions and its examples are gray code, cyclic code, aroma code, etc.

Why are Different Number Systems Used in Computers?

Computers cannot understand human languages, so to understand the commands and instructions given to the computers by programmers, different number systems are used such as the binary system, the octal system, the decimal system, and so on.

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5.2: Algebraic Expressions

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Learning Objectives

After completing this section, you should be able to:

• Convert between written and symbolic algebraic expressions and equations.
• Simplify and evaluate algebraic expressions.
• Add and subtract algebraic expressions.
• Multiply and divide algebraic expressions.

Algebraic expressions are the building blocks of algebra. While a numerical expression (also known as an arithmetic expression) like 5 + 3 5 + 3 can represent only a single number, an algebraic expression such as 5 x + 3 5 x + 3 can represent many different numbers. This section will introduce you to algebraic expressions, how to create them, simplify them, and perform arithmetic operations on them.

Algebraic Expressions and Equations

Xavier and Yasenia have the same birthday, but they were born in different years. This year Xavier is 20 years old and Yasenia is 23, so Yasenia is three years older than Xavier. When Xavier was 15, Yasenia was 18. When Xavier will be 33, Yasenia will be 36. No matter what Xavier’s age is, Yasenia’s age will always be 3 years more.

In the language of algebra, we say that Xavier's age and Yasenia's age are variable and the 3 is a constant. The ages change, or vary, so age is a variable . The 3 years between them always stays the same or has the same value, so the age difference is the constant . In algebra, letters of the alphabet are used to represent variables. The letters most often used for variables are x x , y y , z z , a a , b b , and c c . Suppose we call Xavier's age x x . Then we could use x + 3 x + 3 to represent Yasenia's age, as shown in the table below.

To write algebraically, we need some symbols as well as numbers and variables. The symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division are summarized in Table 5.1, along with words we use for the operations and the result.

In algebra, the cross symbol ( x ) ( x ) is normally not used to show multiplication because that symbol could cause confusion. For example, does 3 x y 3 x y mean 3 × y 3 × y (three times y y ) or 3 • x • y 3 • x • y (three times x x times y y )? To make it clear, use • • or parentheses for multiplication.

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.

• The sum of 5 and 3 means add 5 plus 3, which we write as 5 + 3 5 + 3 .
• The difference of 9 and 2 means subtract 9 minus 2, which we write as 9 − 2 9 − 2 .
• The product of 4 and 8 means multiply 4 times 8, which we can write as 4 • 8 4 • 8 .
• The quotient of 20 and 5 means divide 20 by 5, which we can write as 20 ÷ 5 20 ÷ 5 .

Example 5.1

Translating from algebra to words.

Translate the following algebraic expressions from algebra into words.

• 12 + 14 12 + 14
• ( 30 ) ( 5 ) ( 30 ) ( 5 )
• 64 ÷ 8 64 ÷ 8
• x − y x − y
• According to Table 5.1, this could be translated as 12 plus 14 OR the sum of 12 and 14.
• According to Table 5.1, this could be translated as 30 times 5 OR the product of 30 and 5.
• According to Table 5.1, this could be translated as 64 divided by 8 OR the quotient of 64 and 8.
• According to Table 5.1, this could be translated as x x minus y y OR the difference of x x and y. y.

Your Turn 5.1

Example 5.2, translating from words to algebra.

Translate the following phrases from words into algebraic expressions.

• The difference of 47 and 19
• 72 divided by 9
• The sum of m m and n n
• According to Table 5.1, these words could be translated as 47 − 19 47 − 19 .
• According to Table 5.1, these words could be translated as 72 ÷ 9 72 ÷ 9 .
• According to Table 5.1, these words could be translated as m + n m + n .
• According to Table 5.1, these words could be translated as ( 13 ) ( 7 ) ( 13 ) ( 7 ) .

Your Turn 5.2

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb. In algebra, we have expressions and equations. Example 5.1 and Example 5.2 used expressions. An expression is like an English phrase. Notice that the English phrases do not form a complete sentence because the phrase does not have a verb. The following table has examples of expressions, which are numbers, variables, or combinations of numbers and variables using operation symbols.

Example 5.3

Translating from an english phrase to an expression.

• Seven more than a number n n .
• A number n n times itself.
• Six times a number n n , plus two more.
• The cost of postage is a flat rate of 10 cents for every parcel, plus 34 cents per ounce x x .
• n + 7 n + 7
• n • n n • n or n 2 n 2
• 6 n + 2 6 n + 2
• 10 + 34 x 10 + 34 x

Your Turn 5.3

An equation is two expressions linked with an equal sign (the symbol =). When two quantities have the same value, we say they are equal and connect them with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. So, a = b a = b is read “ a a is equal to b b .” The following table has some examples of equations.

Example 5.4

Translating from an english sentence to an equation.

Translate the following sentences from words into algebraic equations.

• Two times x x is 6.
• n n plus 2 is equal to n n times 3.
• The quotient of 35 and 7 is 5.
• Sixty-seven minus x x is 56.
• 2 x = 6 2 x = 6
• n + 2 = 3 n n + 2 = 3 n
• 35 ÷ 7 = 5 35 ÷ 7 = 5
• 67 − x = 56 67 − x = 56

Your Turn 5.4

The use of variables.

French philosopher and mathematician René Descartes (1596–1650) is usually given credit for the use of the letters x x , y y , and z z to represent unknown quantities in algebra. He introduced these ideas in his publication of La Geometrie , which was printed in 1637. In this publication, he also used the letters a a , b b , and c c to represent known quantities. There is a (possibly fictitious) story that, when the book was being printed for the first time, the printer began to run short of the last three letters of the alphabet. So the printer asked Descartes if it mattered which of x x , y y , or z z were used for the mathematical equations in the book. Descartes decided it made no difference to him; so the printer decided to use x x predominantly for the mathematics in the book, because the letters y y and z z would occur more often in the body of the text (written in French) than the letter x x would! This might explain why the letter x x is still used today as the most common variable to represent unknown quantities in algebra.

Simplifying and Evaluating Algebraic Expressions

To simplify an expression means to do all the math possible. For example, to simplify 4 • 2 + 1 4 • 2 + 1 we would first multiply 4 • 2 4 • 2 to get 8 and then add 1 to get 9. We have introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values. Consider 2 + 7 • 3 2 + 7 • 3 . Do you add first or multiply first? Do you get different answers?

Early on, mathematicians realized the need to establish some guidelines when performing arithmetic operations to ensure that everyone would get the same answer. Those guidelines are called the order of operations and are listed in the table below.

You may have heard about Please Excuse My Dear Aunt Sally or PEMDAS. Be careful to notice in Steps 3 and 4 in the table above that multiplication and division, as well as addition and subtraction, happen in order from LEFT to RIGHT. It is possible, for example, to have PEDMAS or PEMDSA. The PEMDAS trick can be misleading if not fully understood!

Example 5.5

Making a numerical equation true using the order of operations.

Use parentheses to make the following statements true.

• 17 − 10 + 3 = 10 17 − 10 + 3 = 10
• 2 • 26 − 7 = 38 2 • 26 − 7 = 38
• 8 + 12 ÷ 5 − 3 = 14 8 + 12 ÷ 5 − 3 = 14
• 5 + 2 3 • 7 = 91 5 + 2 3 • 7 = 91
• Add the parentheses around the 17 − 10 17 − 10 . Then you have ( 17 − 10 ) + 3 = 7 + 3 = 10 ( 17 − 10 ) + 3 = 7 + 3 = 10 .
• Add the parentheses around the 26 − 7 26 − 7 . Then you have 2 • ( 26 − 7 ) = 2 • 19 = 38 2 • ( 26 − 7 ) = 2 • 19 = 38 .
• Add the parentheses around the 5 − 3 5 − 3 . Then you have 8 + 12 ÷ ( 5 − 3 ) = 8 + 12 ÷ 2 = 8 + 6 = 14 8 + 12 ÷ ( 5 − 3 ) = 8 + 12 ÷ 2 = 8 + 6 = 14 .
• Add the parentheses around the 5 + 2 3 5 + 2 3 . Then you have ( 5 + 2 3 ) • 7 = ( 5 + 8 ) • 7 = 13 • 7 = 91 ( 5 + 2 3 ) • 7 = ( 5 + 8 ) • 7 = 13 • 7 = 91 .

Your Turn 5.5

In the last example, we simplified expressions using the order of operations. Now we'll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

Example 5.6

Evaluating and simplifying an expression.

• Evaluate 3 x + 5 3 x + 5 when x = 2 x = 2 .
• Evaluate x 2 + 3 x + 1 x 2 + 3 x + 1 when x = 2 x = 2 .
• To evaluate, let x = 2 x = 2 in the expression, and then simplify: 3 ( 2 ) + 5 = 6 + 5 = 11 3 ( 2 ) + 5 = 6 + 5 = 11 .
• To evaluate, let x = 2 x = 2 in the expression, and then simplify: 2 2 + 3 ( 2 ) + 1 = 4 + 6 + 1 = 11 2 2 + 3 ( 2 ) + 1 = 4 + 6 + 1 = 11 .

Your Turn 5.6

Operations of algebraic expressions.

Algebraic expressions are made up of terms . A term is a constant or the product of a constant and one or more variables. Examples of terms are 7, y y , 5 x 2 x 2 , 9 a a , and b 5 b 5 . The constant that multiplies the variable is called the coefficient . Think of the coefficient as the number in front of the variable. Consider the algebraic expressions 5 x 2 x 2 , which has a coefficient of 5, and 9 a a , which has a coefficient of 9. If there is no number listed in front of the variable, then the coefficient is 1 since x = 1 • x x = 1 • x .

Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are like terms . If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.

Example 5.7

Adding algebraic expressions.

Add ( x 2 + 4 x − 9 ) + ( 3 x 2 − x + 12 ) ( x 2 + 4 x − 9 ) + ( 3 x 2 − x + 12 ) .

Step 1: Add the terms in any order and get the same result (think: 2 + 3 = 3 + 2 2 + 3 = 3 + 2 ) and drop the parentheses:

x 2 + 4 x − 9 + 3 x 2 − x + 12 x 2 + 4 x − 9 + 3 x 2 − x + 12

Step 2: Group like terms together:

x 2 + 3 x 2 + 4 x − x − 9 + 12 x 2 + 3 x 2 + 4 x − x − 9 + 12

Step 3: Combine the like terms:

4 x 2 + 3 x + 3 4 x 2 + 3 x + 3

Your Turn 5.7

Example 5.8, subtracting algebraic expressions.

Subtract ( 5 x 2 + 4 x − 9 ) − ( 3 x 2 − x + 12 ) ( 5 x 2 + 4 x − 9 ) − ( 3 x 2 − x + 12 ) .

Step 1: Distribute the negative inside the parentheses (think: 2 − ( 3 − 4 ) = 2 − 3 + 4 = − 1 + 4 = 3 2 − ( 3 − 4 ) = 2 − 3 + 4 = − 1 + 4 = 3 , which is the correct answer). You cannot just drop the parentheses (for example, 2 − 3 − 4 = − 1 − 4 = − 5 2 − 3 − 4 = − 1 − 4 = − 5 , which is not correct as we have already verified the answer is 3):

5 x 2 + 4 x − 9 − 3 x 2 + x − 12 5 x 2 + 4 x − 9 − 3 x 2 + x − 12

5 x 2 − 3 x 2 + 4 x + x − 9 − 12 5 x 2 − 3 x 2 + 4 x + x − 9 − 12

2 x 2 + x − 21 2 x 2 + x − 21

Your Turn 5.8

Before looking at multiplying algebraic expressions we look at the Distributive Property , which says that to multiply a sum, first you multiply each term in the sum and then you add the products. For example, 5 ( 4 + 3 ) = 5 ( 4 ) + 5 ( 3 ) = 20 + 15 = 35 5 ( 4 + 3 ) = 5 ( 4 ) + 5 ( 3 ) = 20 + 15 = 35 can also be solved as 5 ( 4 + 3 ) = 5 ( 7 ) = 35 5 ( 4 + 3 ) = 5 ( 7 ) = 35 . If we use a variable, then 5 ( x + 3 ) = 5 x + 15 5 ( x + 3 ) = 5 x + 15 .

We can extended this example to ( 5 + 2 ) ( 4 + 3 ) = ( 5 ) ( 4 ) + ( 5 ) ( 3 ) + ( 2 ) ( 4 ) + ( 2 ) ( 3 ) = 20 + 15 + 8 + 6 = 49 ( 5 + 2 ) ( 4 + 3 ) = ( 5 ) ( 4 ) + ( 5 ) ( 3 ) + ( 2 ) ( 4 ) + ( 2 ) ( 3 ) = 20 + 15 + 8 + 6 = 49 , which can also be solved as ( 5 + 2 ) ( 4 + 3 ) = ( 7 ) ( 7 ) = 49 ( 5 + 2 ) ( 4 + 3 ) = ( 7 ) ( 7 ) = 49 . If we use variables, then ( x + 5 ) ( x + 4 ) = ( x ) ( x ) + ( x ) ( 4 ) + ( 5 ) ( x ) + ( 5 ) ( 4 ) = x 2 + 4 x + 5 x + 20 = x 2 + 9 x + 20 ( x + 5 ) ( x + 4 ) = ( x ) ( x ) + ( x ) ( 4 ) + ( 5 ) ( x ) + ( 5 ) ( 4 ) = x 2 + 4 x + 5 x + 20 = x 2 + 9 x + 20 .

Distributive Property: a ( b + c ) = a b + a c a ( b + c ) = a b + a c

Example 5.9

Simplifying an expression using the order of operations.

Simplify each expression.

• ( x − 3 ) 5 ( x − 3 ) 5
• ( − 3 ) ( x + y − 2 ) ( − 3 ) ( x + y − 2 )
• 5 2 ( 7 + 3 ) ( x ) 5 2 ( 7 + 3 ) ( x )
• 4 + x • 5 4 + x • 5
• ( 4 + x ) • 5 ( 4 + x ) • 5
• 5 x − 3 • 5 = 5 x − 15 5 x − 3 • 5 = 5 x − 15
• ( − 3 ) • x + ( − 3 ) • y − ( − 3 ) • 2 = − 3 x − 3 y + 6 ( − 3 ) • x + ( − 3 ) • y − ( − 3 ) • 2 = − 3 x − 3 y + 6
• 25 ( 7 + 3 ) ( x ) = 25 ( 10 ) ( x ) = 250 x 25 ( 7 + 3 ) ( x ) = 25 ( 10 ) ( x ) = 250 x
• 4 + 5 x 4 + 5 x
• ( 4 ) • ( 5 ) + ( x ) • ( 5 ) = 20 + 5 x ( 4 ) • ( 5 ) + ( x ) • ( 5 ) = 20 + 5 x

Your Turn 5.9

Example 5.10, multiplying algebraic expressions.

Multiply ( 4 x − 9 ) ( x + 2 ) ( 4 x − 9 ) ( x + 2 ) .

Step 1: Use the Distributive Property:

( 4 x ) ( x ) + ( 4 x ) ( 2 ) − ( 9 ) ( x ) − ( 9 ) ( 2 ) ( 4 x ) ( x ) + ( 4 x ) ( 2 ) − ( 9 ) ( x ) − ( 9 ) ( 2 )

Step 2: Multiply:

4 x 2 + 8 x − 9 x − 18 4 x 2 + 8 x − 9 x − 18

4 x 2 − x − 18 4 x 2 − x − 18

Your Turn 5.10

You may have heard the term FOIL which stands for: First, Outer, Inner, Last. FOIL essentially describes a way to use the Distributive Property if you multiply a two-term expression by another two-term expression, but FOIL only works in that specific situation. For example, suppose you have a two-term expression multiplied by a three-term expression, such as ( x + 2 ) ( x + y − 5 ) ( x + 2 ) ( x + y − 5 ) . What terms qualify as inner terms and what terms qualify as outer terms? In this particular situation, FOIL cannot possibly work; the multiplication of ( x + 2 ) ( x + y − 5 ) ( x + 2 ) ( x + y − 5 ) should yield six terms, where FOIL is designed to only give you four! The Distributive Property works regardless of how many terms there are. FOIL can be misleading and applied inappropriately if not fully understood!

Example 5.11

Dividing algebraic expressions.

Divide ( 8 x 2 + 4 x − 16 ) ÷ ( 4 x ) ( 8 x 2 + 4 x − 16 ) ÷ ( 4 x ) .

Divide EACH term by 4 x x :

( 8 x 2 ÷ 4 x ) + ( 4 x ÷ 4 x ) − ( 16 ÷ 4 x ) = 2 x + 1 − 4 x ( 8 x 2 ÷ 4 x ) + ( 4 x ÷ 4 x ) − ( 16 ÷ 4 x ) = 2 x + 1 − 4 x

Your Turn 5.11

Be careful how you divide! Sometimes students incorrectly divide only one term on top by the bottom term. For example, 8 x 2 + 6 x − 3 2 x 8 x 2 + 6 x − 3 2 x might turn into 4 x + 3 x − 3 = 7 x − 3 4 x + 3 x − 3 = 7 x − 3 if done incorrectly. When we divide expressions, EACH term is divided by the divisor. So, 8 x 2 + 6 x − 3 2 x = 8 x 2 2 x + 6 x 2 x − 3 2 x = 4 x + 3 − 3 2 x . 8 x 2 + 6 x − 3 2 x = 8 x 2 2 x + 6 x 2 x − 3 2 x = 4 x + 3 − 3 2 x . If you forget, it is always a good idea to check these rules by creating an example using numerical expressions. For example, 9 + 6 + 3 3 = 18 3 = 6 9 + 6 + 3 3 = 18 3 = 6 . Dividing each term on top by 3 would yield 9 + 6 + 3 3 = 9 3 + 6 3 + 3 3 = 3 + 2 + 1 = 6 9 + 6 + 3 3 = 9 3 + 6 3 + 3 3 = 3 + 2 + 1 = 6 , which is the correct answer. However, if you just divided the 9 on top by the 3 on the bottom, getting 9 + 6 + 3 3 = 3 + 6 + 3 = 12 9 + 6 + 3 3 = 3 + 6 + 3 = 12 , this does not result in the correct answer.

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Al-khwarizmi.

Abu Ja’far Muhammad ibn Musa Al-Khwarizmi was born around 780 AD, probably in or around the region of Khwarizm, which is now part of modern-day Uzbekistan. For most of his adult life, he worked as a scholar at the House of Wisdom in Baghdad, Iraq. He wrote many mathematical works during his life, but is probably most famous for his book Al-kitab al-muhtasar fi hisab al-jabr w’al’muqabalah , which translates to The Condensed Book on the Calculation of al-Jabr ( completion ) and al’muqabalah ( balancing ). The word al-jabr would eventually become the word we use to describe the topic that he was writing about in this book: algebra . From another book of his, with the Latin title Algoritmi de numero Indorum ( Al-Khwarizmi on the Hindu Art of Reckoning ), our word algorithm is derived. In addition to writing on mathematics, Al-Khwarizmi wrote works on astronomy, geography, the sundial, and the calendar.

In 2012, Andrew Hacker wrote an opinion piece in the New York Times Magazine suggesting that teaching algebra in high school was a waste of time. Keith Devlin, a British mathematician, was asked to comment on Hacker's article by his students in his Stanford University Continuing Studies course "Mathematics: Making the Invisible Visible" on iTunes University. Devlin concludes that Hacker was displaying his ignorance of what algebra is.

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Numerical Methods and Modelling for Engineering pp 13–30 Cite as

Numerical Representation

• Richard Khoury 3 &
• Douglas Wilhelm Harder 4  
• First Online: 12 May 2016

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The numerical system used in the Western World today is a place-value base-10 system inherited from India through the intermediary of Arabic trade; this is why the numbers are often called Arabic numerals or more correctly Indo-Arabic numerals. However, this is not the only numerical system possible. For centuries, the Western World used the Roman system instead, which is a base-10 additive-value system (digits of a number are summed and subtracted from each other to get the value represented), and that system is still in use today, notably in names and titles. Other civilizations experimented with other bases: some precolonial Australian cultures used a base-5 system, while base-20 systems arose independently in Africa and in pre-Columbian America, and the ancient Babylonians used a base-60 counting system. Even today, despite the prevalence of the base-10 system, systems in other bases continue to be used every day: degrees, minutes, and seconds are counted in the base-60 system inherited from Babylonian astrologers, and base-12 is used to count hours in the day and months (or zodiacs) in the year.

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Home / United States / Math Classes / 4th Grade Math / The Different Forms of Representing Numbers

The Different Forms of Representing Numbers

The value of each digit in a multi-digit number is known as place value. The place value is determined by the position o f the digits in the number. Learn how to use the place value system to express numbers in the standard form, word form, and the expanded form with the help of some solved examples. ...Read More Read Less

Table of Contents

The Standard Form

The word form, the expanded form.

• Solved Examples
• Frequently Asked Questions

There are different ways of writing a number. We can represent a number in three different forms: 

Representation deals with the method that allows us to express a number to someone else, and the number should be exactly understandable to others as stated.

In our daily life, if someone says the price of a toy is twenty-five dollars, it means he said the number in word form, and this way of expressing a number could be understood. Again, if a price tag on a shirt says ‘$15’, it is said to be in the standard form which is easily understandable.

If we write a number in digits , separating the groups by commas, the number is said to be in the standard form. We often use the standard form to write the numbers in mathematics. It is also the easiest way of expressing the number. So, it is the most common or obvious way of expressing numbers.

For example, 

789,450, and so on.

If one writes or represents a standard number in words, the form is called the word form . This is the same as when we tell a number to others.

For example, 399 is stated as three hundred ninety-nine.

586,256 in the word form is five hundred eighty-six thousand, two hundred fifty-six.

If the number is written as the summation of the place value of each digit , the number is said to be  in the expanded form.

356 in the expanded form is 300 + 50 + 6.

Here, each digit is separated according to its place value. Then the ‘addition’ symbol is mentioned between them. This representation helps us analyze a number easily.

Example: If we take the number 842,062 and make a place value chart as,

This number can be written in three forms as,

The standard form: 842,062.

The word form: Eight hundred forty-two thousand, sixty-two.

The expanded form: 800000 + 40000 + 2000 + 60 + 2.

Solved Examples on Number Representation

Example 1: Write the number in three different forms.

The standard form: 8,563.

The word form: Eight thousand, five hundred sixty-three.

The expanded form: 8,000 + 500 + 60 + 3 . 

Example 2: Write the number in the standard and expanded forms:

Eighty-six thousand, seven hundred and three.

The word form is eighty-six thousand, seven hundred three.

The standard form:

The standard form: 86,703.

The expanded form: 80,000 + 6,000 + 700 + 3 .

Example 3: Write in the standard and word forms.

100,000 + 30,000 + 6,000 + 500 + 50  

The expanded form is,

100,000 + 30,000 + 6,000 + 500 + 50

The standard form: 136,550

The word form: One hundred thirty-six thousand, five hundred fifty.

Example 4: Use the number 700,000 + 5,000 + 20 + 1 to complete the check.

700,000 + 5,000 + 20 + 1

The standard form: 705,021

The word form: Seven hundred five thousand, twenty one.

Example 5: Ava asks Liam and Mia to write “ninety-nine thousand, three hundred thirty-four” in the standard form. Who wrote the correct number? What mistake did the other make?

The word form is ninety-nine thousand, three hundred thirty- four.

The standard form : 99,334

Therefore, Liam wrote the correct answer.

Mia has made a mistake in the tens place where instead of writing 3 she has written as 0.

Write the three forms for representing a number.

The three forms to represent a number are:

The standard form: For example, 78,562.

The word form: For example, Six hundred fifty six.

The expanded form: For example, 500 + 20 + 3.

If a number has a 0 in the middle, what is done with that place value?

If there is a zero in between a number while writing the number in the expanded form or in the word form, the place is skipped.

What is the use of the expanded form of a number?

The expanded form of a number helps us understand the place value of digits clearly. It also helps while comparing two or more numbers.

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Numbers – Definition, Types, Examples, Practice Problems, FAQs

What are numbers, cardinal numbers and ordinal numbers, types of numbers.

• Fraction and Decimal Numbers

Solved Examples on Numbers

Practice problems on numbers, frequently asked questions on numbers.

A number is an arithmetic value used to represent quantity. Hence, a number is a mathematical concept used to count, measure, and label. Thus, numbers form the basis of mathematics.

For example , this is 1 butterfly and these are 4 butterflies.

History of Numbers

The inscriptions found at archaeological sites show that early humans used various symbols to show numbers. For example, ancient farmers, traders, and merchants used tally marks to show quantities. In tally marks, a standing line is drawn for each count and the fifth count is shown by striking off the four lines. This, however, was a tedious way and it was not feasible to show quantities.

Different ways of writing numbers were brought to use with the development of early civilizations. They used different symbols to show larger quantities. But even with these systems, it was not easy to show large quantities.

Around the seventh century, a decimal (or base ten) positional method, was perfected in India. This method used ten unique symbols to represent any number or quantity. These symbols are 0, 1, 2, 3, 4, 5 6, 7, 8, and 9.

This system was spread across Europe by the Arab merchants, scholars, and conquerors.

This system is called the Hindu–Arabic numeral system, and it remains the most common system for representing numbers to date.

Related Worksheets

Numbers in Everyday Life

Numbers are used everywhere around us. Your birthday has numbers that tell the day, month, and year you were born. 

Numbers are used to keep track of time. We use clocks that show us time. We plan our day and events according to time. 

Numbers are involved in buying and selling too. To count money and the units of an item, we use numbers. 

Numbers are used for measurement . Temperature, weight , length , capacity , speed, distance, area , volume , and so on are measured using numbers. 

Numbers play an important role in our body too. We have 2 eyes, 2 ears, 1 nose, 2 hands, 2 legs, and an adult body has 206 bones. 

Our houses have numbers, bank accounts have numbers, and so do our cars, buses, trains, and flights.

Number Representation

A number system is a writing system for denoting numbers using digits or symbols in a logical manner.

We use the digits from 0 to 9 to form all other numbers.

With the help of these digits, we can create infinite numbers.

For example , 121; 34,987; 2,987,633; 459,227,904; …

This number system using 10 digits is called the Decimal Number System.

• Alphabetical Form of Number

Number words are the alphabetical form of numbers. As the name suggests, these are numbers written in words.

For example:

1  One

33 Thirty-three 

• Symbolically, Using Numerals

Numerical symbols are numerals, such as Hindu-Arabic numerals (for example: 112, 415, 999) or Roman numerals (I, II, V, VIII). 

Cardinal numbers are counting numbers . The numbers that we use for counting are called cardinal numbers.

Cardinal numbers tell us how many things, items, or objects are there.

Example: 1, 2, 3, 10, 158

Ordinal numbers give us the exact position of a thing, item, or an object in the list. Ordinal numbers tell the position of an object rather than its quantity. 

Example: 1st, 2nd, 3rd, 9th, 150th 

Apart from the above, there exist other numbers, namely even and odd numbers , prime numbers , and composite numbers . These can be defined as follows:

Fraction and Decimal Numbers:

Example 1: Classify the given set of numbers as fractions or decimals .

7/12; 0.0008; 1.52; 100/10; 4 1/2; 7555.0

Example 2: Write the numerator and denominator of given rational numbers .

• 17/21 (b) 4/5 (c) 25/22

Example 3: Write the numbers in words.

• Five hundred and forty-eight.
• One thousand six hundred and sixty.

Attend this Quiz & Test your knowledge.

How many odd numbers are there between 64 and 90?

Sum of the numbers of primes between 10 to 20 and 30 to 40 is, four thousand eight hundred and eight in numeral form is written as:, what type of number is –5.

How do you find if a number is odd or even?

If a number is divisible by 2 with no remainder , then it is an even number. If a number is divided by 2 and leaves the remainder 1, then it is an odd number.

Is zero an even number?

When zero is divided by 2, the quotient is 0 and remainder is also 0. So, zero is an even number .

Can rational numbers be negative?

Yes, rational numbers are classified as positive, zero, or negative rational numbers.

Do fractions count as whole numbers?

Whole numbers do not include fractions or decimals.

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Numerical representations and error

Introduction.

There are many ways to represent a number within a digital computer. In this section we will examine some of the more useful and common types of these representations, as well as their strengths and weaknesses in terms of error analysis and efficiency.

and the relative error is

Floating-point representations

The most common way of representing numbers in computations is to use a floating-point representation. We will focus on the IEEE (Institute of Electrical and Electronics Engineers) double precision standard, which is currently the most often used (but some other types are important in specialized areas such as audio processing and machine learning).

A floating-point number is represented with a sign (either positive or negative), a mantissa of some number of digits in a particular base (almost always base-2 on a computer), and an exponent. In base-2 most numbers will be represented in the form

So for example the decimal number 1,073,741,824.5, which in exponential binary form is

and as it is positive its sign bit would be 0, so the 64-bit word would be

For more information on floating point numbers, the Wikipedia entry is quite good. There is also a page on "minifloats" , which use a small number of bits and are mainly used as examples, since it is awkward to do examples with 64 or 32 bits.

Special and subnormal numbers

Most numbers cannot be exactly represented in floating-point, and need to be rounded. The IEEE standard rounds them up or down to the nearest representable number, unless it is exactly halfway between the nearest two representable numbers. In that special case, the rounding is chosen so that the last digit of the mantissa will be zero. This helps avoid consistently rounding up or down for the special halfway case.

Machine epsilon

Lets compute

To make the bit representations more clear we will put in some commas between the different segments for sign, exponent, and the significand.

To see what how the number 6/5 is represented in this system, we first compute the binary representation:

This needs to be rounded to have 10 bits after the point; since the remainder is over halfway to the next largest number, we round up to

which rounds down to 1.

and we get zero.

Understanding and Representing Functions: Verbal, Numerical, Graphical, and Algebraic Approaches

Four ways to represent a function.

There are four common ways to represent a function: verbally, numerically, graphically, and algebraically.

1. Verbal representation: This involves describing the function using words. You can explain what the function does, its input and output values, and any patterns or relationships it exhibits. For example, you might say “This function doubles any input number” or “The function returns the square root of the input number.”

2. Numerical representation: This method involves listing the input-output pairs of the function in a table. You would provide a set of input values and their corresponding output values. For example, if the function is f(x) = 2x, you might list the values as:

x | f(x) ——— 0 | 0 1 | 2 2 | 4 3 | 6

3. Graphical representation: This involves plotting the input-output pairs on a graph. The input values are placed on the x-axis and the output values on the y-axis. Each point represents a pair of input and output values, and connecting the points creates a visual representation of the function. For example, if the function is f(x) = 2x, the graph would be a straight line passing through the origin, with a slope of 2.

4. Algebraic representation: This method involves expressing the function using algebraic equations or formulas. You might use variables, symbols, and mathematical operations to represent the function. For example, if the function is f(x) = 2x, the algebraic representation would be an equation stating that the output value (f(x)) is equal to twice the input value (2x).

Overall, these four ways of representing a function provide different perspectives and tools for understanding and analyzing its behavior. It is often helpful to utilize multiple representations in order to gain a comprehensive understanding of a function and its properties.

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Mathematical Representations Series Part 4: Verbal Representation

Welcome back to our deep dive into mathematical representations! Today, we are taking a look at symbolic representations and how we can translate between symbolic, concrete, and visual representations. First, let’s do a two-sentence recap of this series so far:

We have already focused on concrete representations and the immense value of manipulatives, the range of visual representations we want to encourage with our students, and how we can use numerals and operations to represent thinking symbolically . We are centering our conversation around Lesh’s Translation Model, which encompasses the range of ways we represent our thinking, and stresses the importance of making connections between representations.

Today we are talking about verbal representations. While it’s an essential form of representation for our students, it is often less discussed. This is likely due to the fact that it is not explicitly called out in the Concrete-Pictorial Abstract model . This is just another reason why I love introducing teachers to Lesh’s translation model alongside the CPA (often called CRA) model.

Verbal Representation

The language we use to communicate our thoughts and ideas is another equally important representation. This can be oral, written, signed, or any way that a student would look to communicate language. James Heddens writes that students “need to be given opportunities to verbalize their thought processes: verbal interaction with peers will help learners clarify their own thinking.”

If we go back to our previous examples from concrete, visual, and abstract thinking, we have a student with five yellow counters and four red counters. The student then sketched their counters and wrote the number sentence 5+4=9 on their paper.

So how does verbal representation come into play? Perhaps after the activity, a student shows you their sketch of the counters. When you ask them about their drawing, they may share “I had nine counters. Four of them were red and five of them were yellow, and that makes nine.” That statement is a verbal representation of the concept. They have also just translated their visual representation to a verbal representation.

Connecting Two Verbal Representations

If wanted, we could take it a step further by asking the student to write their thoughts down. This will require the student to revisit their thoughts communicated orally and condense them into a written description, like “Four counters and five counters make nine counters.” This extra step of condensing their language into a second form, allowed students to connect two verbal representations. WOW!

Verbal representation is essential to our work, especially in the early grades. Our students who may not have the ability to write words or numbers will often communicate their understanding orally. This NEEDS to be a part of the discussion when we talk about deepening student understanding, and it’s a huge reason why I make sure to consider Lesh’s Translation Model in addition to the Concrete-Pictorial-Abstract model.

What’s Up Next?

This series is going to dive deep into each of the representations discussed in Lesh’s Translation Model, and then we are going to put it all together so we can make a big impact on your math teaching this year.

If you missed Part One about Concrete Representations , Part Two about Visual Representations , or Part Three about Symbolic Representations , check them out so you have all of the info you need before we move on!

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