Number System Study Guide

Introduction

The number system is a way of classifying and representing numbers according to their properties. Numbers are used to count objects, measure quantities, describe positions, perform calculations, and solve problems in mathematics, science, engineering, finance, and computing.

The main groups in the number system are natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. These sets are related to one another, and understanding their relationships makes many mathematical concepts easier.

For example, 5 is a natural number, whole number, integer, rational number, and real number. In contrast, √2 is an irrational real number because it cannot be expressed as a fraction of two integers.

This guide explains the major types of numbers, their properties, number-line representation, operations, factors and multiples, fractions and decimals, and common mistakes.

Learning Objectives

After studying this guide, you should be able to:

• Identify the different types of numbers.

• Explain the relationship between natural, whole, integer, rational, irrational, and real numbers.

• Distinguish rational numbers from irrational numbers.

• Perform operations involving different types of numbers.

• Understand factors, multiples, prime numbers, HCF, and LCM.

• Convert between fractions, decimals, and percentages.

• Represent numbers on a number line.

• Apply the properties of numbers to solve mathematical problems.

What is a Number System?

A number system is a collection of numbers grouped according to common mathematical properties.

The most commonly studied number system is the real number system, which contains both rational and irrational numbers.

The major classification can be shown as:

Real Numbers

├── Rational Numbers

│   ├── Integers

│   │   ├── Whole Numbers

│   │   │   └── Natural Numbers

│   │   └── Negative Integers

│   └── Non-integer Fractions/Decimals

│

└── Irrational Numbers

The sets are nested in this way:

Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real

Natural Numbers

Natural numbers are the numbers commonly used for counting.

Depending on the mathematical convention, natural numbers may begin with either 1 or 0.

The convention most commonly used in elementary mathematics is:

N = {1, 2, 3, 4, 5, ...}

Some textbooks define natural numbers as:

N = {0, 1, 2, 3, 4, ...}

Therefore, when answering a question about whether 0 is a natural number, check the convention being used.

Examples of natural numbers:

• 1

• 7

• 25

• 100

• 1,000

Negative numbers and fractions are not natural numbers under the usual definition.

Whole Numbers

Whole numbers are the non-negative integers.

They include zero and all positive counting numbers:

W = {0, 1, 2, 3, 4, 5, ...}

Examples:

• 0

• 4

• 19

• 250

Whole numbers do not include negative numbers, fractions, or irrational numbers.

Integers

Integers include positive whole numbers, negative whole numbers, and zero.

They are represented by:

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Examples:

• -10

• -3

• 0

• 6

• 42

Integers do not include numbers such as 1/2, 3.7, or √2.

Rational Numbers

A rational number is any number that can be written as:

p/q

where p and q are integers and q ≠ 0.

Examples include:

• 1/2

• 3/4

• -7/5

• 8

• 0

• 2.5

An integer is rational because it can always be written over 1.

For example:

7 = 7/1

Rational Numbers and Decimals

Rational numbers have decimal representations that either:

1. Terminate, or

2. Repeat indefinitely

Examples of terminating decimals:

• 1/2 = 0.5

• 3/4 = 0.75

• 1/8 = 0.125

Examples of repeating decimals:

• 1/3 = 0.333...

• 2/11 = 0.181818...

The repeated digits are called a repeating decimal or recurring decimal.

Irrational Numbers

An irrational number cannot be expressed as a fraction of two integers.

Its decimal representation:

• Never terminates.

• Never repeats in a fixed pattern.

Common examples include:

• √2

• √3

• √5

• π

• e

For example:

√2 ≈ 1.414213562...

The digits continue indefinitely without forming a repeating pattern.

Rational vs. Irrational Numbers

A useful test is:

Terminating or repeating decimal → Rational

Non-terminating and non-repeating decimal → Irrational

Real Numbers

Real numbers include all rational and irrational numbers.

Therefore:

Real numbers = Rational numbers + Irrational numbers

Examples of real numbers include:

• -5

• 0

• 3/7

• 2.75

• √2

• π

Real numbers can be represented on a number line.

The Number Line

A number line represents numbers according to their position.

← negative                         positive →

----|----|----|----|----|----|----|----

   -3   -2   -1    0    1    2    3

Numbers increase as you move to the right and decrease as you move to the left.

For example:

-2 < 1

because -2 is located to the left of 1.

Absolute Value

The absolute value of a number is its distance from zero on the number line.

It is written using vertical bars.

Examples:

|5| = 5

|-5| = 5

|0| = 0

Absolute value is always non-negative.

Positive and Negative Numbers

Positive numbers are greater than zero.

Examples:

• 1

• 8

• 25

Negative numbers are less than zero.

Examples:

• -1

• -8

• -25

Zero is neither positive nor negative.

Comparing Numbers

When comparing numbers, use:

• > greater than

• < less than

• = equal to

• ≥ greater than or equal to

• ≤ less than or equal to

Examples:

8 > 3

-2 < 5

-7 < -3

The last example can be confusing. On the number line, -7 is farther to the left than -3, so it is smaller.

Factors

A factor of a number divides that number exactly without leaving a remainder.

For example, the factors of 12 are:

1, 2, 3, 4, 6, 12

because each divides 12 evenly.

Factors usually occur in pairs:

1 × 12 = 12

2 × 6 = 12

3 × 4 = 12

Multiples

A multiple is obtained by multiplying a number by a whole number.

The multiples of 5 include:

5, 10, 15, 20, 25, 30, ...

Every number has infinitely many multiples.

Prime Numbers

A prime number is a positive integer greater than 1 that has exactly two positive factors:

1 and itself.

Examples:

2, 3, 5, 7, 11, 13, 17, 19

The number 2 is the only even prime number.

Composite Numbers

A composite number is a positive integer greater than 1 that has more than two positive factors.

Examples:

• 4

• 6

• 8

• 9

• 10

• 12

For example, 12 has factors:

1, 2, 3, 4, 6, 12

Therefore, it is composite.

Is 1 Prime or Composite?

1 is neither prime nor composite.

A prime number must have exactly two distinct positive factors. The number 1 has only one positive factor: itself.

This is a common examination question.

Prime Factorization

Prime factorization expresses a number as a product of prime numbers.

For example:

60 = 2 × 2 × 3 × 5

or:

60 = 2² × 3 × 5

Prime factorization is useful for calculating the HCF and LCM.

Highest Common Factor

The Highest Common Factor (HCF), also called the Greatest Common Factor (GCF), is the largest factor shared by two or more numbers.

For example, consider 12 and 18.

Factors of 12:

1, 2, 3, 4, 6, 12

Factors of 18:

1, 2, 3, 6, 9, 18

The common factors are:

1, 2, 3, 6

Therefore:

HCF = 6

Lowest Common Multiple

The Lowest Common Multiple (LCM) is the smallest positive number that is a multiple of two or more numbers.

For 4 and 6:

Multiples of 4:

4, 8, 12, 16, 20, ...

Multiples of 6:

6, 12, 18, 24, ...

Therefore:

LCM = 12

HCF and LCM Relationship

For two positive integers a and b:

HCF(a,b) × LCM(a,b) = a × b

For example, for 12 and 18:

HCF = 6

LCM = 36

Therefore:

6 × 36 = 216

and:

12 × 18 = 216

Even and Odd Numbers

An even number is divisible by 2.

Examples:

0, 2, 4, 6, 8, 10

An odd number is not divisible by 2.

Examples:

1, 3, 5, 7, 9, 11

Zero is an even number because it is divisible by 2:

0 ÷ 2 = 0

Divisibility Rules

Divisibility rules provide quick ways to determine whether a number can be divided evenly by another number.

These rules are especially useful when solving problems without a calculator.

Fractions

A fraction represents a part of a whole or a ratio between quantities.

It has two main parts:

Numerator / Denominator

For:

3/5

3 is the numerator and 5 is the denominator.

The denominator cannot be zero.

Proper and Improper Fractions

A proper fraction has a numerator smaller than its denominator.

Examples:

• 2/5

• 3/8

• 7/10

An improper fraction has a numerator greater than or equal to its denominator.

Examples:

• 7/4

• 9/5

• 8/8

A mixed number combines a whole number and a proper fraction.

For example:

1 3/4

Decimals

A decimal represents a number using a decimal point.

Examples:

• 0.5

• 2.75

• 10.125

Place values include:

ones → tenths → hundredths → thousandths

For example:

4.237

contains:

• 4 ones

• 2 tenths

• 3 hundredths

• 7 thousandths

Converting Fractions to Decimals

To convert a fraction into a decimal, divide the numerator by the denominator.

Example:

3/4 = 3 ÷ 4 = 0.75

Another example:

1/8 = 1 ÷ 8 = 0.125

Some fractions produce repeating decimals:

1/3 = 0.333...

Converting Decimals to Fractions

A terminating decimal can be converted into a fraction using its place value.

For example:

0.75 = 75/100

Simplify:

75/100 = 3/4

Therefore:

0.75 = 3/4

Percentages

A percentage means "per hundred."

For example:

25% = 25/100 = 1/4

To convert a decimal into a percentage, multiply by 100:

0.6 × 100 = 60%

To convert a percentage into a decimal, divide by 100:

60% ÷ 100 = 0.6

Exponents

An exponent indicates how many times a number is multiplied by itself.

For example:

2³ = 2 × 2 × 2 = 8

Here:

• 2 is the base.

• 3 is the exponent.

Important rules include:

a¹ = a

a⁰ = 1, when a ≠ 0

aᵐ × aⁿ = aᵐ⁺ⁿ

aᵐ ÷ aⁿ = aᵐ⁻ⁿ, when a ≠ 0

Square Roots

A square root of a number is a value that produces the original number when multiplied by itself.

For example:

√25 = 5

because:

5 × 5 = 25

Perfect squares include:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Not every square root is rational.

For example:

√4 = 2

is rational, but:

√2

is irrational.

Order of Operations

When an expression contains multiple operations, follow the conventional order of operations.

A common memory aid is PEMDAS:

• P — Parentheses

• E — Exponents

• M — Multiplication

• D — Division

• A — Addition

• S — Subtraction

Multiplication and division have the same priority and are evaluated from left to right. Addition and subtraction also have the same priority and are evaluated from left to right.

For example:

8 + 2 × 3 = 14

not 30, because multiplication is performed before addition.

Properties of Numbers

Commutative Property

Changing the order does not change the result for addition and multiplication.

a + b = b + a

a × b = b × a

For example:

3 + 5 = 5 + 3

However, subtraction and division are not generally commutative.

Associative Property

Changing the grouping does not change the result for addition and multiplication.

(a + b) + c = a + (b + c)

(a × b) × c = a × (b × c)

Subtraction and division are not generally associative.

Distributive Property

Multiplication can be distributed across addition or subtraction.

a(b + c) = ab + ac

For example:

3(4 + 5) = 3×4 + 3×5

= 12 + 15

= 27

Identity Properties

For addition:

a + 0 = a

So 0 is the additive identity.

For multiplication:

a × 1 = a

So 1 is the multiplicative identity.

Additive and Multiplicative Inverses

The additive inverse of a number is the number that produces zero when added to it.

For example:

5 + (-5) = 0

Therefore, -5 is the additive inverse of 5.

The multiplicative inverse of a nonzero number is its reciprocal.

For example:

3 × 1/3 = 1

Therefore, 1/3 is the multiplicative inverse of 3.

Real Numbers and the Number Line

Every real number corresponds to a position on the number line.

Rational and irrational numbers are both included.

For example:

-2       -1        0        1        √2        2

|---------|---------|---------|----------|--------|

                              ↑

                           ≈ 1.414

The irrational number √2 lies between 1 and 2.

Common Mistakes

Mistake 1: Thinking Zero Is Positive

Zero is neither positive nor negative.

Mistake 2: Thinking 1 Is Prime

1 is neither prime nor composite because it has only one positive factor.

Mistake 3: Assuming Every Decimal Is Irrational

A terminating decimal such as 0.25 is rational because:

0.25 = 25/100 = 1/4

Mistake 4: Forgetting That Repeating Decimals Are Rational

0.333... is rational because it equals 1/3.

Mistake 5: Dividing by Zero

Division by zero is undefined.

There is no real number x for which:

x × 0 = 5

because every number multiplied by zero equals zero.

Mistake 6: Assuming √2 Is a Fraction

√2 cannot be represented as a ratio of two integers, so it is irrational.

Quick Classification Examples

Summary

The number system organizes numbers into different sets according to their properties.

Natural numbers are used for counting, while whole numbers include zero. Integers add negative numbers. Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot. Together, rational and irrational numbers form the real numbers.

Factors, multiples, prime numbers, HCF, and LCM are important concepts for understanding whole numbers and integers. Fractions, decimals, percentages, exponents, and square roots extend these ideas into more advanced calculations.

The most important classification to remember is:

Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real

and:

Rational + Irrational = Real

FAQ

What are the main types of numbers?

The main types are natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

Is zero a natural number?

It depends on the definition being used. Some conventions include zero in the natural numbers, while others begin with 1.

Is zero a whole number?

Yes. Whole numbers include 0, 1, 2, 3, ....

Is zero an integer?

Yes. Integers include negative numbers, zero, and positive numbers.

Is every integer a rational number?

Yes. Every integer can be written as a fraction with denominator 1. For example, -6 = -6/1.

Is π rational or irrational?

π is irrational. Its decimal expansion is non-terminating and non-repeating.

Is √9 rational?

Yes. √9 = 3, and 3 is rational.

Is √2 irrational?

Yes. √2 cannot be expressed as a ratio of two integers.

Is 1 a prime number?

No. 1 has only one positive factor, while a prime number has exactly two positive factors.

Can a number be both rational and irrational?

No. A real number is either rational or irrational, but not both.

Key Takeaways

• Natural numbers are counting numbers, with conventions differing over whether 0 is included.

• Whole numbers include zero and positive integers.

• Integers include positive and negative whole numbers and zero.

• Rational numbers can be written as p/q, where p and q are integers and q ≠ 0.

• Irrational numbers have non-terminating, non-repeating decimal representations.

• Real numbers consist of all rational and irrational numbers.

• 1 is neither prime nor composite, while 2 is the only even prime number.

• Zero is neither positive nor negative and division by zero is undefined.

• HCF is the greatest shared factor, while LCM is the smallest positive shared multiple.

• The number system provides the foundation for algebra, geometry, statistics, science, and computing.

References

1. NIST Digital Library of Mathematical Functions — authoritative mathematical reference covering numbers, functions, and mathematical notation.
   NIST Digital Library of Mathematical Functions

2. OpenStax — Prealgebra 2e — educational textbook covering whole numbers, integers, fractions, decimals, properties, and real numbers.
   OpenStax Prealgebra 2e

3. OpenStax — College Algebra 2e — educational reference covering real numbers, radicals, exponents, and algebraic properties.
   OpenStax College Algebra 2e

4. Encyclopaedia Britannica — Number — reference material on numbers and their mathematical classification.
   Britannica: Number

5. Wolfram MathWorld — Integer — mathematical reference covering integers and related number concepts.
   Wolfram MathWorld: Integer

6. Wolfram MathWorld — Rational Number — reference material on rational numbers and their mathematical properties.
   Wolfram MathWorld: Rational Number

7. Wolfram MathWorld — Irrational Number — reference material on irrational numbers and their properties.
   Wolfram MathWorld: IrrationalNumber.html