Real Numbers: Symbol, Properties & Chart

Real numbers are all the numbers on the number line. It includes the rational numbers like the integers or fractions and the irrational numbers like √2 or π. Real numbers are used in mathematics and in everyday life to measure, count and solve problems . Learning of real numbers helps you to understand algebra, geometry, and other mathematical ideas. This article describes real numbers in detail, their types, properties, representation and real life examples.

Table Of Contents

• What Are Real Numbers?
• Real Numbers Chart
• Real Number System
• Types of Real Numbers
• Real Numbers on a Number Line
• Properties of Real Numbers
• Operations on real numbers
• Rational Numbers and Irrational Numbers
• Difference Between Rational and Irrational Numbers
• Real Numbers Formula
• Examples of Real Numbers
• Solved Examples on Real Numbers
• Real Numbers Questions and Answers

What Are Real Numbers?

Real numbers are all the numbers you can place on a number line. This includes positive numbers, negative numbers, zero, fractions, decimals, and even numbers like π (pi). In simple terms, if you can measure it, count it, or find it on a number line, it is a real number.

A real number can be defined as any number that belongs to the set of rational or irrational numbers.

Examples: 5, −3, 0, ½, 0.75, √2, π all of these are real numbers.

Real Number Symbol (ℝ)

The symbol for the set of real numbers is ℝ (a bold or double-struck letter R). When mathematicians write ℝ, they mean the complete collection of all real numbers stretching from negative infinity to positive infinity.

Importance of Real Numbers

Real numbers form the foundation of almost all mathematics. Whether you are solving an equation, measuring a distance, calculating interest, or studying physics, real numbers are involved. Without understanding the real number system, topics like algebra, calculus, and statistics would not make sense.

Real Numbers Chart

The diagram at the top of this article is your real numbers chart. It shows clearly how every type of number fits inside a larger category. Here is a quick summary in text form:

Real Number System

Understanding the Real Number System

The real number system is a well-organised family of numbers arranged in groups, where each group is contained inside a larger group. Think of it like nested boxes the smallest box sits inside a bigger one, and that sits inside an even bigger one.

The chart above shows this nesting clearly. Natural numbers sit inside whole numbers, whole numbers sit inside integers, integers sit inside rational numbers, and rational numbers sit alongside irrational numbers all together inside the big family of real numbers.

Classification of Real Numbers

Real numbers are mainly split into two broad groups:

• Rational numbers: numbers that can be written as a fraction p/q (where q ≠ 0)

• Irrational numbers: numbers that cannot be written as a fraction and have non-repeating, non-terminating decimals

Types of Real Numbers

Natural Numbers

Natural numbers are the counting numbers you learned as a child: 1, 2, 3, 4, 5…

They start at 1 and go on forever. They do not include zero, fractions, or negative numbers.

Symbol: ℕ

Examples: 7, 15, 100, 1000

Whole Numbers

Whole numbers are just like natural numbers, but they also include zero: 0, 1, 2, 3, 4…

The only difference between natural numbers and whole numbers is the inclusion of 0.

Symbol: W

Examples: 0, 4, 23, 500

Integers

Integers include all whole numbers and their negative counterparts: …−3, −2, −1, 0, 1, 2, 3…

They do not include fractions or decimals.

Symbol: ℤ

Examples: −10, −1, 0, 5, 99

Rational Numbers

A rational number is any number that can be expressed as a fraction p/q, where both p and q are integers and q is not zero.

Symbol: ℚ

Examples: ½, −¾, 0.5 (= ½), 0.333… (= ⅓), 7 (= 7/1)

Key point: decimal expansions of rational numbers either terminate (end) or repeat in a pattern.

Irrational Numbers

Irrational numbers cannot be written as a simple fraction. Their decimal forms go on forever without repeating.

Symbol: ℝ  ℚ

Examples: √2 = 1.41421356…, π = 3.14159265…, e = 2.71828182…, √3, √5

Real Numbers on a Number Line

Representation of Rational Numbers

Every rational number occupies an exact, findable spot on the number line. For example, ½ sits exactly halfway between 0 and 1. The number −3/2 sits halfway between −1 and −2. You can always locate a rational number precisely.

Representation of Irrational Numbers

Irrational numbers also have a definite place on the number line, even though we cannot write their exact decimal value. For instance, √2 sits between 1 and 2 (closer to 1.4), and π sits between 3 and 4 (closer to 3.14).

Locating Real Numbers on a Number Line

Every real number, without exception, corresponds to exactly one point on the number line. This is why the number line is sometimes called the "real line." No real number is left out, and no point on the line is left uncovered.

Properties of Real Numbers

Closure Property

If you add or multiply any two real numbers, the result is always a real number. The set is "closed" under these operations. For example, 3 + 7 = 10 (still a real number) and 2 × 9 = 18 (still a real number).

Commutative Property

The order in which you add or multiply two real numbers does not change the answer.

• Addition: a + b = b + a → 4 + 6 = 6 + 4 = 10

• Multiplication: a × b = b × a → 3 × 5 = 5 × 3 = 15

Note: This property does not apply to subtraction or division.

Associative Property

When adding or multiplying three or more numbers, the way you group them does not matter.

• Addition: (a + b) + c = a + (b + c) → (1 + 2) + 3 = 1 + (2 + 3) = 6

• Multiplication: (a × b) × c = a × (b × c) → (2 × 3) × 4 = 2 × (3 × 4) = 24

Distributive Property

Multiplication distributes over addition: a(b + c) = ab + ac

Example: 5 × (3 + 4) = 5×3 + 5×4 = 15 + 20 = 35

This property connects multiplication and addition together.

Identity Property

• The additive identity is 0: a + 0 = a (adding zero changes nothing)

• The multiplicative identity is 1: a × 1 = a (multiplying by one changes nothing)

Inverse Property

• Every real number a has an additive inverse −a such that a + (−a) = 0

• Every non-zero real number a has a multiplicative inverse 1/a such that a × (1/a) = 1

Table of properties on Real Numbers

Operations on Real Numbers

Operations on real numbers are the four basic math operations addition, subtraction, multiplication, and division done with all real numbers such as integers, fractions, decimals, and numbers like √2 and π. When you add or subtract numbers with the same sign, you add them and keep that sign. 

Rational Numbers and Irrational Numbers

What Are Rational Numbers?

A rational number is any number you can write in the form p/q, where p and q are integers and q ≠ 0. The decimal form either ends (0.5) or repeats in a cycle (0.333…).

Common examples of rational numbers: ¾, −2, 0, 1.25, 5/7, 0.666…

What Are Irrational Numbers?

An irrational number cannot be expressed as p/q for any integers p and q. The decimal digits go on forever with no repeating pattern.

Common examples of irrational numbers: π, √2, √3, √5, e (Euler's number), the golden ratio φ

Examples of Rational and Irrational Numbers

Difference Between Rational and Irrational Numbers

Rational Numbers vs Irrational Numbers

The key difference lies in how their decimal expansions behave. Rational decimals end or follow a repeating cycle. Irrational decimals continue forever in a pattern that never repeats.

Comparison Table

Real Numbers Formula

Real Number Set Formula

The complete set of real numbers is written as:

ℝ = Q ∪ Q'

Where Q = the set of rational numbers and Q' = the set of irrational numbers. This simply says that real numbers are everything rational and irrational put together.

Important Formulas Related to Real Numbers

These algebraic identities hold true for all real numbers a and b:

• (a + b)² = a² + 2ab + b²

• (a − b)² = a² − 2ab + b²

• (a + b)(a − b) = a² − b² (difference of squares)

• |a| ≥ 0 for all real numbers (absolute value is never negative)

• |a × b| = |a| × |b| (absolute value of a product)

Examples of Real Numbers

Positive Real Numbers Examples

Positive real numbers are all real numbers greater than zero: 1, 2.5, ¾, √9 = 3, π ≈ 3.14159, 100, 0.001

Negative Real Numbers Examples

Negative real numbers are all real numbers less than zero: −1, −2.5, −¾, −√2, −100, −0.001

Rational Real Numbers Examples

These are real numbers that can be written as fractions: 0, ½, −4, 0.333…, 7/3, 1.75, −99

Irrational Real Numbers Examples

These are real numbers that cannot be written as fractions: √2, √3, √5, √7, π, e, φ (golden ratio = 1.618…)

Solved Examples on Real Numbers

Basic Real Numbers Examples

Question 1: Is 0.7 a real number? What type?

Answer: Yes. 0.7 = 7/10, which is rational.

Since all rational numbers are real, 0.7 is a real number.

Question 2: Is √16 rational or irrational?

Answer: √16 = 4, and 4 = 4/1 is rational.

So √16 is a rational real number.

Question 3: Classify −5 in the real number system.

Answer: −5 is a negative integer.

It is also rational (−5/1) and therefore a real number.

Number Line Examples

Question 4: Between which two integers does √10 lie?

Answer: 3² = 9 and 4² = 16.

Since 9 < 10 < 16, we get 3 < √10 < 4.

So √10 lies between 3 and 4 on the number line.

Question 5: Place ¾ on the number line.

Answer: ¾ = 0.75.

This lies between 0 and 1, exactly three-quarters of the way from 0 to 1.

Rational and Irrational Number Examples

Question 6: Is the sum of two irrational numbers always irrational?

Answer: Not always.

For example, √2 + (−√2) = 0, which is rational.

However, √2 + √3 is irrational.

Question 7: Prove that √2 is irrational (brief version).

Answer: Assume √2 = p/q in lowest terms.

Then 2q² = p², meaning p² is even, so p is even.

Write p = 2k. Then 2q² = 4k², giving q² = 2k², so q is also even.

But this contradicts our assumption that p/q is in lowest terms.

Therefore √2 cannot be rational it is irrational.

Real Numbers Questions and Answers

Practice Questions

1. Which of the following are irrational? 0.5, √3, 9/4, π, −7

2. Identify the additive inverse of −8.

3. State the multiplicative identity for real numbers.

4. Is every integer a real number? Justify.

5. Between which two whole numbers does √50 lie?

Answers:

1. √3 and π are irrational; the rest are rational.

2. The additive inverse of −8 is 8, since −8 + 8 = 0.

3. The multiplicative identity is 1.

4. Yes, because integers ⊂ rational numbers ⊂ real numbers.

5. 7² = 49 and 8² = 64, so √50 lies between 7 and 8.

Read more: Real Numbers Questions

Exam Oriented Questions

Question 1: Is every real number a rational number? 

No. Irrational numbers like π and √2 are real but not rational.

Question 2: Can a number be both rational and irrational? 

No. A real number is either rational or irrational, never both.

Question 3: What is the difference between real numbers and imaginary numbers?

Real numbers can be placed on a number line. Imaginary numbers, like √(−1) = i, cannot. They form a separate category outside the real number system.