Real Numbers

Real numbers are a fundamental concept in mathematics, encompassing both rational and irrational values. This set forms the backbone of many mathematical principles and applications.

Table Of Contents

What are Real Numbers ?
What are the different types of real numbers?
What are the properties of real numbers ?
Is 0 a real number?
Is Pi a real number?
Are negative numbers real numbers?
What is the difference between rational and irrational numbers?
What is the difference between real and non-real numbers?
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What are Real Numbers ?

Real numbers ( $\mathbb{R}$ ) are the set of all numbers that can be found on the number line. They include both rational and irrational numbers, making them the most extensive and commonly used number system in mathematics. Real number can be positive, negative, or zero.

What are the different types of real numbers?

Natural Numbers

Symbol: ( $\mathbb{N}$ )
These are the counting numbers starting from 1, 2, 3, …
Example: 1, 2, 3, …

Whole Numbers

Symbol: ( $\mathbb{N}_\mathbf{0}$ )
This set includes all natural numbers plus zero.
Example: 0, 1, 2, 3, …

Integers

Symbol: ( $\mathbb{Z}$ )
Integers expand whole numbers to include their negative counterparts.
Example: … , -3 , -2, -1, 0, 1, 2, 3, …

Rational Numbers

Symbol: ( $\mathbb{Q}$ )
These are numbers that can be expressed as a ratio of two integers $\frac {a} {b}$ , where ${a}$ and ${b}$ are integers and ${b} \neq{0}$ .
Example: $\frac{1}{2}; 2,3333; -\frac{5}{3}; 2 \frac{1}{2}; 0,25; -3,6$
Decimal representations of rational number either terminate or repeat.

Irrational Numbers

The number that cannot be expressed as a fraction or ratio of two integers.
Decimal representations of irrational number is non-terminating and non-repeating.
Example: $\pi; \sqrt{2}; 0,1010010001...$

What are the properties of real numbers ?

Closure: Real number is closed under addition, subtraction, multiplication and division (except by zero).
Commutative Property: ${a} + {b} = {b} + {a}$ and ${a} \times {b} = {b} \times {a}$
Associative Property: $({a} + {b})+{c} = {a} + ({b}+{c})$ and $({a} \times {b}) \times {c} = {a} \times ({b} \times {c})$
Distributive Property: ${a} \times ({b}+{c})$ = ${a}\times {b} + {a} \times {c}$ .
Identity Elements: ${a}+{0}={a}$ and ${a}\times{1}={a}$
Inverse Elements: For every number ${a}$ , there exists ${-a}$ such that ${a}+({-a})={0}$

Is 0 a real number?

Rational Number: 0 is a rational number because it can be expressed as a fraction $\frac{0}{1}$ , which is valid since the numerator is 0 and the denominator is non-zero.

Position on the Number Line: 0 lies at the origin of the number line, separating positive numbers from negative numbers. This makes it part of the real number set.

Neutral Element in Addition: 0 is the additive identity, meaning that adding 0 to any real number does not change its value ${a}+{0}={a}$ .

Neither Positive nor Negative: 0 is neither positive nor negative, but it is part of the real number set.

In conclusion, 0 is unequivocally a real number.

Is Pi a real number?

${\pi}$ belongs to the real number because:

It can be placed on the number line: Even though ${\pi}$ is irrational, it has a precise location between 3.141 and 3.142, making it a real number.

Decimal Expansion: The value of ${\pi}$ is approximately 3.14159, and its infinite, non-repeating decimal satisfies the definition of real number.

Are negative numbers real numbers?

Real number is the number that can be represented on the number line, includes both positive and negative numbers, as well as zero.

What is the difference between rational and irrational numbers?

What is the difference between real and non-real numbers?

Real number include all numbers that can be represented on the number line, such as rational number (fractions, integers, and decimals) and irrational number (non-terminating, non-repeating decimals like √2 and π). They are used to represent measurable quantities like distances and temperatures.

Non-real number, or imaginary number, cannot be represented on the number line and include number like √-1 , denoted as ( ${i}$ ). Non-real numbers are used in the complex number system, which combines real and imaginary numbers (e.g., ${3} + {2i}$ ), and they are critical for solving equations and modeling scenarios that cannot be addressed with real number alone.