Last modified on August 3rd, 2023

chapter outline

 

Real Numbers

Think of any number, and it is possibly a real number. Real numbers can be integers, whole numbers, natural naturals, fractions, or decimals. Real numbers can be positive, negative, or zero.

Thus, real numbers broadly include all rational and irrational numbers. They are represented by the symbol ${\mathbb{R}}$ and have all numbers from negative infinity, denoted -∞, to positive infinity, denoted ∞, written in interval notation as (-∞, ∞).

List

Listed below is a chart showing all the subsets or types of real numbers in the number system:

Real Numbers Chart
  • Natural numbers: These are all the counting numbers starting from 1. The set is denoted by ${\mathbb{N}}$ and includes {1, 2, 3, 4, …}.
  • Whole numbers: 0, along with all the natural numbers, constitute the set of whole numbers ${\mathbb{W}}$. It includes ${\mathbb{W}}$ = {0, 1, 2, 3, 4, …}.
  • Integers: The set of integers ${\mathbb{Z}}$ contains 0, all the natural numbers, and their additive inverses. Thus ${\mathbb{Z}}$ = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}.
  • Rational Numbers: All the integers, fractions (with non-zero denominator), and decimal numbers (terminating and repeating) such as ${\dfrac{7}{9}}$, – 2.5, 0.3333…, ${-\dfrac{1}{2}}$ constitute the set of rational numbers. This set is denoted by ${\mathbb{Q}}$ = {${\dfrac{p}{q}}$: p and q are integers and q ≠ 0} 
  • Irrational Numbers: Numbers which are not rational, e.g., ${-\sqrt{13}}$, -e, ${-\dfrac{π}{2}}$, 0, 3.3278964205…, ${\sqrt{5}}$ forms the set of irrational numbers. It is denoted by ${\overline{\mathbb{Q} }}$ = {x: x is not rational}

We can also represent the above chart in the form of a Venn diagram, as shown:

Real Numbers Venn Diagram

However, real numbers do not include:

  • Imaginary or Complex Numbers: Numbers that are not real. ${\sqrt{-1}}$, ${\sqrt{-7}}$, 3 + 2i, -3i are few such examples.
  • Infinity

Real Numbers on a Number Line

Every real number has a unique position on a number line. Again, each point on a number line corresponds to exactly one real number. Thus, representing real numbers on a number line helps us organize them according to their values (lesser or greater), find a number, and perform various arithmetic operations.

The origin or zero lies in the middle, with positive numbers to the right of zero and negative numbers to the left. Consecutive integers are placed at a unit distance from each other, and the fractions and decimal numbers lie between the integers.

The values increase from left to right and decrease from right to left.

Here we have graphed the integers from (-4) to 4 with rational and irrational numbers between them by comparing and ordering their values from smallest to largest.

Real Number Line

Properties

Real numbers hold some crucial properties in mathematics:

Identity Property

Real numbers have 0 as their additive identity and 1 as their multiplicative identity. If we consider a real number ‘m’,

  • For addition: m + 0 = m = 0 + m, e.g., 3 + 0 = 3 = 3 + 0
  • For multiplication: m × 1 = m = 1 × m, e.g., (-8) × 1 = (-8) = 1 × (-8)

Closure Property

Real numbers are closed under the arithmetic operations of addition, subtraction, multiplication, and division. In other words, addition, subtraction, multiplication, and division of two real numbers, ‘m’ and ‘n’, always give a real number.

For example,

  • 2 + 5 = 7
  • 0.9 – 0.6 = 0.3
  • ${\dfrac{3}{5}}$ × ${\dfrac{2}{9}}$ = ${\dfrac{2}{15}}$
  • (-20) ÷ 5 = (-4)

Commutative Property

Real numbers are commutative under the operations of addition and multiplication. If we consider any two real numbers, m, and n,

  • For addition: m + n = n + m, e.g., 2 + 4 = 6 = 4 + 2
  • For multiplication: m × n = n × m, e.g., ${\dfrac{1}{2}}$ × ${\dfrac{4}{5}}$ = ${\dfrac{2}{5}}$ = ${\dfrac{4}{5}}$ × ${\dfrac{1}{2}}$  

But this property does not hold for subtraction and division as

  • For subtraction: m – n ≠ n – m, e.g., 1.2 – 0.8 ≠ 0.8 – 1.2
  • For division: m ÷ n ≠ n ÷ m, e.g., (-16) ÷ 4 ≠ 4 ÷ (-16) 

Associative Property

Real numbers are associative only under the operations of addition and multiplication. If m, n, and p are any three real numbers,

  • For addition: m + (n + p) = (m + n) + p, e.g., (5 + 4) + 2 = 11 = 5 +( 4 + 2)
  • For multiplication: m × (n × p) = (m × n) × p, e.g., (${\dfrac{2}{3}}$ × ${\dfrac{1}{2}}$) × ${\dfrac{4}{7}}$ = ${\dfrac{4}{21}}$ = ${\dfrac{2}{3}}$ × (${\dfrac{1}{2}}$ × ${\dfrac{4}{7}}$)

Distributive Property 

Multiplication of real numbers satisfies the distributive property over addition and subtraction. For example, if m, n, and p are any three real numbers,

  • For multiplication over addition: m × (n + p) = (m × n) + (m × p), e.g., 2 × (5 + 1) = 12 = (2 × 5) + (2 × 1)
  • For multiplication over subtraction: : m × (n – p) = (m × n) – (m × p), e.g., 9 × (4 – 7) = -27 = (9 × 4) – (9 × 7)

Real Numbers vs. Rational Numbers

Real numbers are a set of all rational and irrational numbers. In contrast, rational numbers are those real numbers that is represented in the form of a fraction, the denominator being non-zero.

Thus, all rational numbers are real numbers. However, all real numbers are not rational numbers.

Real Numbers vs. Imaginary Numbers

Real numbers can be whole, natural, integers, rational, or irrational numbers. In contrast, an imaginary or complex number is the square root of a negative number and does not have a tangible value.

Solved Examples

Identify the real number(s) from the following list.
${\sqrt{-14}}$, ${-\dfrac{9}{10}}$, 0.0001, ${\sqrt{7}}$ and 11.

Solution:

${\sqrt{-14}}$ is a complex number. Thus it is not a real number.
${-\dfrac{9}{10}}$, 0.0001 are rational numbers, while 11 is an integer.
So the real numbers from the above list are ${-\dfrac{9}{10}}$, 0.0001, and 11.

Is ${-\dfrac{5}{3}}$ a real number ?

Solution:

${-\dfrac{5}{3}}$ can be written as non- terminating decimal -1.66666…… or ${-1\cdot \overline{6}}$ . Since the decimal is repeating, it is a rational number. Hence, ${-\dfrac{5}{3}}$ is a real number.

Which of the given choices are real numbers? Check all that apply.
(a) ${\left( -9\right) ^{\dfrac{1}{3}}}$
(b) ${\left( -25\right) ^{\dfrac{1}{2}}}$
(c) ${\left( -45\right) ^{\dfrac{1}{5}}}$
(d) ${\left( -39\right) ^{\dfrac{1}{6}}}$

Solution:

As we know, nth root of a negative number is not a real number when n is even (e.g., 2, 4, 6, 8), (b) ${\left( -25\right) ^{\dfrac{1}{2}}}$ and (d) ${\left( -39\right) ^{\dfrac{1}{6}}}$ are not real numbers.
Thus the real numbers are (a) ${\left( -9\right) ^{\dfrac{1}{3}}}$ and (c) ${\left( -45\right) ^{\dfrac{1}{5}}}$.

Last modified on August 3rd, 2023

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