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What Are Imaginary Numbers?(Definition & Properties)

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In mathematics, you have solved different types of equations. The solutions of these equations can be natural numbers or whole numbers, integers, fractional or decimal numbers, rational numbers, or irrational numbers.  All these sets together form a set of real numbers. But there are certain equations whose solutions do not lie in the set of real numbers. These numbers are called imaginary numbers.

What are Imaginary Numbers?

An imaginary number is a number that is a square root of a negative number. When an imaginary number is squared, it results in a negative number.

An imaginary number consists of two parts – a real part and an imaginary unit denoted by the symbol $i$.

For example, $5i$ is an imaginary number and square of $5i$, i.e., $\left(5i \right)^{2} = -25$.

Similarly, $\sqrt3i$ is an imaginary number and $\left(\sqrt{3}i \right)^{2} = -3$.

What is an Imaginary Unit?

The imaginary number $i = \sqrt{-1}$, i.e., the square root of $-1$. The imaginary unit is denoted and commonly referred to as “$i$”, (which is also known as “iota“).

The imaginary unit $i$ is a solution of the equation $x^{2} + 1 = 0$.

Solving $x^{2} + 1 = 0$, we get $x^{2} = -1 => x = \pm \sqrt{-1} => x = \pm i$.

Although there are two possible square roots of any number, the square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point $+i$ and $-i$ can then be distinguished. Since either choice is possible, there is no ambiguity in defining $i$ as the square root of $-1$.

Imaginary Number Rules

The imaginary number rules are associated with the powers of $i$, which are as follows

$\left(i \right)^0 = 1$

$\left(i \right)^1 = i$

$\left(i \right)^2 = -1$

$\left(i \right)^3 = \left(i \right)^2 \times i = -i$

$\left(i \right)^4 =\left (i^{2} \right)^{2} = \left(-1 \right)^{2} = 1$

$\left(i \right)^5 = \left(i \right)^4 \times i = 1 \times i = i$

$\left(i \right)^6 = \left (i^{2} \right)^{3} = \left(-1 \right)^{3} = -1$

$\left(i \right)^7 = \left(i \right)^6 \times i = -1 \times i = -i$

If you observe that there is a cycle of numbers $1$, $i$, $-1$, and $-i$ that repeats after every $4$ step. For any value of $k$ this cycle can be expressed as

$i^{4k} = 1$

$i^{4k + 1} = i$

$i^{4k + 2} = -1$

$i^{4k + 3} = -i$

(where $k$ is any integer)

Using these rules, you can calculate the powers of $i$.

Examples

Let’s consider some examples to understand the cycle of powers of $i$.

Ex 1: Find the value of $i^{19}$

$i^{19} = i^{4 \times 4 + 3} = -i$

Alternatively, $i^{19} = i^{18} \times i = \left(i^{2}\right)^9 \times i = \left(-1\right)^9 \times i = -1 \times i = -i$.

Ex 2: Find the value of $i^{60}$

$i^{60} = i^{4 \times 15} = 1$

Alternatively, $i^{60} = \left(i^{2} \right)^{30} = \left(-1 \right)^{30} = 1$

Ex 3: Find the value of $i^{53}$

$i^{53} = i^{4 \times 13 + 1} = i$

Alternatively, $i^{53} = i^{52} \times i = \left(i^{2} \right)^{26} \times i = \left(-1 \right)^{26} \times i = 1 \times i = i$

Ex 4: Find the value of $i^{102}$

$i^{102} = i^{4 \times 25 + 2} = -1$

Alternatively, $i^{102} = \left(i^{2} \right)^{51} = \left(-1 \right)^{51} = -1$

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Arithmetic Operations With Imaginary Numbers

As you can perform the four arithmetic operations – addition, subtraction, multiplication, and division with any real numbers, you can also perform these operations with imaginary numbers.

Adding and Subtracting Imaginary Numbers

Adding and subtracting imaginary numbers is just like how you combine the like terms in algebra. If $ai$ and $bi$ are any two imaginary numbers then 

  • $ai + bi = \left(a + b \right)i$
  • $ai – bi = \left(a – b \right)i$

Examples

Let’s consider some examples to understand the addition and subtraction of imaginary numbers.

Ex 1: Add $2i$ and $7i$

$2i + 7i = \left(2 + 7 \right)i = 9i$

Ex 2: Add $6i$ and $-4i$

$6i + \left(-4i \right) = \left(6 + \left(-4 \right) \right)i = \left(6 – 4 \right)i = 2i$

Ex 3: Add $-9i$ and $-13i$

$-9i + \left(-13i \right) = -9i – 13i = \left(-9 – 13 \right)i = -\left(9 + 13 \right)i = -22i$

Ex 4: Subtract $5i$ from $17i$

$17i – 5 i = \left(17 – 5 \right)i = 12i$

Ex 5: Subtract $21 i$ from $11i$

$11 i – 21i = \left(11 – 21 \right)i = -10i$

Ex 6: Subtract $-2i$ from $7i$

 $7i – \left(-2i \right)  = 7i + 2i = \left(7 + 2 \right)i = 9i$

Ex 7: Subtract $8i$ from $-15i$

$-15i – 8i = \left(-15 – 8 \right)i = -\left(15 + 8 \right)i = -23i$

Multiplying Imaginary Numbers

You can multiply the imaginary numbers just like how you multiply the terms in algebra. Here, you may have to use the rule of exponents $a^{m} \times a^{n} = a^{m + n}$. Along with that, you have to take care of the fact that $i^{2} = -1$.

Examples

Let’s consider some examples to understand the multiplication of imaginary numbers.

Ex 1: Find the product of $5i$ and $8i$

$5i \times 8i = \left(5 \times 8 \right) \times \left(i \times i \right) = 40 \times i^{2} = 40 \times \left(-1 \right) = -40$.

Ex 2: Find the product of $2i$ and $-7i$

$2i \times \left(-7i \right) = \left(2 \times \left(-7 \right) \right) \times \left(i \times i \right) = -14 \times i^{2} = -14 \times \left(-1 \right) = 14$.

Ex 3: Find the product of $-6i$ and $-11i$

$-6i \times \left(-11i \right) = \left(-6 \times \left(-11 \right) \right) \times \left(i \times i \right) = 66 \times i^{2} = 66 \times \left(-1 \right) = -66$

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Dividing Imaginary Numbers

Before moving on to the division of imaginary numbers, let’s first understand what is reciprocal of $i$, i.e., $\frac {1}{i}$.

Multiplying the numerator and the denominator with $i$.

 $\frac {1}{i} = \frac {1}{i} \times \frac {i}{i} = \frac {1 \times i}{i \times i} = \frac {i}{i^{2}} = \frac {i}{-1} = -i$.

Note: Reciprocal of $i$, i.e., $\frac {1}{i} = -i$.

While dividing imaginary rules, you use the rule of exponents $\frac {a^{m}}{a^{n}} = a^{m – n}$. In the result after division, $i$ is not kept in the denominator. If you get so, then use the rule $\frac {1}{i} -i$ discussed above.

Examples

Let’s consider some examples to understand the division of imaginary numbers.

Ex 1: Divide $6i$ by $3i$

$6i \div 3i = \frac {6}{3} \times \frac {i}{i} = 2 \times 1 = 2$

Ex 2: Divide $12i$ by $4$

$12i \div 4 = \frac {12i}{4} = \frac {12}{4} \times \frac {i}{i} = 3 \times i = 3i$

Ex 3: Divide $15$ by $5i$

$15 \div 5i = \frac {15}{5i} = \frac {15}{5} \times \frac {1}{i} = 3 \times \left(-i \right) = -3i$ 

Conclusion

An imaginary number is a square root of a negative number and is denoted by $ai$, where $a$ is a real number and $i$ is the imaginary unit whose value is $\sqrt{-1}$. The imaginary numbers are used in complex numbers and you can perform any of the four basic operations with these numbers.

Practice Problems

  1. Write True or False
    • The value of $i$ is $0$
    • The value of $i$ is $1$
    • The value of $i$ is $-1$
  2. Evaluate the following
    • $i^{15}$
    • $i^{27}$
    • $i^{30}$
    • $i^{44}$
    • $i^{19} + i^{17}$
    • $i^{28} + i^{53}$
    • $i^{42} – i^{38}$
    • $i^{57} – i^{98}$
    • $i^{13} \times i^{18}$
    • $i^{59} \times i^{43}$
    • $i^{123} \div i^{87}$
    • $i^{0} \div i^{23}$
  3. Perform the following operations
    • $7i + 9i$
    • $28i + \left(-13i \right)$
    • $12i \times 6i$
    • $15i \times 7$
    • $28i \div 4i$
    • $52 \div 13i$

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FAQs

What is an imaginary number?

An imaginary number is a number whose square results in a negative number. An imaginary number is written in the form $ai$, where $a$ is a real number and $i$ is an imaginary unit whose value is $\sqrt{-1}$.

For example, $8i$ is an imaginary number and $\left(8i \right)^{2} = -64$.

What are the rules for imaginary numbers?

The rules for imaginary numbers are
$i = \sqrt{-1}$
$i^{2} = -1$
$i^{3} = -i$
$i^{4} = 1$
In general, 
$i^{4k} = 1$
$i^{4k + 1} = i$
$i^{4k + 2} = -1$
$i^{4k + 3} = -i$

Why is there an imaginary number?

The imaginary unit $i$ allows us to find solutions to many equations that do not have real number solutions. For example, imaginary numbers are used in finding the solutions of quadratic equations where the discriminant $b^{2} – 4ac \lt 0$.

Is $0$ an imaginary number?

Though $0$ can be written as $0i$, it is not an imaginary number as it is not associated with the square root of any negative number.

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