Complex Numbers and Quadratic Equations

Operations on Complex Numbers

What happens when we add two Imaginary Numbers? Is there even a way to do it? The basic algebraic operations on imaginary numbers like addition, subtraction, multiplication and division open up various new possibilities. As a result of these properties, imaginary numbers are very important in many branches of  Physics and Chemistry.

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The Complex Algebra

In basic algebra of numbers, we have four operations namely – addition, subtraction, multiplication and division. As we will see in a bit, we can combine complex numbers with them. Let z1 and z2 be any two complex numbers and let, z1 = a+ib and z2 = c+id.

Imaginary Numbers

Example: Schrodinger Equation which governs atoms is written using complex numbers

Addition and Subtraction of Imaginary Numbers

The addition of two complex numbers is defined as:

z1 + z2 = (a + ib) ± (c + id) = (a + c) ± i(b + d)

Which gives another complex number whose real part is Re(z1) + Re(z2) = a + c and imaginary part of the new complex number = Im(z1) + Im(z2) = b + d. For example, on adding 2 and 3 + 4i, we can write 2 as 2 + 0*i and therefore, 2 + (3+4i) = (2+3) + i(0+4) = 5 + i.4

Browse more Topics under Complex Numbers And Quadratic Equations

Properties of Addition and Subtraction

The addition or subtraction of complex numbers always results in a complex number. This operation follows the following rules *:

  • The Closure law or the closure property: Addition or subtraction of complex numbers yields a complex number.
  • The Commutative Law: Any two complex numbers commute with respect to addition or subtraction i.e. z1 + z2 = z2 + z1
  • The Associative Law: If z1, z2, z3 are any three complex numbers then we have  (z1 + z2) + z3 = z1 + (z2+z3)
  • Additive Identity: Additive identity is such a complex number that will return the original number upon addition. For example for real numbers, 0 is the additive identity. Similarly, we see for complex numbers 0 + i.0 is the additive identity (We will just denote it by 0).
  • Additive Inverse: For each operation involving the combination of complex numbers through addition or subtraction, there exists an inverse such that the addition or subtraction of a complex number with it, yields the additive identity. For every complex number z = a + ib, there exists a complex number – z = -a + i(-b) such that z + (-z) = 0 or the additive identity.

*These properties are very important. Any collection of mathematical objects (set) that follows the above-mentioned properties under an operation (here ‘+’ and ‘-‘) is said to be a Group. You will learn about Groups and Fields in higher algebra. Groups are used in the development of almost all modern scientific theories.

Imaginary Numbers

Complex Numbers are used to study the shapes of atoms and molecules

Multiplication and Division of Imaginary Numbers

The multiplication of two complex numbers is defined as:

(z1 × z2) = (a + ib) × (c + id) = (ac – bd) + i(ad + bc)

Similar \( \frac{z_1}{z_2} \) = z1 × \( \frac{1}{z_2} \) ; we can use cross multiplication and the multiplication of complex numbers for division. The multiplication and division also form a group i.e. they have similar properties as addition and subtraction.

You can download Complex Numbers Cheat Sheet by clicking on the download button below
Operations on Complex Numbers

Properties of Multiplication And Addition

  • The Closure law or the closure property: (z1 × z2) is always a complex number.
  • The Commutative Law: If z1 and z2 are two  complex numbers the, (z1 × z2) = (z2 × z1)
  • The Associative Law: If z1, z2, z3 are any three complex numbers then we have  (z1 × z2) × z3 = z1 × (zz3).
  • Multiplicative Identity: For each complex number there exists a number 1 + i.0 such that z × ( 1 + i.0) = z ; where z is a complex number.
  • Multiplicative Inverse: For each complex number z there exists a number 1/z such that z×(1/z) = 1 + i.0. This is known as the Multiplicative Inverse.
  • The Distributive Law: If z1, z2, z3 are any three complex numbers then we have z1 × (z2 + z3) = z1 z2 + z1 z3 (Left distributive law) and ( z1 + z) × z3 = z1 z3 + z1 z2

The division also follows the same properties.

Solved Examples For You

Question 1: Find the smallest integer n such that \( \frac{(1+i)^n}{(1-i)^n} \) is

A) 16                           B) 12

C) 8                             D) 4

Answer :D). Here we use a little trick. We have \( \frac{1}{i} \) = \( \frac{1}{i} \) × \( \frac{i}{i} \)

Remember that i×i = -1 and therefore we have, \( \frac{1}{i} \) × \( \frac{i}{i} \) = -i. Hence using it in the denominator of the given equation, we have: \( \frac{(1+i)^n}{((1+i)/i)^n} \) = (i)n. Hence, for (i)n = 1, n should be atleast 4.

Question 2: Why do we use imaginary numbers?

Answer: Also known as complex numbers they are used in real-life applications, such as electricity, as well as quadratic equations. Moreover, AC electricity changes between positive and negative in a sine wave. However, combining AC currents can be very difficult because they may not match properly on the waves.

Question 3: What is an imaginary number?

Answer: The numbers, which by their nature are impossible and ordinary, are known as imaginary or fancied numbers. The main reason for them being imaginary is that they exist only in our imagination.  However, it is a matter of acceptance of or the lack of it, which gave them the name.

Question 4: What do you mean by real and imaginary numbers?

Answer: Complex or imaginary numbers refer to the quantity of form ix, where x is the real number and ‘i’ is the positive square root of -1. Besides, imaginary numbers are no less ‘real’ than the real numbers. Furthermore, the quantity ‘i’ is called the unit imaginary number.

Question 5: Are imaginary numbers positive or negative?

Answer: They refer to that squared number that gives a negative result. Generally, when you square a real number you always get a positive number. But when you square imaginary number their answer is negative.

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