Operations on Complex Numbers Mathematics Notes | EduRev

Algebra for IIT JAM Mathematics

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Mathematics : Operations on Complex Numbers Mathematics Notes | EduRev

The document Operations on Complex Numbers Mathematics Notes | EduRev is a part of the Mathematics Course Algebra for IIT JAM Mathematics.
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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.
Operations on Complex Numbers Mathematics Notes | EduRev
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

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.
Operations on Complex Numbers Mathematics Notes | EduRev
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 z1/z2 = z1 × 1/z2; 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.

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, (z,×z2) = (z2×z1)
• The Associative Law: If z1, z2, z3 are any three complex numbers then we have  (z1×z2) × z3 = z1 × (z2×z3).
• 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 + z2 ) × z3 = z1 z3 + z1 z2
The division also follows the same properties.

Solved Examples For You
Question: Find the smallest integer n such that Operations on Complex Numbers Mathematics Notes | EduRev is
A) 16
B) 12
C) 8
D) 4
Solution:(D). Here we use a little trick. We have Operations on Complex Numbers Mathematics Notes | EduRev
Remember that i×i = -1 and therefore we have, Operations on Complex Numbers Mathematics Notes | EduRev. Hence using it in the denominator of the given equation, we have:Operations on Complex Numbers Mathematics Notes | EduRevHence, for (i)n = 1, n should be atleast 4.

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