# Addition of Complex Numbers

**Addition of complex numbers :**The sum z

_{1}+ z

_{2}of two complex numbers z

_{1}= a

_{1}+ ib

_{1}and z

_{2}= a

_{2}+ ib

_{2}is defined as the complex number (a

_{1}+ a

_{2}) + i(b

_{1}+b

_{2}), i.e

z

_{1}+ z

_{2}= (a

_{1}+ ib

_{1}) + (a

_{2}+ ib

_{2})

It is therefore, observed that while adding two complex numbers the real and imaginary parts of the system is obtained by adding the real and imaginary parts of the summands.

**Examples :**

1) (3 + i7) + (4 + i8) = (3 + 4) + i(7 + 8) = 7 + i15

2) (12- i7) + i4 = 12 + i ( -7 + 4) = 12 - i3

**Properties of addition of complex numbers**

**Closure :**The sum of two complex numbers is , by definition , a complex number. Hence, the set of complex numbers is closed under addition.

**Example :**(5+ i2) + 3i = 5 + i(2 + 3) = 5 + i5 <

From the above we can see that 5 + i2 is a complex number, i3 is a complex number and the addition of these two numbers is 5 + i5 is again a complex number.

**Commutative property:**For two complex numbers z

_{1}= a + ib and z

_{2}= c + id

z

_{1}+ z

_{2}= (a + ib) +(c +id) = (a + c) + i ( b + d)

z

_{2}+ z

_{1}= (c + id) +(a + ib)= (c + a) + i(d + b)

But we know that, a + c = c + a and b + d = d + b

∴ z

_{1}+ z

_{2}= z

_{2}+ z

_{1}

**Associative Property :**Consider three complex numbers,

z

_{1}= a + ib , z

_{2}= c + id and z

_{3}= e + if

(z

_{1}+ z

_{2})+ z

_{3}= z

_{1}+(z

_{2}+ z

_{3})

(a + ib + c + id ) + (e + if) = (a + ib) + ( c +id + e + if)

[(a + c) + i( b +d)] + (e + if) = (a + ib) +[(c + e) + i( d +f)]

(a + c + e ) + i(b + d + f ) = ( a + c + e) + i(b + d + f)

**Additive Identity:**Let a + ib be the identity for addition. Then

(x + iy) + (a + ib) = x + iy

⇒ (x + a) + i( y + b) = x + iy

⇒ x + a = x and y + b = y

⇒ a = 0 and b = 0

Hence, the additive identity is the complex number

**0 + i0**, written simply as 0.

**Additive Inverse:**

z = a + ib so its additive inverse will be -z which -(a + ib) = - a - ib

**Examples on Addition of Complex Numbers**

1) Add : 5 + 3i and -8 + 2i

**Solution :**5 + 3i + (-8 + 2i)

= ( 5 + (-8) + i(3 + 2)

= - 3 + i5

2) Find the additive inverse of - 5 + i7 .

**Solution :**z = -5 + i7 so additive inverse will be

**-z**

so -z = - (z)

= - ( -5 + i7)

= 5 - i7

3) Find the sum of 2/3 + i5/3 ; -2/3i and -5/4 - i

**Solution :**Using associative property, we have

[( 2/3 + i5/3)+(0-2/3i)] + (-5/4 - i)

= (2/3 + i) +(-5/4 - i)

= -7/12

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