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High School Mathematics - 2
2.3 Addition of Complex Numbers

Addition of Complex Numbers
The sum z1 + z2 of two complex nujmbers z1 = a1 + ib1 and z2 = a2 + ib2 is defined as the complex number (a1 + a2) + i(b1 + b2). i.e
Closure Property: The sum of two complex numbers is a complex number. Hence, the set of complex numbers is closed under addition.

Commutative Property: For two complex numbers z1 = a+ib and z2 = c+id, we have
z1 + z2 = (a+ib) + (c+id) = (a+c) + i(b+d)
z2 + z1 = (c+id) + (a+ib) = (c+a) + i(d+b)
Thus addition of two complex numbers is commutative.

Associative Property: Consider three complex numbers z1 = a+ib , z2 = c+id, z1 = e+if
We have
z1 + z2 = (a+c) + i(b+d)
z2 + z3 = (c+e) + i(d+f)
(z1 + z2) + z3 = [(a+c)+e] + i[(b+d)

The Additive Identity: Let a+ib be the identity for addition. Then
(x+iy) + (a+ib) = x + iy
This gives
(x+a) + i(y+b) = x+iy
x+a = x, y+b = y
a = 0, b = 0

Additive Inverse: Let z = a+ib be a complex number and let w = c+id be its additive inverse then
z+w = 0 i.e (a+ib) + (c+id) = 0
(a+c) + i (b+d) = 0 + i0
a + c = 0 and b+d = 0
c = -a and d = -b
Hence w = c+id = -a+i(-b) = -a-ib = -z
Thus z + (-z) = -z+z = 0


Directions: Answer the following questions. Also write at least 5 examples of each of the property above.
Q 1: What is the additive inverse of 2/3i?
2/3i
-2/3i
0

Q 2: (1/3+i7/3) + (4+i/3) - (-4/3+i)
0
17/3-5/3i
17/3+5/3i

Q 3: Find the additive inverse of -5+i7
5-i7
0
5+i7

Q 4: Find the sum of 2/3+i5/3, -2/3 i and -5/4-i
-7/12
0
7i/12

Q 5: What should be added to -3-85i to make it 12-45i?
0
15-40i
15+40i

Q 6: (5+i4) + (5-i4)
10
0
2i

Q 7: (1/5+i2/5) - (4+i5/2)
0
19/5+21/10i
-19/5-21/10i

Q 8: (-3-6i) + (7-8i)
4-14i
0
8-4i

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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