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### High School Mathematics - 22.3 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?-2/3i2/3i0 Q 2: Find the additive inverse of -5+i75+i75-i70 Q 3: Find the additive inverse of 3 - (-6i)03-6i-3-i6 Q 4: (5+i4) + (5-i4)0102i Q 5: (1/3+i7/3) + (4+i/3) - (-4/3+i)17/3-5/3i17/3+5/3i0 Q 6: What should be added to -3-85i to make it 12-45i?15-40i15+40i0 Q 7: Find the sum of 2/3+i5/3, -2/3 i and -5/4-i07i/12-7/12 Q 8: (7-i2) - (4+i) + (-3+i5)0+2i0-2i0 Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!