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Problem type

Add, subtract, multiply and divide complex numbers

What it solves
All four operations on two complex numbers, each worked from the single fact that i squared is minus one: parts added side by side, brackets multiplied out term by term, and division done by the conjugate and then checked by multiplying back.
Subject
Complex Numbers
Topic
Arithmetic with complex numbers
Also called
add complex numbers, multiply complex numbers, divide complex numbers, conjugate, rationalise the denominator
Your problem

Work out z+wz + w, z−wz - w, zwzw and zw\dfrac{z}{w}.

z=4+7i,w=−5+8iz = 4 + 7 i, \qquad w = -5 + 8 i
Invent a new one:
Femtobytes
(4 + 7i) and (-5 + 8i): sum, difference, product and quotient, step by step
femtobytes.com/solve/complex-arithmetic/4_7_-5_8
The problem

Work out z+wz + w, z−wz - w, zwzw and zw\dfrac{z}{w}.

z=4+7i,w=−5+8iz = 4 + 7 i, \qquad w = -5 + 8 i
The result
z+w=−1+15i,z−w=9−i,zw=−76−3i,zw=3689−6789iz + w = -1 + 15 i, \quad z - w = 9 - i, \quad zw = -76 - 3 i, \quad \dfrac{z}{w} = \frac{36}{89} - \frac{67}{89} i

The quotient was checked by multiplying it by ww, which gives back zz exactly.

checked 27 lines of this solution were evaluated with exact arithmetic before it was shown to you. A solution with a false line is never served.
Picture 1 of 2Step 1: Add: real part with real part, imaginary part with imaginary part
55i-5-5i1010i-10-10i1515i-15-15ireal axisimaginary axiszwz + w
z, w and their sum on the plane. Adding w carries z along the arrow of w: across by the real part of w, then up by its imaginary part.
1

Add: real part with real part, imaginary part with imaginary part

Why this step

A complex number is two ordinary numbers travelling together: how far across, and how far up. No amount of adding ordinary numbers ever produces a multiple of ii, so the two parts cannot leak into each other. That is why addition needs no new rule at all: it is two ordinary additions done side by side.

(4+7i)+(−5+8i)\displaystyle \left( 4 + 7 i \right) + \left( -5 + 8 i \right)==(4−5)+(7+8)i\displaystyle \left( 4 - 5 \right) + \left( 7 + 8 \right) i
Drop the brackets and put like with like: the two real parts together, the two numbers in front of ii together. checked
==−1+15i\displaystyle -1 + 15 i
Two ordinary additions. checked
55i-5-5i1010i-10-10i1515i-15-15ireal axisimaginary axiszwz + w
z, w and their sum on the plane. Adding w carries z along the arrow of w: across by the real part of w, then up by its imaginary part.
2

Subtract: the minus sign reaches both parts of w

Why this step

Subtracting ww means adding its opposite, and the opposite of c+dic + di is −c−di-c - di: BOTH parts change sign. The slip to guard against is changing the sign of the real part only and leaving the ii part as it was.

(4+7i)−(−5+8i)\displaystyle \left( 4 + 7 i \right) - \left( -5 + 8 i \right)==(4−(−5))+(7−8)i\displaystyle \left( 4 - \left( -5 \right) \right) + \left( 7 - 8 \right) i
Real part minus real part, and the number in front of ii minus the number in front of ii. checked
==9−i\displaystyle 9 - i
Two ordinary subtractions. checked
55i-5-5i1010i-10-10ireal axisimaginary axiszw
Subtracting reverses the arrow of w before carrying z along it, which is why the minus sign has to reach both parts.
3

Multiply: every part of z meets every part of w

4

Divide: clear i from the bottom with the conjugate

5

Check the division by multiplying back

6

Two facts worth seeing on these numbers: the mirror and the ruler