Advanced algebra
Complex-number arithmetic
ALEKS placement
Treat i like a variable, then replace every i² with -1.
What this covers
i2=−1,(a+bi)(a−bi)=a2+b2- Add and subtract by combining real parts and imaginary parts separately.
- Multiply by expanding as binomials.
- Replace each i² with -1 and recombine.
- Divide by multiplying top and bottom by the conjugate of the denominator.
Worked example
Worked example
Multiply (3 + 2i)(1 - 4i).
- 3−12i+2i−8i2Expand all four products.
- 3−10i+8-8i² = -8(-1) = +8.
- 11−10iCombine real parts.
Another worked example
Find (4 + 7i) + (1 − 4i), then (4 + 7i) − (1 − 4i).
- (4+7i)+(1−4i)=5+3iReal parts: 4 + 1 = 5. Imaginary parts: 7i − 4i = 3i. They stay separate, like x and y.
- (4+7i)−(1−4i)=3+11iSubtract every part of the second number: 4 − 1 = 3 and 7i − (−4i) = 11i.
A common mistake
A mistake Lemma catches
From 3−12i+2i−8i2
3−10i−83−10i+8
“i² = −1, so −8i² = −8 · (−1) = +8. The sign flips the moment i² becomes −1; writing −8 treats i² as if it were +1.”
Try one
Sample problem
Add and write the result in a + bi form.
(4+7i)+(1+4i)Practice complex numbers free
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Where this fits
Before this