Complex numbers
Numbers with an imaginary part: arithmetic, size, and powers.
Pen and paper is fine · no calculator needed why?
Opens at level 53. You're level 1. You can read and practice here now.
the idea
A complex number a + bi has a real part a and an imaginary part b, where i is a number whose square is −1. All the usual rules of algebra still work.
Picture a + bi as the point (a, b). Its modulus, written |a + bi|, is its distance from 0. Its angle is measured from the positive real axis. Multiplying multiplies sizes and adds angles, which makes big powers quick.
techniques
Count around the cycle of 4
- The powers of i repeat every 4: i, −1, −i, 1.
- Divide the power by 4 and keep the remainder.
- Remainder 1 gives i, 2 gives −1, 3 gives −i, and 0 gives 1.
worked example
What is i⁴²?
- 42 = 4 × 10 + 2, so the remainder is 2.
- i⁴² = i² = −1.
Answer: −1
Multiply out, then use i² = −1
- Multiply every part by every part, as with (a + b)(c + d): four products.
- Replace i² with −1, so a term like −3i² becomes +3.
- Collect the plain numbers into the real part. The numbers in front of i add up to the imaginary part.
worked example
(2 + 3i)(4 − i) = a + bi, where a and b are real. What is a?
- Four products: 8 − 2i + 12i − 3i².
- Since i² = −1, the term −3i² becomes +3.
- Real part: 8 + 3 = 11. Imaginary part: −2 + 12 = 10.
Answer: 11
Size and angle
- Size: |a + bi| is √(a² + b²), the distance from 0 by Pythagoras.
- Angle: read it off the point (a, b). 1 + i sits at 45°, and √3 + i at 30°.
- For the nth power, raise the size to the nth power and multiply the angle by n.
- An angle of 180° points along the negative real axis, and 360° along the positive one.
worked example
(1 + i)⁴ is a real number. What is it?
- 1 + i has size √(1 + 1), which is √2, and angle 45°.
- Size: (√2)⁴ is 4. Angle: 4 × 45° is 180°.
- 180° points along the negative real axis, so the answer is −4.
Answer: −4
watch out for
- Treating i² as +1. It is −1, so a term like −8i² adds 8 to the real part.
- Adding the parts for the modulus. |3 + 4i| is 5, the square root of 9 + 16, not 7.
- Raising the wrong size to the power. 1 + i has size √2, not 2, so (1 + i)⁸ has size 16, not 256.
practice
Powers of i
worked example
What is i⁷¹?
- +i
- −i
- −1
- 1
Answer: −i
- Powers of i repeat every 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1.
- 71 = 4 × 17 + 3, so i⁷¹ = i³ = −i.
Multiply
worked example
(3 − 5i)(4 − 3i) = a + bi, where a and b are real. Enter a, b.
Answer: (-3, -29)
- Multiply out: 12 − 9i − 20i + 15i².
- Since i² = −1, 15i² = −15.
- Real part: 12 − 15 = −3. Imaginary part: −9 − 20 = −29.
- So the product is −3 − 29i: enter −3, −29.
Modulus
worked example
What is the modulus |−8 + 6i|?
Answer: 10
- |a + bi| = √(a² + b²), the distance from 0 to the point (a, b).
- √((−8)² + 6²) = √(64 + 36) = √100 = 10.
Powers by angle
worked example
(1 − i)¹² is a real number. What is it?
Answer: -64
- 1 − i has length √2 and angle −45°.
- The 12th power has length (√2)¹² = 64 and angle 12 × (−45°) = −540°.
- −540° = −3 × 180° points along the negative real axis, so (1 − i)¹² = −64.