The Imaginary Unit and Complex Arithmetic
Quadratics with no real roots stop being dead ends: i is defined so that i squared is -1, and numbers of the form a + bi add, multiply, conjugate and divide by the ordinary rules of algebra.
What a learner can do afterwards
- Solve a quadratic with negative discriminant and write both roots in a + bi form
- Add, multiply and divide two complex numbers, using the conjugate for division
- Explain why the rules of real algebra carry over once i squared is replaced by -1
- Plot a + bi as the point (a, b) and find its distance from the origin with Pythagoras
1 · Read
Some quadratics have no real roots, and i rescues them so every quadratic has an answer. Define i by i squared equals minus one. Then x squared plus 4 equals zero gives x squared equals minus 4, so x is 2i or minus 2i. Each answer fits a plus bi form, with real and imaginary parts.
Combine real and imaginary parts separately: (3 + 2i) plus (1 + 4i) is 4 + 6i, and (2 + 3i) minus (5 + i) is minus 3 + 2i. To multiply, expand term by term and swap i squared for minus one. A conjugate pair (flip the i sign) gives a real answer: (3 + i) times (3 - i) is 9 + 1, which is 10.
Divide by multiplying top and bottom with the bottom's conjugate. Then (1 + i) over (1 - i) becomes 2i over 2, which is i. Powers cycle too: i to the 4th is i squared times i squared, or minus one times minus one, which is 1. The roots 2i and minus 2i multiply to 4.
Treat complex numbers with ordinary algebra, then swap every i squared for minus one at the end. Expand brackets fully: (1 + i) squared gives 1 + 2i + i squared, which is 2i. Dropping the middle term or leaving i squared in the answer are the two classic slips.
Plot a plus bi as the point (a, b): across for the real part, up for the imaginary part. Its distance from the origin uses Pythagoras, since the gaps a and b are legs of a right triangle. For 6 + 8i the point is (6, 8). Square each gap: 36 and 64. Add: 36 + 64 = 100. The root of 100 is 10, so 6 + 8i sits 10 from the origin.
With i squared as minus one, complex numbers add, multiply, and divide by the ordinary rules of algebra.
Readings 'Why i exists', 'Multiplying and the conjugate', 'Dividing with the conjugate' and 'Powers of i and the complex plane' are original to this dataset. All 8 extra questions are original to this dataset.
2 · Watch
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Where it sits
Where this leads
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16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.