--- title: "The Imaginary Unit and Complex Arithmetic" description: "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." canonical: https://lightmysky.com/learn/mathematics/the-imaginary-unit-and-complex-arithmetic-mt_Faemafn550 source: https://lightmysky.com/learn/mathematics/the-imaginary-unit-and-complex-arithmetic-mt_Faemafn550.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Algebra · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/the-imaginary-unit-and-complex-arithmetic-mt_Faemafn550 ## Ready when they can - 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 ## Lesson: Numbers beyond the number line 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. **Example.** Add parts with parts: (3 + 2i) plus (1 + 4i) joins 3 with 1 and 2i with 4i, giving 4 + 6i. Subtraction matches: (2 + 3i) minus (5 + i) is minus 3 + 2i. For products, use a conjugate (flip the i sign). So (3 + i) times (3 - i) expands to 9 + 1, which is 10, since i squared is minus one. 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. **Example.** 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. **Recap.** With i squared as minus one, complex numbers add, multiply, and divide by the ordinary rules of algebra. ## Practice 16 questions on this page, each with its working shown. ## Needs first - [The Quadratic Formula](https://lightmysky.com/learn/mathematics/the-quadratic-formula-mt_C-4brnRYHY) ## Opens up - [The Wavefunction and the Born Probability Rule](https://lightmysky.com/learn/science/the-wavefunction-and-the-born-probability-rule-mt_elzmbBzmKE) - [Complex Functions and the Complex Plane as a Domain](https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs) - [Eigenvalues and Eigenvectors](https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa) - [The Argand Plane, Modulus and Argument](https://lightmysky.com/learn/mathematics/the-argand-plane-modulus-and-argument-mt_Wq01c5UN9e)