Key takeaway: A complex number pairs an ordinary number with a multiple of i, written p + qi, so it has a horizontal real part p and a vertical imaginary part q on the complex plane.
Grade 11 – Algebra 2N-CN.A.1Numbers and Operations
Definition of Complex Number
Each complex number carries two pieces of information. In \(p + qi\), the piece \(p\) is the real part, and \(q\), the multiplier of \(i\), is the imaginary part. A plain 8 is complex with \(q = 0\), and a number like \(5i\) is complex with \(p = 0\).
Addition and subtraction combine like pieces. Multiplication works like expanding two binomials, plus the extra rule \(i^2 = -1\). Each number can be drawn as a point: real part along the horizontal axis, imaginary part up the vertical axis.
Formula and notation
A complex number in standard form, and a sum of two of them (real parts \(p, r\), imaginary parts \(q, s\)):
\[ z = p + qi, \qquad (p + qi) + (r + si) = (p + r) + (q + s)i \]
The product uses distribution and \(i^2 = -1\): \((p+qi)(r+si) = (pr – qs) + (ps + qr)i\).
Key facts
- The set of complex numbers contains all real numbers and all imaginary numbers.
- Two complex numbers are equal only when their real parts and imaginary parts both match.
- Addition works like vector addition on the complex plane.
- Quadratics with a negative discriminant have complex solutions.
Complex Number example
Let \(z = 3 + 5i\) and \(w = -7 + 2i\). Find \(z + w\), then find the product \((2 + 3i)(4 – 5i)\).
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Add the real parts and the imaginary parts: \((3 – 7) + (5 + 2)i = -4 + 7i\).
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Multiply: \((2 + 3i)(4 – 5i) = 8 – 10i + 12i – 15i^2\).
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Replace \(i^2\) with \(-1\): \(8 + 2i + 15 = 23 + 2i\).
The sum is \(-4 + 7i\) and the product is \(23 + 2i\).
Common mistake
Watch out
Zoe leaves the product as \(8 – 10i + 12i – 15i^2 = 8 + 2i – 15i^2\) and writes the answer as \(8 + 2i – 15i^2\).
Why it is wrong
The answer is not in standard form until \(i^2\) is replaced by \(-1\). Then \(-15i^2 = +15\), and the real parts combine: \(8 + 15 = 23\). The final form is \(23 + 2i\).
Related terms and lessons
- Imaginary Unit (i)
- Complex Conjugate
- Real Numbers
- Quadratic Formula
- Discriminant
- Polar Form of a Complex Number
- Algebra 2 · Real and Complex NumbersLessonWorksheetQuiz
- Algebra 2 · Quadratic Equations with Complex SolutionsLessonWorksheetQuiz
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