Complex numbers extend the real number line into a plane, giving mathematics a way to describe quantities with both real and imaginary parts. They are written in the form a + bi, where a and b are real numbers and i is the imaginary unit. This makes it possible to solve equations such as x^2 + 1 = 0, which have no real-number solutions.
Complex numbers are essential in algebra, geometry, physics, engineering, and signal processing.
Key Facts
- Standard form: z = a + bi, where a is the real part and b is the imaginary part.
- Imaginary unit: i^2 = -1, so sqrt(-1) = i.
- Complex plane: a + bi is plotted as the point (a, b).
- Addition: (a + bi) + (c + di) = (a + c) + (b + d)i.
- Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
- Conjugate and modulus: if z = a + bi, then z* = a - bi and |z| = sqrt(a^2 + b^2).
Vocabulary
- Complex number
- A number of the form a + bi, where a and b are real numbers and i^2 = -1.
- Imaginary unit
- The number i, defined by the property i^2 = -1.
- Real part
- The real number a in a complex number a + bi.
- Imaginary part
- The real number b in a complex number a + bi, not including the symbol i.
- Complex conjugate
- The number formed by changing the sign of the imaginary part, so the conjugate of a + bi is a - bi.
Common Mistakes to Avoid
- Treating i as a variable is wrong because i is a specific number with the fixed property i^2 = -1.
- Writing the imaginary part of 3 + 2i as 2i is wrong in standard vocabulary because the imaginary part is the real coefficient 2.
- Adding complex numbers by combining all numbers together is wrong because real parts must be added to real parts and imaginary parts to imaginary parts.
- Forgetting that i^2 = -1 during multiplication is wrong because it changes terms like 6i^2 into -6, which affects the real part of the answer.
Practice Questions
- 1 Plot z = 3 + 2i on the complex plane and find its modulus |z|.
- 2 Compute (4 - 3i) + (-2 + 5i), then compute (4 - 3i)(-2 + 5i).
- 3 Explain geometrically why the conjugate of z = a + bi is a reflection of z across the real axis.