Indices, Radicals and Rational Expressions
Complex Numbers: Arithmetic and the Conjugate
The imaginary unit, why it was needed, arithmetic with a + bi, the powers of i, and dividing by a complex number using its conjugate.
What this page covers
- State the definition of i and write square roots of negatives in terms of it
- Add, subtract and multiply complex numbers
- Reduce any power of i using its four-step cycle
- Divide by a complex number by multiplying by the conjugate
The equation that forced the extension
The source introduces complex numbers by putting two nearly identical equations side by side. The comparison is the clearest possible motivation.
| Equation | Rearranged | Over the reals |
|---|---|---|
| x2 - 9 = 0 | x2 = 9 | x = ±3 |
| x2 + 9 = 0 | x2 = -9 | No solution: a real square is never negative |
With i available the second equation is solved exactly as the first, by splitting the radical:
Any number of the form a + bi, with a and b real. a is the real part and b the imaginary part. Taking b = 0 recovers the real numbers, which is why every real number is also complex.
The extension turns out to be far more powerful than the motivating example suggests. Admitting i does not merely solve x2 = -9; it makes every polynomial equation solvable, and no further extension is ever needed for algebra.
Square roots of negative numbers
Split the radical into √-1 and the positive part, then replace √-1 by i.
Worked example — the source's case
The rule √xy = √x√y requires non-negative radicands. Applied carelessly to negatives it gives √-4√-9 = √36 = 6, whereas the correct value is (2i)(3i) = 6i2 = -6. Always convert each radical to i form before multiplying.
The powers of i
Everything follows from i2 = -1, and the powers repeat with period four.
| Power | Value | Working |
|---|---|---|
| i0 | 1 | Any non-zero base to the power 0 |
| i1 | i | — |
| i2 | -1 | The definition |
| i3 | -i | i2 · i = -1 · i |
| i4 | 1 | i2 · i2 = (-1)(-1) |
| i5 | i | The cycle restarts |
Divide the exponent by 4 and keep the remainder. i27: 27 = 4 × 6 + 3, so i27 = i3 = -i.
Arithmetic
Addition and subtraction combine the real and imaginary parts separately, exactly as like terms are collected.
Multiplication is ordinary expansion followed by replacing i2 with -1.
Worked example — squaring, both signs
The source works both, on different pages, which makes the sign pattern clear.
The real parts agree and the imaginary parts are opposite. That is not a coincidence: the two inputs are conjugates, and squaring conjugates gives conjugates.
(5i)2 = 25i2 = -25, not +25. The square of the coefficient is positive but i2 contributes the minus. Missing this is the commonest error in complex arithmetic.
The conjugate and division
The conjugate of a + bi is a - bi. Their product is always a non-negative real number:
This is the same identity used to rationalise a surd denominator, and it is used for the same purpose: to clear an unwanted quantity out of a denominator.
Worked example — the source's division
Written in standard form the answer is 1213 + 813i. Note the third line: -4i2 becomes +4, so the denominator is 13, not 5.
Multiply back: (3 - 2i) · 12 + 8i13 = 36 + 24i - 24i - 16i213 = 36 + 1613 = 4. Correct.
- Write the conjugate of the denominator.
- Multiply top and bottom by it. This is multiplication by 1.
- Expand, replacing every i2 by -1.
- Separate into reald + imaginarydi.
Where complex roots come from
A quadratic with negative discriminant has a conjugate pair of complex roots, and the source works one in full.
Worked example — x2 + 4x + 5 = 0
The two roots -2 + i and -2 - i are conjugates. This always happens when the coefficients are real: the only complex part comes from the radical, which enters with both signs.
The roots should sum to -4 and multiply to 5. Sum: -4. Product: (-2 + i)(-2 - i) = 4 - i2 = 5. Both correct.
The same pattern appears in the source's cubic x3 - 8 = 0, which factors as (x - 2)(x2 + 2x + 4) and yields x = 2 together with the conjugate pair -1 ± i√3.
Because complex roots of a real polynomial always arrive in conjugate pairs, a polynomial of odd degree with real coefficients must have at least one real root — there is no way to pair up an odd number of roots entirely.
Common mistakes
| Mistake | Correct | Why |
|---|---|---|
| (5i)2 = 25 | -25 | i2 = -1 |
| √-4√-9 = 6 | -6 | Convert to i form first: (2i)(3i) = -6 |
| 9 - 4i2 = 5 | 13 | -4i2 = -4(-1) = +4 |
| Conjugate of 3 - 2i given as -3 + 2i | 3 + 2i | Only the imaginary part changes sign |
| Leaving i in a denominator | Multiply by the conjugate | Standard form is a + bi |
| Comparing complex numbers with < | Not defined | No ordering of ℂ respects the arithmetic |
Frequently asked questions
Is i a real thing?
It is as real as any other number, in the sense that it is a well-defined object obeying consistent rules. The name 'imaginary' is a historical accident from a period when the numbers were distrusted. They describe alternating current, wave phase and rotation perfectly concretely.
Why is √-9 = 3i rather than -3i?
By convention the principal square root of -a for positive a is i√a. Both 3i and -3i square to -9, exactly as both 3 and -3 square to 9; the symbol names one of them, and solving x2 = -9 gives both.
Can complex numbers be ordered?
No. There is no way to define < on the complex numbers that respects the arithmetic. This is why inequalities are a real-number topic only, as the source notes state.
What is a complex number good for?
Every polynomial equation has a solution once they are admitted, which makes the algebra complete. In application they encode rotation and phase compactly, which is why they run through alternating-current analysis, signal processing, control theory and vibration.
Source. Handwritten teaching notes, Week 1 page 11, Week 2 pages 1 and 11, and Week 3 page 7.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.