x² = -1 has no real solution, so a number i with i² = -1 is defined and joined to the reals. A complex number is a + bi with a and b real; every real number is one of them, and the system is built so that all the algebra of the courses from Foundations of Algebra to Rational and Radical Expressions keeps working unchanged.
Core ideas
Extending the number system to solve an equation is not a new move. It has happened three times already on this path, and every time for the same reason.
i² = -1, and nothing else about it is assumed. Everything else — how to add, multiply and divide with it — follows from that one equation together with the ordinary laws of algebra, which is what “Operations with Complex Numbers” works out.
In a + bi, both a and b are real numbers. The imaginary part of 3 - 7i is -7, not -7i: it is a real number by universal convention, and the convention exists so that "imaginary part" names a coordinate you can compare and equate.
5 is 5 + 0i. The reals are not replaced or contradicted; they sit inside the complex numbers as the ones with imaginary part 0. Everything proved in the courses from Foundations of Algebra to Rational and Radical Expressions goes on holding, for exactly the numbers it held for before.
The material
Why the extension is legitimate, what the objects are, and how to write a root of a negative number.
“The Discriminant” ended with D < 0 and the answer "no real solution". The word carrying the weight in that phrase is real. x² + 1 = 0 needs a number whose square is -1, and no point on the number line has one, because the square of a real number is 0 or positive.
Each earlier number system was extended for the same reason: an equation with no solution inside it.
At each stage the new numbers were called unnatural, then absurd, then irrational, then imaginary, and each time the name stuck while the objection did not. "Imaginary" is a seventeenth-century insult that hardened into a technical term. It does not mean these numbers are less legitimate than the others, and nothing in the arithmetic treats them as such.
i is the number defined by the property i² = -1. It is not a real number.
That i is not real is a consequence rather than an assumption: every real number r satisfies r² ≥ 0, and -1 < 0.
A complex number is one that can be written a + bi with a and b real. a is the real part and b is the imaginary part, and both of them are real numbers.
a + bi is called standard form. A complex number with b = 0 is real; one with a = 0 and b ≠ 0 is called pure imaginary.
So 3 - 7i has real part 3 and imaginary part -7. Calling the imaginary part -7i is a common slip and a consequential one: the two coordinates of a complex number are both real numbers, and that is what makes the next theorem sayable.
For real a, b, c and d: a + bi = c + di if and only if a = c and b = d.
If a = c and b = d the two expressions are identical, so they are equal.
Conversely, suppose a + bi = c + di. Rearranging gives a - c = (d - b)i. If d - b were not 0, dividing by it would give i = (a - c)/(d - b), a quotient of two real numbers with a non-zero denominator and therefore itself real — and i is not real. So d - b = 0, that is b = d; and then a - c = 0, that is a = c.
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One equation between complex numbers is therefore two equations between real numbers. That is used constantly from here on, and it is why complex numbers behave like points in a plane: a + bi corresponds to the point (a, b), with the real numbers lying along the horizontal axis and the pure imaginary ones up the vertical.
The plane picture also explains an absence. No ordering of the complex numbers is compatible with their arithmetic, so the words "positive", "greater than" and "between" are not available for them, and neither are the signs < and ≤. Points in a plane do not come in an order. Those words are used only where the numbers involved are real.
For a positive real number k, √(-k) is defined to be i√k.
One line checks it: (i√k)² = i²(√k)² = (-1)k = -k. The other square root of -k is -i√k, exactly as a positive number has two real square roots and the symbol names one of them.
Convert to i before doing anything else with the expression. The rule √a · √b = √(ab) is stated for non-negative a and b and does not survive down here, as “Operations with Complex Numbers” shows in detail.
Applied to an equation with D < 0, the formula hands back something like (-6 + √(-8))/2. Convert the root first: √(-8) = 2i√2. Then (-6 + 2i√2)/2 = -3 + i√2, which is standard form with real part -3 and imaginary part √2.
Both terms of the numerator were divided by 2, exactly as in “The Quadratic Formula”. “Complex Roots of Quadratics” does this systematically; the point here is only that the result is an ordinary complex number, of the shape this lesson defined.
Interactive
Parts, conjugate, modulus, and the powers of i found by multiplying
Method
Standard form is a + bi with a and b real. Four moves get anything there.
√(-12) becomes i√12. Do this before any multiplying: the radical laws of Rational and Radical Expressions assume non-negative radicands, and they stop being reliable the moment a negative goes under a root.
√12 = 2√3 by Rational and Radical Expressions, so i√12 becomes 2i√3. An unsimplified radical is as unfinished here as it was there.
Collect everything without an i into a and everything with one into bi. A real number is a + 0i and a pure imaginary is 0 + bi; both are in standard form once written that way.
(-6 + 2i√2)/2 is -3 + i√2. Every term of the numerator gets divided, which is the same reduction “The Quadratic Formula” needed and the same place it goes wrong.
Each one ends as a + bi, with a and b named and both of them real.
Line 2 is the one worth pausing on. 7 is a complex number, with imaginary part 0. Nothing about it changed in this lesson; the definition was written so as to include it.
Line 5 is the square root property of “The Square Root Property”, applied where it previously reported nothing. x² = -49 has two solutions among the complex numbers, and they are negatives of one another, exactly as x² = 49 had two real ones. “Complex Roots of Quadratics” shows that the pairing is not a coincidence.
For a faded standard-form pass, simplify (8 + √(-32))/4. The supplied first move is √(-32) = i√32; simplify that radical and divide both numerator terms. You should reach 2 + √2 i, with real part 2 and imaginary part √2. If the imaginary part is reported as √2 i, revisit the definition: the part is the real coefficient of i.
Convert any root of a negative number to i immediately, simplify the radical, and name the real and imaginary parts as real numbers. You should also be able to say what i is defined by — one equation, i² = -1 — and resist the idea that anything further about it has been assumed.
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