The Imaginary Unit and Complex Numbers

The equation the discriminant refused

Take the simplest quadratic with no real roots:

Here $a = 1$, $b = 0$, $c = 1$, so the discriminant is $\Delta = 0^2 - 4(1)(1) = -4 < 0$, and the previous chapter’s biconditional delivers its verdict: no real solutions. The parabola $y = x^2 + 1$ has its vertex at height $1$ and floats entirely above the $x$-axis. But the discriminant is a summary of a deeper fact, and since this whole chapter is built on that fact, it deserves its own proof from first principles.

The proof is airtight, so read its conclusion carefully. The word doing all the work is real. The argument used facts about real numbers, their signs and their order, and it rules out a real solution only. It leaves open a subtler question: could there be a larger supply of numbers, obeying the same laws of arithmetic, in which $x^2 = -1$ does have an answer?

Enlarging the number system, one more time

You have watched this exact situation resolve itself several times already, and the resolution was the same every time. Within the whole numbers, $3 - 5$ has no answer, so the integers were admitted and subtraction became possible without restriction. Within the integers, $2x = 3$ has no answer, so the rationals were admitted and division became possible. Within the rationals, $x^2 = 2$ has no answer, since no fraction squares exactly to $2$, so the irrationals joined and the reals were complete. Each extension shared three features: the old numbers were kept, sitting inside the new system unchanged; the rules of arithmetic, the commutative, associative, and distributive laws, kept holding. The third feature is that the equation that forced the extension became solvable.

So we make the same move for $x^2 = -1$, and the entire construction is one sentence. The imaginary unit $i$ is a new number defined by the single property

and we insist that the ordinary rules of algebra continue to apply to it. That is the complete definition. Every other fact in this chapter is deduced from that one equation plus the laws of arithmetic you already own.

If inventing a number by decree feels like cheating, notice that every extension before it was the same decree. Nobody can hold $-2$ apples either, yet negative numbers earned their place because they obey the rules and settle real questions. What makes the decree mathematics rather than wishful thinking is consistency: no contradiction ever falls out of computing with $i$. And a later lesson in this chapter will even show you where these numbers live geometrically, which is when most people stop feeling uneasy about them.

The payoff is immediate. The equation $x^2 = -1$ now has a solution, namely $i$, and in fact it has a second one: $-i$. To check it, square $-i$ using the sign rules, which still apply:

So $x^2 = -1$ has the two solutions $x = \pm i$, a mirror pair, exactly the way $x^2 = 2$ has the mirror pair $\pm\sqrt{2}$.

Square roots of negative numbers

With $i$ in hand, every negative number acquires a square root. For a positive real number $k$, we define

The definition has to earn the radical sign, so check that the right side really does square to $-k$, using the commutative law to square each factor separately:

It checks. So $\sqrt{-25} = i\sqrt{25} = 5i$, and $\sqrt{-7} = i\sqrt{7}$. Two habits of notation are worth adopting on day one. Write the $i$ in front of the radical, as in $i\sqrt{7}$, or after a whole coefficient, as in $5i$. Either placement keeps the $i$ from ever being misread as sitting underneath the radical. And remember that the radical symbol names a single number, just as it did for positive inputs: $\sqrt{-25}$ is $5i$, while the equation $x^2 = -25$ has the two solutions $\pm 5i$.

One familiar rule does not survive the trip. For nonnegative $a$ and $b$ you proved $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$, and that hypothesis was not decoration. Feed the rule two negative inputs and it fails:

The left computation is the correct one; the right one applies a theorem outside the territory it was proved on. The safe procedure is mechanical: convert every square root of a negative into its $i$-form first, and only then multiply.

Equations with no real roots, solved

The whole chapter opened because $x^2 + 1 = 0$ had no real solution. Equations of that shape now surrender completely.

The same two steps, isolate the square and attach $\pm i$, solve every equation of the form $x^2 = -k$ with $k$ positive: the solutions are $x = \pm i\sqrt{k}$. What this lesson does not yet do is solve a general quadratic whose discriminant is negative. That harvest, where the quadratic formula runs unchanged and simply outputs complex numbers, is the final lesson of this chapter.

The powers of i cycle

The definition hands you $i^2$. Multiplying by $i$ again and again hands you everything else, and a striking pattern appears within four steps:

The fourth power is $1$, so the fifth power is $i^4 \cdot i = 1 \cdot i = i$ and the tape starts over. The powers of $i$ march through the same four values forever: $i$, $-1$, $-i$, $1$, then again $i$, $-1$, $-i$, $1$, with no exceptions.

The cycle turns the scariest-looking problems into division with remainder.

Complex numbers and their two parts

Multiples of $i$ alone are not the end of the construction, because the rules of arithmetic let you add a real number to one of them. The result, something like $3 + 5i$, cannot be compressed any further: it is not a real number, and it is not a plain multiple of $i$. It is a genuinely two-part number, and numbers of this shape get the chapter’s name.

A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers. The real number $a$ is called its real part, the real number $b$ is called its imaginary part, and the expression $a + bi$ is called standard form. Note the fine print carefully: the imaginary part of $3 + 5i$ is $5$, the real coefficient of $i$, and not $5i$. Signs ride along with the parts, so $-2 - i = -2 + (-1)i$ has real part $-2$ and imaginary part $-1$.

The definition quietly swallows everything that came before it. Choose $b = 0$ and $a + 0i$ is just the real number $a$: every real number is a complex number. So the reals sit inside the new system unchanged, exactly as the integers sit inside the rationals. Choose $a = 0$ with $b \ne 0$ and you get numbers like $4i$, called pure imaginary numbers. Choose both parts nonzero and you get the new two-part numbers like $3 + 5i$. The name complex is the old sense of the word, a whole built out of parts, as in a building complex; it is a description of structure, not of difficulty.

How to add, multiply, and divide these two-part numbers is the entire next lesson. This lesson’s job is to know what they are, and to settle the one operation that needs no new machinery at all: deciding when two of them are equal.

One real-number habit, though, does not survive the extension, and it is worth seeing why before it bites. Real numbers are ordered: any two of them can be compared with $<$. The previous chapters’ inequality rules relied on that, in particular the rule that multiplying an inequality by a positive number preserves it. Try to fit $i$ into such an ordering. If $i > 0$, then multiplying both sides of $i > 0$ by the positive number $i$ must preserve the inequality, giving $i^2 > 0$, that is, $-1 > 0$, which is false. If instead $i < 0$, then $-i > 0$, and multiplying $-i > 0$ by the positive number $-i$ gives $(-i)^2 > 0$, which is again $-1 > 0$, false. And $i = 0$ fails immediately, since $0^2 = 0 \ne -1$. No placement is consistent, so the complex numbers cannot be ranked on a number line: a question like “is $2 + 3i$ bigger than $3 + 2i$” has no meaning. Complex numbers are compared for equality, never for size.

Equality matches the parts

Two complex numbers in standard form are equal exactly when they match part by part. This sounds too obvious to need proof, but it is a genuine theorem, and its proof leans on the very first proof of the lesson in a satisfying way.

The theorem is a workhorse, because it means one complex equation silently carries the information of two real equations: match the real parts, then match the imaginary parts. Whole problems fall to that single move.

Common mistakes