Invent one new number, i² = −1, and every equation finally has a place to live.
Every number system you have met was born from a question the old numbers could not answer. Subtraction asked for 3 − 5 and the negatives appeared; division asked for 1 ÷ 3 and the fractions appeared; taking a root asked for √2 and the irrationals appeared. Each time, arithmetic outran the numbers we had, and each time we extended the system rather than declaring the question forbidden. This lesson takes the next step. The equation x² = −1 has no answer on the real line — so we invent exactly one new number, give it the single rule i² = −1, and watch a whole new world, the complex numbers z = a + b i, open up around it.
Start with the equation that has no home: x² = −1. It is asking a very physical question — which length, squared, gives −1? On the real line that is impossible, because squaring can never produce a negative number. A positive times a positive is positive; a negative times a negative is also positive; and 0² = 0. So x² is never less than 0, and the value −1 is simply out of reach.
This is exactly the situation our predecessors faced with 3 − 5 and with √2: the answer is not there, on the numbers we currently own. The honest response is not "no solution forever," but "we need a bigger number system." We are about to build it — and remarkably, it takes only one new ingredient.
Name a square root of −1 with a single symbol, the imaginary unit i, defined by the one rule
i² = −1.
That is the entire definition. The number i is not on the real line — it is a brand-new object — but every ordinary rule of algebra (commuting, associating, distributing) is kept exactly as before. Once we have i, square roots of negatives become easy. For instance,
√(−9) = √(9 · (−1)) = √9 · √(−1) = 3i, and you can check it: (3i)² = 9 · i² = 9 · (−1) = −9. ✓
The familiar identity √a · √b = √(ab) is for non-negative a and b. It fails for negatives. Watch:
√(−1) · √(−1) = i · i = i² = −1, but √((−1)(−1)) = √1 = 1.
Those are different answers, so the shortcut is illegal here. Always pull the negative out as i first — write √(−9) = 3i — and only then multiply.
With i available, the most general number we can build by ordinary arithmetic is a real number plus a real multiple of i. We give it a name. A complex number is anything of the form
z = a + b i, with a and b real.
Here a is the real part, written Re z = a, and b is the imaginary part, written Im z = b. Note carefully: the imaginary part is the real number b that multiplies i — it is b, not b i. A few examples make the shape concrete:
| z | Re z | Im z | kind |
|---|---|---|---|
| 3 + 2i | 3 | 2 | a genuine complex number |
| −1 − 4i | −1 | −4 | a genuine complex number |
| 7 | 7 | 0 | real (7 = 7 + 0i) |
| 5i | 0 | 5 | purely imaginary (5i = 0 + 5i) |
A complex number carries two independent real numbers — an amount on the real axis and an amount on the imaginary axis. That is exactly why, in 31.2, each z gets an address as the point (a, b) on a plane. For now, every z is just the pair "real part, imaginary part."
When are two complex numbers the same? Since each carries two independent reals, they agree only when both pieces agree:
a + b i = c + d i ⇔ a = c and b = d.
This single line is more powerful than it looks: one complex equation is secretly two real equations, one for the real parts and one for the imaginary parts. That trick — match the parts — solves a great many problems, and you'll meet it again throughout the stage. Classification falls straight out of the form:
On the real line we can always say which of two numbers is bigger. On ℂ we cannot. A statement like "3 + 2i < 4 + i" is meaningless — there is no consistent way to rank complex numbers. You can compare their sizes (their moduli |z|, the arrow lengths of 31.2), but never the numbers themselves. Whenever you feel the urge to write "<" between two non-real complex numbers, stop: that inequality does not exist.
A whole stage rests on this small vocabulary. Keep it close:
| Idea | What it says |
|---|---|
| The imaginary unit | i² = −1 — one new number, one rule; √(−k) = i√k for k > 0. |
| Complex number | z = a + b i, a and b real; Re z = a, Im z = b (the real number, not b i). |
| Equality | a + b i = c + d i ⇔ a = c and b = d — one complex equation is two real ones. |
| Kinds | b = 0 ⇒ real; a = 0, b ≠ 0 ⇒ purely imaginary; so ℝ ⊂ ℂ. |
| No order | You cannot write z₁ < z₂ for non-real z — only sizes |z| compare. |
Next, in 31.2 The Complex Plane, we give every z an address: the point (a, b), the arrow from the origin, its length |z|, and its direction θ.
Simplify √(−25) and check your answer by squaring.
√(−25) = √(25 · (−1)) = √25 · √(−1) = 5i. Check: (5i)² = 25 · i² = 25 · (−1) = −25. ✓
For z = −2 + 7i, state Re z and Im z. Is z real, purely imaginary, or neither?
Re z = −2 and Im z = 7 (the real number multiplying i, not 7i). Since both parts are nonzero, z is neither real nor purely imaginary.
A classmate writes "√(−4) · √(−4) = √16 = 4." Find the error and the correct value.
The rule √a·√b = √(ab) is only valid for non-negative a, b, so it cannot be used here. Do it honestly: √(−4) = 2i, so √(−4) · √(−4) = (2i)(2i) = 4i² = −4, not 4.
Find real numbers x and y so that (x − 1) + (y + 2)i = 4 − 3i.
Match the parts: real gives x − 1 = 4 ⇒ x = 5; imaginary gives y + 2 = −3 ⇒ y = −5. One complex equation became two real ones.
Classify each number as real, purely imaginary, or neither: (a) 6, (b) −3i, (c) 2 − i, (d) 0.
(a) 6 = 6 + 0i — real. (b) −3i = 0 − 3i — purely imaginary. (c) 2 − i — neither. (d) 0 = 0 + 0i — real (it has no imaginary part; it is the one number that is both real and "imaginary-free").
Explain why the statement "1 + i < 1 + 2i" is meaningless, and what comparison is allowed.
ℂ carries no order: there is no consistent < that ranks non-real complex numbers, so "1 + i < 1 + 2i" has no meaning. What you may compare is size — the moduli |1 + i| = √2 and |1 + 2i| = √5 — but that compares lengths, not the numbers themselves. (Moduli are the subject of 31.2.)
Six questions to lock it in. Tap the answer you think is right.
This lesson opens CCSS HSN-CN.A.1 (the imaginary unit i with i² = −1, and the form a + b i with a, b real) and HSN-CN.A.2's groundwork (treating a + b i as a number you can later add and multiply). It also rehearses the recurring idea that each new equation we cannot solve — like x² = −1 — motivates an extension of the number system, the same move that gave us negatives, fractions, and irrationals.
Two points reward emphasis at home. First, the imaginary part of a + b i is the real number b, not b i — a frequent slip. Second, the complex numbers are not ordered: it is correct to compare sizes (moduli) but meaningless to write one complex number as "less than" another. The geometry that makes all of this visible — the plane, the arrow, the length, the angle — arrives in 31.2.