Why You Can't Take the Square Root of a Negative Number — Or Can You?
Ask a calculator for √−9 and you'll probably get an error. Ask a math teacher and you might hear "you can't." Ask an electrical engineer and they'll say "3i" without blinking.
All three are right, depending on which number system you're working in. The story of how mathematicians moved from "impossible" to "indispensable" is one of the strangest in mathematics, and it started with people who didn't believe these numbers existed but needed them anyway.
"Impossible" Roots Were First Used to Find Real Answers
You might assume mathematicians invented square roots of negatives to solve equations that have no real solutions. That's not how it happened.
They were forced to use them to find solutions that were perfectly real. The imaginary parts appeared in the middle of the calculation and then canceled out. Ignoring them meant you couldn't get the real answer at all.
Why It's Impossible With Real Numbers
A square root of a number x is a value that, multiplied by itself, gives x. The problem with negative x comes from the sign rules:
positive × positive = positive 3 × 3 = 9
negative × negative = positive (−3) × (−3) = 9
zero × zero = zero
Every real number squared is zero or positive. So no real number squares to −9. Among the real numbers, √−9 simply doesn't exist. That's the true meaning of "you can't." Browse the list of square roots and you'll only find non-negative inputs.
Cardano's Uncomfortable Problem
In 1545, Gerolamo Cardano published Ars Magna, a landmark book of algebra. In it he posed a puzzle: divide 10 into two parts whose product is 40.
The parts must satisfy x + y = 10 and xy = 40, which leads to:
x = 5 + √−15, y = 5 − √−15
Check: (5 + √−15) + (5 − √−15) = 10, and (5 + √−15)(5 − √−15) = 25 − (−15) = 40. It works, if you're willing to handle √−15 with the usual rules. Cardano called the manipulation "as subtle as it is useless."
An Insider Reference: Bombelli's Breakthrough
The turning point came from Rafael Bombelli, an Italian hydraulic engineer. In his 1572 book L'Algebra, he studied the cubic equation:
x³ = 15x + 4
It has an obvious real solution: x = 4 (since 64 = 60 + 4). But Cardano's formula for cubics produces:
x = ∛(2 + √−121) + ∛(2 − √−121)
The formula demanded √−121, even though the answer was real. Bombelli boldly treated √−1 as a new kind of quantity and noticed that:
(2 + √−1)³ = 2 + 11√−1 = 2 + √−121
So ∛(2 + √−121) = 2 + √−1, and ∛(2 − √−121) = 2 − √−1. Adding them:
(2 + √−1) + (2 − √−1) = 4
The imaginary parts canceled, leaving the real answer 4. This is the case mathematicians call the casus irreducibilis: when a cubic has three real roots, the standard formula must pass through square roots of negative numbers.
Giving It a Name: i
If √−1 is going to be used, it deserves a symbol. Leonhard Euler introduced the letter i for √−1 in a paper written in 1777:
i² = −1
Then every negative number has square roots:
√−9 = √9 × √−1 = 3i (and −3i also squares to −9)
√−2 = √2 · i ≈ 1.414i
Numbers of the form a + bi, where a and b are real, are called complex numbers. René Descartes had earlier used "imaginary" as a dismissive label. The name stuck, but the dismissal didn't.
A Trap: Root Rules Break
With real non-negative numbers, √a × √b = √(ab). Apply that blindly to negatives and you get nonsense:
−1 = i × i = √−1 × √−1 = √((−1)(−1)) = √1 = 1 ✗
The rule √a × √b = √(ab) isn't valid when a and b are both negative. That's one reason careful mathematicians write i, 3i and so on, instead of √−9.
Where It Made Sense: A New Dimension
Complex numbers truly clicked when mathematicians like Caspar Wessel (1799), Jean-Robert Argand (1806) and Carl Friedrich Gauss pictured them as points on a plane. The real numbers run left-right, and the imaginary numbers run up-down.
In that picture, multiplying by i means rotating 90° counterclockwise:
1 → i → −1 → −i → 1
Do it twice and you've rotated 180°, turning 1 into −1. That's exactly why i² = −1. A square root of −1 is simply "half of a half-turn." Explore that geometry with the polar to rectangular converter.
Two Concepts Worth Knowing
Principal Square Root
Every non-zero number has two square roots. The principal square root is the one we pick by convention: the non-negative one for positive numbers (√9 = 3), and the one with positive imaginary part for negative numbers (√−9 = 3i).
The Fundamental Theorem of Algebra
With complex numbers, every polynomial equation of degree n has exactly n roots, counting repeats. Gauss gave an early proof in 1799. The real numbers alone can't make that promise: x² + 1 = 0 has no real roots at all.
Quick Answer: Can You Take the Square Root of a Negative Number?
Not with real numbers, since every real number squared is zero or positive. But mathematicians define i as a number whose square is −1. Then negative numbers have square roots in the complex numbers: for example, √−9 = 3i, since (3i)² = 9 × i² = −9.
Try Them Yourself
- Square Roots List: the real square roots, before i enters
- Square Numbers List: every square is non-negative
- Polar to Rectangular Converter: see i as a 90° rotation
- Algebra Formulas: the quadratic formula and its discriminant
- Scientific Calculator: verify Bombelli's (2 + i)³ numerically
- Unlocking the Power of Quadratic Functions: where negative discriminants show up
Expand (2 + i)³ by hand using i² = −1. When you land on 2 + 11i, you'll have repeated the calculation that convinced Europe that imaginary numbers were worth taking seriously.