How this instrument works
Square any real number and the outcome never falls below zero: 3² = 9, and (−3)² = 9 as well, since two negative factors cancel into a positive one. That single fact leaves every negative number without a real square root — there is no ordinary number that, multiplied by itself, lands back on −9 or −2 or any other value below zero. Rather than declare the question unanswerable, mathematicians in the sixteenth century invented a new quantity built specifically to answer it: the unit i, defined by one property alone, i² = −1. Nothing mystical is happening — i is simply a bookkeeping device whose entire job description is squaring to a negative one, filling the exact gap the real numbers leave open.
Once that definition exists, the square root of any negative x can be written down directly: strip the sign from x, take the ordinary square root of what remains, and pair it with the imaginary unit — coefficient = √|x|, so √x = coefficient × i. Feed in x = −9 and the sheet strips the sign to get 9, takes its root to get 3, and reports a coefficient of 3.0, meaning √−9 = 3i. Squaring that result checks it instantly: (3i)² = 9 × (−1) = −9, exactly the starting value recovered.
This page's scope differs from two related pages elsewhere on this site. The sheet covering powers of the imaginary unit already treats that unit as given and asks where its repeated powers land, cycling through four fixed points as the exponent climbs. The complex-root sheet goes the other direction, starting from a full quadratic equation and using this same coefficient idea to build both of its two roots at once. This page sits underneath both: it is the single conversion step — negative input in, imaginary coefficient out — that those other tools assume is already understood.
The output is not always a tidy whole number. A radicand of −2 produces a coefficient of roughly 1.4142135623730951, the square root of 2 carried out to its usual irrational digits, since stripping the sign from −2 leaves the same 2 whose root never terminates. And the boundary sits at zero: feed in x = 0 and the coefficient comes back as 0.0, the one point where the negative and non-negative cases touch and neither a real nor an imaginary part is needed at all.
- Enter a value at or below zero into the Negative number (x) field — positive entries fall outside what this sheet is built to convert.
- Read Imaginary coefficient (√x = coefficient × i) for the number that pairs with i to form the answer.
- Multiply the displayed coefficient by i in your own notes to write the full result, such as 3i or 1.4142135623730951i.
- Square the coefficient and negate it as a check — it should reproduce your original x exactly, within rounding.
- Try x = 0 once to see the boundary case, where the coefficient drops to zero and no imaginary part remains.
Worked example — the square root of −9
Enter x = −9. Stripping the sign leaves 9, and √9 = 3 exactly, so the sheet reports a coefficient of 3.0, meaning √−9 = 3i. Squaring that answer confirms it belongs to −9 and nothing else: (3i)² = 9 × (−1) = −9, the original input recovered digit for digit with no rounding anywhere in the check.
Two other entries show the range this same rule covers. At x = 0, the coefficient falls to exactly 0.0 — the one input where positive and negative meet and neither a real nor an imaginary answer is required. At x = −2, stripping the sign leaves 2, and √2 has no exact decimal ending, so the coefficient comes back as 1.4142135623730951 — an irrational multiplier attached to i, for the same reason √2 itself never terminates in the ordinary positive case.
Questions
What is an imaginary number?
A quantity built from the unit i, defined by the single property i² = −1, multiplied by a real coefficient. It exists because no real number squares to a negative result, so a square root of a negative x is written as that coefficient times i rather than left unanswered.
Why isn't there a real square root of a negative number?
Because squaring any real number, positive or negative, always produces zero or a positive result — (−3)² and 3² both equal 9. Nothing on the ordinary number line can square down to a negative value, which is the entire reason the unit i was invented in the first place.
What does i² = −1 actually mean?
It is the defining rule of the imaginary unit, not a derived fact — i is simply the quantity chosen to make that equation true, filling the one gap real numbers leave open. Every property of imaginary and complex numbers traces back to that single starting rule.
What is the square root of −9?
3i — strip the negative sign to get 9, take its ordinary square root to get 3, and pair that 3 with the imaginary unit. Squaring the result confirms it: (3i)² = 9 × (−1) = −9, matching the original input exactly.
Why is the coefficient for √−2 an irrational number?
Because stripping the sign from −2 leaves 2, and √2 ≈ 1.4142135623730951 never ends in a repeating or terminating decimal — a fact proven since antiquity for the ordinary positive root. Attaching i to that same irrational value doesn't change its digits at all.
How is this different from the site's calculator for powers of the imaginary unit?
That sheet treats the imaginary unit as already defined and raises it to a whole-number exponent, tracking a four-step cycle as the power climbs. This sheet does the opposite job: it starts from a negative number and produces the coefficient that pairs with that same unit to answer the number's square root in the first place.
What happens when x equals zero?
The coefficient comes back as exactly 0.0. Zero sits on the boundary between positive and negative, and its square root is zero itself, needing no imaginary part at all — the one case where this sheet's answer matches what a plain square-root calculator would also report.