An imaginary number is a real number multiplied by i, the imaginary unit, where i = √−1 and i2 = −1. Examples are 3i, −12i and √−25 = 5i. Imaginary numbers give square roots of negative numbers a value, so equations like x2 = −9 have solutions: x = ± 3i.
In Spanish
número imaginario
Also called
imaginary unit, i, pure imaginary number, square root of a negative number
Multiplying by i is a quarter turn, so four turns bring you back to 1. That is why the powers of i repeat every 4.
Formal definition and notation
The imaginary unit i is defined by i2 = −1. A pure imaginary number has the form bi with b real and b ≠ 0. For a > 0, the principalsquare root of a negative number is √−a = i√a. Adding a real number gives a complex numbera + bi; many textbooks (including OpenStax) call any a + bi with b ≠ 0 an imaginary number and reserve “pure imaginary” for a = 0. Imaginary numbers are not on the real number line; they lie on the vertical axis of the complex plane.
Symbol
Read it as
Meaning
i
i
the imaginary unit, i = √−1, so i2 = −1
bi
b i
a pure imaginary number, such as 4i
a + bi
a plus b i
a complex number with real part a and imaginary part b
Key facts
No real number squared is negative, which is why √−1 needs a new symbol, i.
Multiply imaginary numbers like variables, then replace i2 with −1: (5i)(2i) = 10i2 = −10.
The powers of i repeat every 4: i10 = i2 = −1 because 10 has remainder 2 when divided by 4.
A quadratic with a negative discriminant has two imaginary (non-real) solutions, such as x2 + 16 = 0, x = ± 4i.
“Imaginary” is a historical name. Imaginary numbers are exact and useful: electrical engineers use them to describe alternating current.
Rules for i
Rules for i
√−a = i√a for a > 0, and i2 = −1
powers of i
i1 = i, i2 = −1, i3 = −i, i4 = 1, then the cycle repeats
in
divide n by 4 and use the remainder: remainder 1 gives i, 2 gives −1, 3 gives −i, 0 gives 1
Take the i out first: √−4 · √−9 = 2i · 3i = 6i2 = −6, not √36 = 6.
Irrational numbers like √2 are real numbers with non-repeating decimals. √−2 = i√2 is imaginary.
Quick check: 3 questions
Question 1 of 3
What is √−36?
Show answer and explanation
Answer: C.√−36 = √36 · √−1 = 6i. 6 and −6 are real and square to 36, not −36. −6i also squares to −36, but the principal root is 6i.
Question 2 of 3
What is i2?
Show answer and explanation
Answer: B. By definition i = √−1, so i2 = −1. 1 is i4, and −i is i3.
Question 3 of 3
Simplify (3i)(4i).
Show answer and explanation
Answer: D.(3i)(4i) = 12i2 = 12(−1) = −12. 12i forgets one i, 12 forgets that i2 = −1, and 7i adds instead of multiplying.
Imaginary Number: frequently asked questions
Is i a real number?
No. i is defined by i2 = −1, and every real number squared is 0 or positive, so i cannot be real. It sits one unit up the vertical axis of the complex plane. Real numbers and imaginary numbers together make the complex numbers, a + bi.
What is the square root of -1?
The square root of −1 is the imaginary unit i. Both i and −i square to −1, and √−1 names the principal one, i. Using it, any negative number has square roots: √−50 = i√50 = 5i√2. Real numbers alone cannot do this, because a real number squared is never negative.
Why are they called imaginary numbers?
The name comes from mathematicians who first used square roots of negatives as a calculation trick and did not think of them as real quantities. The word stuck, but imaginary numbers follow exact rules and describe real things, such as alternating current in circuits and waves in physics. “Imaginary” is a label, not a judgment.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSN-CN.A.1, CCSS HSN-CN.C.7
Cite this entry
Effortless Math Team. “Imaginary Number.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/imaginary-number/
Flashcard
TermImaginary Numberbuilt on i = √−1
DefinitionA real number times i, where i2 = −1. Example: √−25 = 5i, and x2 = −9 gives x = ± 3i.
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