Principal Square Root of a Negative Number

Quadratic Equations and the Fundamental Theorem of AlgebraSquare Roots of a Complex Number

Principal Square Root of a Negative Number

For any positive real number a , the principal square root of − a is defined as:

− a = i a

This follows from i 2 = − 1 . Be careful: the usual product rule x ⋅ y = x y does not apply when both x and y are negative.

Quick Reference

Expression Value Reasoning
− 1 i Definition: i 2 = − 1
− 4 2 i 4 ⋅ i = 2 i
− 9 3 i 9 ⋅ i = 3 i
− 7 i 7 7 ⋅ i
− 4 ⋅ − 9 − 6 = 2 i ⋅ 3 i = 6 i 2 = − 6 (not + 6 )

Definition

Since i 2 = − 1 and ( − i ) 2 = ( − 1 ) 2 ⋅ i 2 = − 1 , both i and − i are square roots of − 1 .

Definition. The principal square root of − 1 is i , written − 1 = i .

For any positive real number a , the principal square root of − a is:

− a = i a

For example:

− 25 = i 25 = 5 i , − 3 = i 3 , − 1 4 = 1 2 i

Example 1. Simplify 3 − 16 − 2 − 9 .

Solution.

\begin{aligned} 3\sqrt{-16} - 2\sqrt{-9} &= 3(4i) - 2(3i) \\ &= 12i - 6i \\ &= 6i \end{aligned}

A Critical Warning: Products of Negative Square Roots

For positive numbers a and b :

− a ⋅ − b = ( i a ) ( i b ) = i 2 a b = − a b

Warning. The product rule x ⋅ y = x y is only valid when x ≥ 0 and y ≥ 0 . It fails when both numbers are negative:

− a ⋅ − b ≠ ( − a ) ( − b ) = a b

The correct result is − a ⋅ − b = − a b .

Example 2. Compute − 4 ⋅ − 9 .

Correct approach:

− 4 ⋅ − 9 = ( 2 i ) ( 3 i ) = 6 i 2 = 6 ( − 1 ) = − 6

Incorrect approach (do not use):

− 4 ⋅ − 9 ≠ ( − 4 ) ( − 9 ) = 36 = 6

The correct answer is − 6 , not + 6 .

Example 3. Simplify − 2 ⋅ − 8 .

Solution.

− 2 ⋅ − 8 = ( i 2 ) ( i 8 ) = i 2 16 = − 1 ⋅ 4 = − 4

Alternatively, − 2 ⋅ − 8 = − 2 ⋅ 8 = − 16 = − 4 .

Simplifying Expressions with − a

When simplifying expressions involving square roots of negative numbers, always convert to the i a form first, then carry out arithmetic on i as usual.

Example 4. Compute ( 3 + − 4 ) ( 1 − − 9 ) .

Solution. Convert first: − 4 = 2 i and − 9 = 3 i .

\begin{aligned} (3 + 2i)(1 - 3i) &= 3 - 9i + 2i - 6i^2 \\ &= 3 - 7i - 6(-1) \\ &= 9 - 7i \end{aligned}

Frequently Asked Questions

What is − 1 equal to?

− 1 = i , the imaginary unit. It is defined as the principal square root of − 1 , meaning the one that we designate as positive (by convention). The other square root of − 1 is − i , since ( − i ) 2 = i 2 = − 1 .


Why can't I use x ⋅ y = x y when x and y are negative? The identity x ⋅ y = x y is only valid for non-negative real numbers. It relies on the principal square root being a well-defined non-negative number, which breaks down for negative inputs. Using it with negative numbers leads to contradictions like 1 = 1 = ( − 1 ) ( − 1 ) = − 1 ⋅ − 1 = i ⋅ i = − 1 .

Is − a always equal to i a when a > 0 ? Yes. By definition, the principal square root − a is the value i a (not − i a ). This convention ensures a consistent choice among the two square roots of − a .

How do I simplify − 12 ? Factor out the negative sign and then simplify: − 12 = i 12 = i ⋅ 2 3 = 2 i 3 .
Quadratic Equations and the Fundamental Theorem of AlgebraSquare Roots of a Complex Number
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