Square roots and negative numbers are connected in several ways. In some cases, we might need to bring complex or imaginary numbers into the discussion.
So, can a square root be negative? A square root can be negative – for example, when we take the opposite of the principal square root of a positive number. Any positive number has both positive and negative even roots (square roots, fourth roots, etc.) However, the square root of a negative number is imaginary, not negative.
Of course, if take an odd root (3rd root, 5th root, etc.) of a negative number, there will always be a negative root we can find.
In this article, we’ll talk about square roots and when they are negative. We’ll also answer some other common questions about roots and radicals.
Let’s get going.
Can A Square Root Be Negative?
A square root can be negative if we take the opposite of the principal square root of a positive number. This comes from the fact that every number (except zero) has two square roots that are opposites (negatives) of each other.
Remember that the principal square root a positive number N is the unique positive number S such that N = S2.
For example, the principal square root of 9 is +3.
Likewise, the principal square root of 16 is +4.
To get a negative square root of a positive number, we take the opposite (negative) of the principal square root.
So, -3 is the other square root of 9, since (-3)2 = (-3)(-3) = 9 (a negative times a negative is positive).
Likewise, -4 is the other square root of 16, since (-4)2 = (-4)(-4) = 16.
The table below gives examples of numbers, their principal square roots, and their negative square roots, along with the general formula for a positive number n and the two square roots.
| Number | Principal Square Root | Negative Square Root |
|---|---|---|
| 1 | +1 | -1 |
| 4 | +2 | -2 |
| 9 | +3 | -3 |
| 16 | +4 | -4 |
| 25 | +5 | -5 |
| N > 0 | +N1/2 | -N1/2 |
principal square roots, and their negative
square roots, along with the general formula
for a positive number N and the two square roots.
When we take a square root, the number under the radical is the radicand, and the index is 2. An alternative way to express the principal square root of a positive number x is with a power of ½: x1/2.
If you want the negative square root, you would use the expression –(x1/2), or in words “the negative of the square root of x”. This is not the same thing as (-x)1/2, which in words is “the square root of –x.”
If you are wondering why every positive real number has two roots, here is the explanation.
We are solving for x, the square root of N, and the key equation is:
- x = N1/2
- x2 = N [square both sides]
- x2 – N = 0 [subtract N from both sides]
- (x + N1/2)(x – N1/2) = 0 [factor as a difference of squares]
This gives us solutions of x = N1/2 and x = -N1/2. Each solution is the opposite (negative) of the other one.
Negative Radicand & Even Index (A Negative Number Inside A Square Root)
When the radicand is negative and the index is even, we get a complex number (more sepcifically, an imaginary number) as a result.
Remember that a complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit (that is, i is the square root of -1, or i2 = -1).
More specifically, a negative radicand and an index of 2 will always give us an imaginary number. An imaginary number is a specific type of complex number with a real part that is zero (that is, a = 0).
For example, the square root of -4 is (-4)1/2 = 2i, which is an imaginary number (the index of the radical is 2, and the radicand is -4).
Remember that we can take the square root of any complex number – you can learn more about how to do this in my article here.
Can A Principal Root Be Negative?
A principal square root cannot be negative. By definition, we can only take the principal square root of a positive number, and the result is defined to be the positive square root.
For example, +5 is the principal square root of 25 (since it is the positive square root). Although -5 is also a square root of 25, it is not the principal square root of 25.
Can A Cube Root Be Negative?
A cube root can be negative if the radicand (number under the radical) is also negative.
For example, the cube root of -8 is -2, since (-2)3 = (-2)(-2)(-2) = -8 (a negative raised to an odd power is always negative). Another way to express this with rational exponents is (-8)1/3 = -2.
Likewise, the cube root of -64 is -4, since (-4)3 = )-4)(-4)(-4) = -64. Another way to express this with rational exponents is (-64)1/3 = -4.
If we take the cube root of a positive number, we get a positive number, but never a negative number. For example, the cube root of 125 is +5, since 53 = 5*5*5 = 125.
Of course, there are 3 cube roots of any number: one is real, and the other two are complex (of the form a + bi, with a and b real numbers).
The table below summarizes the sign of the real cube root, based on the sign of the radicand (number we are taking the cube root of). Except for a radicand of zero, there are always two complex cube roots (which are complex conjugates).
| Sign Of Radicand | Sign Of Real Cube Root | Number Of Complex Roots |
|---|---|---|
| positive | positive | 2 |
| zero | zero | none |
| negative | negative | 2 |
cube root, based on the sign of the radicand.
Can A Fourth Root Be Negative?
The fourth root of a positive number can be negative, just like a square root of a positive number can be negative. This would involve taking the negative of the principal (positive) fourth root.
For example, the principal fourth root of 16 is 2, since 24 = 16.
Then -2 is also a fourth root of 16, since (-2)4 = (-2)(-2)(-2)(-2) = 16 (a negative raised to an even power is always positive).
Likewise, the principal fourth root of 81 is 3, since 34 = 81.
Then -3 is also a fourth root of 81, since (-3)4 = (-3)(-3)(-3)(-3) = 81 (a negative raised to an even power is always positive).
Of course, a positive number has 4 fourth roots: two real (one positive, one negative) and two complex (they are complex conjugates).
One way to see this is to take the square roots (positive and negative) of the number. Then, take the two square roots of each of these numbers.
This would give us a positive and negative (the two square roots of the positive square root) and two complex numbers (the two square roots of the negative square root).
For example, let’s find the 4 fourth roots of 16.
First, take the two square roots of 16: +4 and -4.
Next, take the two square roots of +4: +2 and -2.
Then, take the two square roots of -4: +2i and -2i.
So, our 4 fourth roots of 16 are +2, -2, +2i, and -2i.
Another way to see this is by forming a polynomial (a quartic, which as degree 4) and then factoring completely to find the zeros.
Using the same example, let’s say we want the fourth root of 16 (call it x). Then the key equation is:
- x = 161/4
- x4 = 16 [take the 4th power of both sides]
- x4 – 16 = 0 [subtract 16 from both sides]
- (x2 – 4)(x2 + 4) = 0 [factor x4 – 16 as a difference of squares]
- (x – 2)(x + 2)(x2 + 4) = 0 [factor x2 – 4 as a difference of squares]
- (x – 2)(x + 2)(x2 – (-4)) = 0 [rewrite x2 + 4 as x2 – (-4)]
- (x – 2)(x + 2)(x – 2i)(x + 2i) [factor x2 – (-4) as a difference of squares]
This gives us the 4 fourth roots of 16: +2, -2, +2i, and -2i.
*Note: the 4 fourth roots of a negative number will all be complex.
What Is The Square Root Of i? (How To Find The Square Root Of The Imaginary Unit)
We know that the square root of -1 is i, but what is the square root of i?
Let’s call the number z. Then we want to solve z = i1/2.
We know that z will be a complex number, of the form z = a + bi.
Here are the steps:
- z = i1/2 [z is the square root of i]
- z2 = i [square both sides]
- (a + bi)2 = i [since z = a + bi]
- (a + bi)(a + bi) = i
- a2 + abi + bai + b2i2 = i [use FOIL]
- a2 + 2abi +b2i2 = i [combine like terms]
- a2 + 2abi – b2 = i [i2 = -1]
- (a2 – b2) + (2ab)i = 0 + 1i [group real and imaginary parts]
From here, we can create two equations by comparing the real parts on both sides of the equation and the imaginary parts on both sides of the equation:
- Equation 1: a2 – b2 = 0 [compare real parts]
- Equation 2: 2ab = 1 [compare imaginary parts]
For equation 1, we can factor the left side as a difference of squares to get:
- (a + b)(a – b) = 0
This gives us a = b or a = -b.
For equation 2, we want to check both cases: a = b and a = -b.
First, we plug in a = b into Equation 2:
- 2(b)(b) = 1
- 2b2 = 1
- b2 = ½
- b = +(1/2)1/2 or b = -(1/2)1/2
Since a = b, we also get a = (1/2)1/2 or a = -(1/2)1/2.
So, the two square roots of i are:
z1 = (1/2)1/2 + (1/2)1/2i
z2 = -(1/2)1/2 – (1/2)1/2i
Conclusion
Now you know how square roots and negative numbers are connected. You also know a little more about radicals and higher values of the index (3rd and 4th roots) and how negative numbers come into play in those cases.
You can learn how to add, multiply, and divide square roots here.
Learn more about square roots and other radicals in denominators (and how to rationalize them) here.
You can learn more about square roots in my article here.
You can also learn all about what square roots are used for in my article here.
You can learn more about square roots and whether they can be negative in this article.
You can learn how to do square roots by hand in my article here.
You can also learn how to graph square roots in my article here.
Remember that sometimes an equation with a radical will turn out to be an equation with no solution.
I hope you found this article helpful. If so, please share it with someone who can use the information.
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~Jonathon