Number Systems
Square Roots and Irrational Numbers
The principal square root of a nonnegative real number is its nonnegative square root. Integer square roots are rational exactly when the integer's prime exponents are all even.
Reviewed
Principal roots and equation solutions
The symbol sqrt(25) means the principal, nonnegative root 5. The equation x^2=25 has two solutions, x=5 and x=-5. The radical symbol and the solution set answer different questions.
Within the real numbers, sqrt(x) requires x>=0. Negative radicands are handled by the complex number system.
The radical symbol and an equation answer different questions
The principal square root is defined to be nonnegative. Solving an equation requires every number whose square matches the target.
| Expression | Meaning | Result |
|---|---|---|
| sqrt(25) | The principal square root of 25 | 5 |
| x^2 = 25 | All real numbers whose square is 25 | x = 5 or x = -5 |
| sqrt(-25) | Principal square root in the complex system | 5i |
Worked example 1
Simplify sqrt(72)
Use the largest perfect-square factor.
72=36 x 2sqrt(72)=sqrt(36)sqrt(2)sqrt(36)=6
Result: sqrt(72)=6sqrt(2), approximately 8.48528
Simplifying a radical
Factor the radicand into a perfect square times a remaining factor. Pull the square root of the perfect square outside the radical.
The remaining radicand should have no perfect-square factor greater than 1.
sqrt(a^2 b) = |a| sqrt(b) for real a and b>=0A radical product rule needs domain care
Applying sqrt(ab) = sqrt(a)sqrt(b) to any real a and b
Over the real numbers, the familiar product rule is safe when a and b are nonnegative.
Using a = b = -1 would claim sqrt(1) = sqrt(-1)sqrt(-1), or 1 = -1 in complex arithmetic. Principal complex square roots need branch-aware rules.
Why roots can be irrational
If a positive integer is not a perfect square, its square root cannot be a ratio of integers. Its decimal expansion continues without repeating.
A simplified radical is often a more informative exact answer than a rounded decimal.
Questions about square roots and irrational numbers
Is every nonterminating square root irrational?
An irrational root has a nonterminating, nonrepeating decimal, but some displayed nonterminating decimals may only be rounded representations.
Why is sqrt(2) irrational?
A contradiction proof using lowest-term fractions shows no integer ratio can square to 2.