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.
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>=0Why 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.
Worked examples
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
Classify sqrt(0.49)
Write the decimal as a fraction.
0.49=49/100sqrt(49/100)=7/10
Result: sqrt(0.49)=0.7, a rational number
Common mistakes
- Writing sqrt(a+b)=sqrt(a)+sqrt(b).
- Adding a plus-or-minus sign to the principal radical itself.
- Pulling a variable from sqrt(x^2) as x instead of |x| for unrestricted real x.
Where this idea is used
- Distance, diagonal, and Pythagorean calculations.
- Exact solutions of quadratic equations.
- Simplifying formulas before numerical approximation.
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.
Can square roots of negative numbers be calculated?
Yes in the complex system, using sqrt(-a)=i sqrt(a) for positive real a under the principal-root convention.