Finding a square root means working backwards to the number whose square is a given number: since 3² = 9, both 3 and −3 are square roots of 9, and the symbol √9 means the positive one, 3.
Some lengths, like the side of a square of area 2, are real, yet no whole number or fraction squared gives them exactly.
Numbers like that, which cannot be written as fractions, are irrational, and the real numbers, rational and irrational together, fill the number line with no gaps.
Changing n in the picture shows where √n lies on the number line and between which two whole numbers it falls.
What a square root is
To give the side of a square as a number when only its area is known, you need an operation that undoes squaring. An area of 4 gives a side of 2 at once, but for an area of 2 no whole number or fraction squared gives 2. So that length gets a new symbol, √2, and numbers like it that cannot be written as fractions are called irrational. Rational numbers alone leave gaps on the number line; the real numbers, irrationals included, fill those gaps, so that points on the line and real numbers pair up one to one.
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The position of √n on the number line as n changes, its approximate value, and the inequality between neighbouring squares
The first picture marks √n as a point on a number line from 0 to 4, with its approximate value above it and (√n)² = n at the top. For n of 2 or more, a dashed right triangle with legs 1 and √(n − 1) shows that its hypotenuse is √n; it is a sketch with the height shrunk. At the bottom, a perfect square is noted as such; otherwise an inequality shows √n lying between two neighbouring whole numbers. The second picture zooms in near √2: the more it zooms, the more densely the rational points appear, yet none of them ever lands on √2.
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👀 Meaning of a square root
①A number whose square is n is a square root of n. A positive n has two square roots of opposite sign, and the symbol √n means the positive one.
②Numbers like 4 and 9, whose roots √4 = 2 and √9 = 3 are whole numbers, are perfect squares.
③Roots that do not come out exactly, such as √2 and √3, are irrational.
④A number is a perfect square exactly when every exponent in its prime factorisation is even.
Rational and irrational numbers
It is easy to think that a root of a number that is not a perfect square has no value, but √2 is one definite point on the number line; it simply cannot be written as a fraction or a terminating decimal. The symbol √a also means only the one of a's two square roots that is 0 or more, so the square roots of 9 are ±3 while √9 is just 3. No real number squared is negative, so √(−4) is not defined among the real numbers. Remembering that √(a²) is |a| rather than a also prevents many sign mistakes.
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Rational points growing denser as the view zooms in near √2, with √2 among them
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🔍 How the real numbers are built
①Rational numbers can be written as fractions; integers, terminating decimals and recurring decimals all belong here.
②Irrational numbers cannot be written as fractions; like √2 and π, they are non-recurring infinite decimals.
③The real numbers, rational and irrational together, fill the number line with no gaps.
Properties of square roots
Definition
for a ≥ 0, x² = a → x = ±√a
A positive number a has two square roots, √a and −√a.
Properties
(√a)² = a, √(a²) = |a|
Squaring a square root gives the original number, and the square root of a square is the absolute value.
⚠️ Care points
①√9 = 3: the symbol √ means only the positive square root.
②9 has two square roots, 3 and −3.
③√(−4) is not defined among the real numbers.
Comparing square roots
Comparing sizes
if a > b ≥ 0, then √a > √b
For numbers 0 or more, the larger the number under the root, the larger the root.
💡 How to compare
①To compare √5 and 2, write 2 = √4; since 5 > 4, √5 > 2.
②To compare 3 and √10, write 3 = √9; since 9 < 10, 3 < √10.
③For positive numbers, put everything under a root and compare the numbers inside.
Worked examples
Example 1
Find the value of √64.
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Write 64 as the square of a number.
64 = 8²
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8² = 64 and 8 is positive, so √64 = 8.
√64 = 8
▸ 8
√a is the number 0 or more whose square is a.
Example 2
Compare 3 and √8.
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Put 3 under a root: 3 = √9.
3 = √9
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9 > 8, so √9 > √8.
√9 > √8 ⇒ 3 > √8
▸ 3 > √8
For positive numbers, the larger the number under the root, the larger the root.
Exam checklist
Classifying real numbers
real numbers = rational (integers, terminating and recurring decimals) + irrational
Every point on the number line pairs with exactly one real number.
Grade-9 school exam type
What is the integer part of √10 (the greatest integer less than √10)?