Square Roots & Real Numbers

A square root undoes squaring: which number, squared, gives n? Like the side of a square with area 2, some lengths are never a whole number squared, yet they still exist. Numbers you cannot write as a fraction are irrational; with the rationals they form the reals that fill the number line. Here you resize a square and watch where its side lands on the line.

What is a Square Root?

n value2

👀 Square Root Intuition

①A square root is 'a number whose square is n'

②√4 = 2 (2² = 4), √9 = 3 — these are perfect squares

③√2, √3 don't simplify cleanly → irrational numbers

Rationals & Irrationals

Zoom1

🔍 Structure of Real Numbers

①Rational: expressible as a fraction (ints, terminating, repeating decimals)

②Irrational: not a fraction (√2, π — non-repeating infinite decimals)

③Reals = rationals + irrationals → fill the number line completely

Properties of Square Roots

Definition

For a ≥ 0, x² = a → x = ±√a

A positive number has two square roots: ±√a

Properties

(√a)² = a, √(a²) = |a|

Squaring a root returns a; root of a square is the absolute value

⚠️ Watch Out

①√9 = 3 (positive root only!)

②The square roots of 9 are ±3 (two values)

③√(-4) does not exist in the reals

Comparing Square Roots

Order

a > b ≥ 0 → √a > √b

On positives, √ is increasing

💡 Tips for Comparison

①√5 vs 2 → √5 vs √4 → 5 > 4 so √5 > 2

②3 vs √10 → √9 vs √10 → 9 < 10 so 3 < √10

③Compare the numbers inside the radicals!

Work It Out

Example 1

Find the value of √64.

Write 64 as a square.

64 = 8²

√(8²) = 8.

√64 = 8

▸ 8

A square root is the non-negative number whose square is the given number.

Example 2

Compare 3 and √8.

Write 3 with a radical: 3 = √9.

3 = √9

Since 9 > 8, √9 > √8.

√9 > √8 ⇒ 3 > √8

▸ 3 > √8

For positive numbers, a larger radicand means a larger root.

Exam Wrap-up

Classification of Reals

Reals = Rationals (ints + terminating/repeating) + Irrationals

Every point on the number line is a real number

Grade-9 school exam type

What is the integer part of √10 (the greatest integer less than √10)?

①2

②3

③4

④9

⑤10

▸ ② 3

Locate √10 between consecutive square roots: √9 < √10 < √16.

√9 < √10 < √16 ⇒ 3 < √10 < 4

The integer part is 3.

integer part of √10 = 3

🎯 Exam Points

①Inside √a we need a ≥ 0

②√(a²) is |a|, not a

③Perfect square test: prime factorization with all even exponents

④Irrational examples: √2, √3, π, … (non-repeating infinite decimals)

⑤Real numbers ↔ points on the line (1:1 correspondence)

Operations with Radicals

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