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.
Resize a square and watch where its side lands on the line.
What is a Square Root?
When you know a square but cannot recover the number that was squared, you have no name for a side whose area is already given. Some lengths never match a natural number squared, yet they are still there, and those values that will not sit on a fraction are called irrational. Rationals and irrationals together are the reals that fill every gap on the number line, so a point can sit on the line even when the number under the radical is not a perfect square. Reading squaring and square root as one pair makes it clear that a side and an area undo each other.
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See where √n sits on the number line
On a number line marked from 0 to 4, a point sits at √n, a three-decimal approximation is written above it, and (√n)² = n is written higher still. When n is at least 2, a dashed right triangle in the interval from 0 to 1 is labeled with hypotenuse √n; if n is some integer squared, a perfect-square note appears below, and if not, an inequality shows n sitting between two neighboring squares. Changing n from 1 through 16 updates the point, the approximation, the dashed triangle, and that bottom line together. The other picture zooms on a neighborhood of √2, with rational marks above the line and √2 below it; raising the zoom from 1 through 10 narrows the window and packs those rational marks more tightly.
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👀 Square Root Intuition
①A square root is 'a number whose square is n'. The symbol √n is the positive root only; the square roots of n are a signed pair
②√4 = 2 (2² = 4), √9 = 3 (3² = 9) — these are perfect squares. The symbol writes 2 and 3; the roots of 4 and 9 are signed pairs
③√2, √3 don't simplify cleanly → irrational. A number is a perfect square when every prime exponent is even
Rationals & Irrationals
It is easy to decide there is no value unless the number under the radical is a natural number. A square whose side is not a natural number squared still has that length; the length simply does not land on a fraction. Writing a real square root of a negative number leaves the range used here, so the first reading is the positive root of a number at least 0 and where that root sits on the line. From the square side, two roots with opposite signs can appear, but the radical symbol itself names only the non-negative one.
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Zoom in — irrationals live between any two rationals
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🔍 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.
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Write 64 as a square.
64 = 8²
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√(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.
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Write 3 with a radical: 3 = √9.
3 = √9
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Since 9 > 8, √9 > √8.
√9 > √8 ⇒ 3 > √8
▸ 3 > √8
For positive numbers, a larger radicand means a larger root.