The Babylonians who carved √2 to six decimals, 3,700 years ago.
A square plot of area 25 has side 5m. What if the area is 20?
Squaring asked in reverse — the square root.
Experiment
Hands-on experiment
Predict first — a square of area 25 has side 5m. If the area is 20, the side is…?
Squeeze the side
We hunt the number that squares to 20. Press to tighten one step at a time.
The squeeze now = the estimate now
Start squeezing and the estimate of √20 narrows with it.
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Recovering a square's side from its area is a natural question — area 25, side 5. But area 20 lands somewhere between 4 and 5, on no tidy number.
You can squeeze: 4.4²=19.36, 4.5²=20.25, 4.47²=19.98… The unending number needed a name — 'the positive number that squares to 20' became √20.
√ is the inverse of squaring. As the logarithm asks a power's 'count', the root asks its 'base' — the two inverse operations are the side doors of exponentiation.
Insight
Insights from the video
“√ is a question frozen into a symbol.”
√20 declares that 'the number squaring to 20' will be handled as a number itself. Even before any digits, you can multiply and compare √20s — the symbol carries the computation for you.
“The root is the door back from area-world to length-world.”
Wherever quantities pile up as squares — area, energy, variance — √ returns them to the original scale. That's exactly why standard deviation is the square root of variance.
Misconception
Common misconceptions
√16 + √9 = √25 (roots add).
√16+√9 = 4+3 = 7, but √25 = 5. √ does not split over addition — it's friendly only with multiplication (√a×√b=√ab). This is the single most common square-root mistake.
√25 = ±5.
Two numbers square to 25 — 5 and −5 — but the symbol √25 is defined to mean only the positive one, 5. 'The square roots of 25' (both) and '√25' (the positive one) are different phrases.
Formula
Writing it as math
What the square-plot experiment confirmed, in mathematical language.
Definition
(√a)² = a. The symbol picks the positive one — write ±√a to mean both.
Friendly with multiplication
√2×√8 = √16 = 4. Over multiplication it splits and merges freely — simplifications like √12 = 2√3 come from here.
At odds with addition
√ does not split over addition: √(a+b) ≠ √a + √b — the root (pun intended) of the most common wrong answers.
In Real Life
Where you meet it in real life
Sides from areas
A 100m² square garden has 10m sides; a 55-inch screen's true dimensions — whenever you return from area to length, √ is on duty.
Braking distance
Braking distance grows with speed squared. Reversed, investigators estimate crash speed from skid marks with a square root — v = √(255×friction×distance).
Standard deviation
Variance averages squared deviations — returning to the original units takes a root. Statistics' σ = √variance.
Distance in games and GPS
The distance formula √(Δx²+Δy²) — collision checks and GPS compute square roots every moment.
Try Yourself
Test yourself
Q1Which natural numbers lie between √49 and √64?Show answer ▾
√49 = 7 and √64 = 8 — none! Between 7 and 8 live only irrationals like √50, √51, … √63, packed tight.
Q2Simplify √12.Show answer ▾
√12 = √(4×3) = √4×√3 = 2√3. Pulling square factors out through the multiplication rule (√ab = √a√b) is what 'simplifying radicals' means.
Q3Is √5 closer to 2 or 3?Show answer ▾
2.2² = 4.84 and 2.3² = 5.29, so √5 ≈ 2.24 — much closer to 2. Squaring to squeeze builds your sense of root sizes.
Try answering yourself before revealing it — getting it wrong is where learning starts.
Connection
Concepts connect
Previous concept
Leads to next
Irrational Numbers
Squeezing √20 never ended — why these numbers escape fractions entirely is the irrationals' story.
Go to the Irrational Numbers lab →Related
Labs worth exploring together
Related lab
Powers
The root is the inverse of the power — meet the forward direction first.
Go to the Powers lab →Related lab
Irrational Numbers
Where the unending numbers like √20 and √2 reveal their identity.
Go to the Irrational Numbers lab →