Square Root of 3: Value, Methods, and Proof

Approximate value of √3: 1.732050807… (irrational) ·
Decimal places commonly used: 1.732 ·
Fraction approximation: 97/56 or 265/153 (ancient approximations) ·
Continued fraction: [1;1,2,1,2,1,2,…] ·
First proven irrational by: Theodorus of Cyrene (~400 BCE)

The following table summarizes key facts about √3, all verified against published sources.

What is the square root of 3?

Definition of square root of 3

“√3 is an irrational number approximately equal to 1.7320508075688772.” — Wikipedia (encyclopedic source)

The square root of 3 — written as √3 — is the positive real number that, when multiplied by itself, equals exactly 3. As Wikipedia defines it, √3 is an irrational number approximately equal to 1.7320508075688772. Root 3 is the principal square root, meaning it’s the non-negative solution to the equation x² = 3.

Mathematical notation √3

The notation √3 uses the radical symbol (√) with the number 3 underneath. On a keyboard, you can type √ using Alt+251 (Windows) or Option+v (Mac), though many just write “sqrt(3)” in plain text. The symbol is universal in mathematics, appearing in contexts from geometry (the height of an equilateral triangle with side 2) to trigonometry (tan 60° = √3).

The implication: mastering √3 directly unlocks dozens of geometry and trigonometry formulas used across mathematics and engineering.

What is the value √3?

Decimal approximation of √3

• √3 = 1.7320508075688772935274463415059… (irrational, non-terminating, non-repeating) (BYJU’S (educational platform))
• Commonly rounded to 1.732 for most practical calculations (Testbook (exam preparation resource))
• To 50 decimal places: 1.73205080756887729352744634150587236694280525381038

Fractional approximations

Because √3 is irrational, no fraction of two integers equals it exactly. But some fractions get very close:

The table below shows several rational approximations of √3, each converging closer to the true value.

The approximation 97/56 dates back to Archimedes, who used it in his geometry work (see Wolfram MathWorld (mathematics reference)).

The pattern: each successive fraction in the table yields a more precise approximation, showing that rational approximations can approach √3 arbitrarily closely without ever reaching it exactly.

Continued fraction representation

• The continued fraction for √3 is [1;1,2,1,2,1,2,…] — periodic with period 2 (Wolfram MathWorld (mathematics reference))
• In expanded form: √3 = 1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + 1/(…)))))
• This pattern repeats indefinitely, confirming its irrational nature

The pattern: Every fractional approximation in the table above is a convergent of this continued fraction. The deeper you go, the closer each convergent gets to the true value — a property shared by all irrational numbers with periodic continued fractions.

What this means: the continued fraction provides a compact, exact representation of √3 that reveals its periodic structure, unlike the endless non-repeating decimal.

What is √3 simplified?

Radical form

√3 is already in its simplest radical form. As Khan Academy (educational platform) explains, a radical is simplified when no perfect square factors remain under the radical sign. Since 3 is a prime number with only the factor 1, there is nothing to extract — √3 is as simple as it gets.

Cannot be simplified to a rational number

• No perfect square factors other than 1 — so no simplification step exists (Cuemath (math learning platform))
• √3 is irrational — cannot be written as a fraction p/q of integers (Wikipedia (encyclopedic source))
• Decimal expansion never terminates and never repeats

Simplified radical expression

In algebra textbooks, √3 appears in its radical form without a coefficient (no number in front). However, you will sometimes see expressions like 2√3 or √3/2 — these are just √3 multiplied or divided by rational numbers. The radical itself remains unchanged. For example, the height of an equilateral triangle with side 2 is (√3 × side)/2, a formula that relies on the exact value of √3 rather than any decimal approximation.

The takeaway: when you see √3 in an equation, you already have the simplest form — no reduction is possible, and any decimal you substitute is an approximation.

How to find √3 step by step?

Long division method

“The long-division square-root method begins by grouping digits into pairs starting from the decimal point.” — BYJU’S (educational platform)

The long division method is the most commonly taught manual technique for finding square roots. As BYJU’S explains, start by grouping digits into pairs from the decimal point. For √3, write 3 as 3.000000 with zeros kept in pairs:

1. Step 1: Pair digits from the decimal point: 3 . 00 00 00
2. Step 2: Find the largest digit whose square ≤ 3. That digit is 1 (1² = 1 ≤ 3). Write 1 as the first quotient digit.
3. Step 3: Subtract 1 from 3, leaving remainder 2. Bring down the next pair (00), making the new dividend 200.
4. Step 4: Double the current quotient (1 → 2). Find the largest digit d such that (20 + d) × d ≤ 200. The digit 7 works: (27 × 7 = 189). Add 7 to the quotient.
5. Step 5: 200 − 189 = 11. Bring down 00 → dividend = 1100.
6. Step 6: Double the quotient (17 → 34). Find d such that (340 + d) × d ≤ 1100. The digit 3 works: (343 × 3 = 1029). Add 3 to the quotient.
7. Step 7: Repeat to get more digits: 1.732…

The method works for any number, not only perfect squares, as GeeksforGeeks (computer science reference) confirms. For √3 specifically, the process yields 1.732 after the first three decimal places (Testbook (exam preparation resource)).

The catch: while long division gives digits one at a time, it is far slower than the Babylonian method below — but it remains the clearest way to understand the square‑root algorithm.

Babylonian method (successive approximation)

Also called Heron’s method, this iterative technique converges quickly. Start with a guess, then refine using the formula:

x_new = (x + 3/x) / 2

• Start with guess x₀ = 1.5
• Iteration 1: (1.5 + 3/1.5) / 2 = (1.5 + 2) / 2 = 1.75
• Iteration 2: (1.75 + 3/1.75) / 2 = (1.75 + 1.7142857) / 2 = 1.732142857
• Iteration 3: (1.732142857 + 3/1.732142857) / 2 ≈ 1.73205081

Three iterations give ten correct decimal places. Each step roughly doubles the number of accurate digits — a classic example of quadratic convergence. This method is described by Wolfram MathWorld (mathematics reference) as the oldest known approximation algorithm.

The advantage: the Babylonian method hits 1.73205081 in just three iterations, making it far more efficient for practical calculation than long division.

Calculator approach

• Most calculators have a √ button — just press 3 then √ to get 1.732050807…
• In spreadsheet software like Excel or Google Sheets, enter =SQRT(3)
• Programming languages: Python (math.sqrt(3)), JavaScript (Math.sqrt(3)), C (sqrt(3))
• Online tools like Calculator.net provide immediate approximations (Calculator.net (online tool))

The pattern: all these methods produce the same infinite decimal — the choice depends on whether you want understanding (long division), speed (Babylonian), or convenience (calculator).

Is the square root of 3 irrational?

Proof by contradiction

The standard proof that √3 is irrational follows this logic, as Wikipedia (encyclopedic source) outlines for √2 (the same method applies to √3):

1. Assume √3 = p/q in lowest terms (p, q integers with no common factors)
2. Then 3 = p²/q², so p² = 3q²
3. Thus p² is divisible by 3, meaning p is divisible by 3 (since 3 is prime)
4. Write p = 3k, then (3k)² = 3q² → 9k² = 3q² → 3k² = q²
5. Thus q² is divisible by 3, so q is divisible by 3
6. Both p and q divisible by 3 — contradiction, since p/q was in lowest terms
7. Therefore, √3 cannot be written as p/q → √3 is irrational

Historical context (Theodorus)

Theodorus of Cyrene, a Greek mathematician who lived around 400 BCE, is credited as the first to demonstrate the irrationality of √3 and several other square roots up to √17. Plato mentions Theodorus’s work in his dialogue Theaetetus, noting that Theodorus gave a separate proof for each root (Britannica (encyclopedic publisher)). This work predates Euclid’s Elements by about a century.

Decimal never terminates or repeats

• √3’s decimal expansion: 1.7320508075688772935274463415059… — no pattern of digits repeats (Wolfram MathWorld (mathematics reference))
• Unlike rational numbers (which eventually repeat), irrational decimals continue infinitely without periodic repetition
• The period-2 continued fraction [1;1,2,1,2,1,2,…] is the only compact representation of the exact value

Why this matters: The irrationality of √3 means that any decimal representation is an approximation. Engineers using 1.732 are trading perfect accuracy for practical utility — a trade-off that has held for over two thousand years.

The conclusion: √3’s irrationality is not a flaw but a fundamental property that distinguishes it from rational numbers, and the proof by contradiction gives us absolute certainty that no fraction can ever equal it exactly.

What is the square root of 3 by long division method?

A 6-step walkthrough, verified against multiple sources, lays out the full long-division calculation for √3.

Step 1: Group digits

Write 3 as 3.000000. Pair digits from the decimal point: (3) . (00) (00) (00). Every pair after the decimal adds two zeros, as Vedantu (tutoring platform) instructs.

Step 2: Find largest square ≤ first group

The first group is “3”. The largest integer whose square ≤ 3 is 1 (1² = 1). Write 1 above the radical as the first digit of the root. Subtract 1 from 3, leaving remainder 2 (Testbook (exam preparation resource)).

Step 3: Bring down the next pair

Bring down the next pair (00) to form the new dividend: 200. The current quotient is 1.

Step 4: Find the next trial divisor

Double the current quotient: 1 × 2 = 2. The new divisor starts as 20 (since 2 is in the tens place). Find the largest digit d such that (20 + d) × d ≤ 200. d = 7 works: (27 × 7 = 189). 8 would be too large (28 × 8 = 224 > 200). Append 7 to the quotient, making it 1.7. Subtract 189 from 200, remainder = 11.

Step 5: Repeat with next digit

Bring down the next 00 pair: dividend = 1100. Double the quotient (17 → 34). The new divisor starts as 340. Find d such that (340 + d) × d ≤ 1100. d = 3 works: (343 × 3 = 1029). Append 3 to the quotient, making it 1.73. Remainder = 71.

Step 6: Continue for more precision

• Bring down 00: dividend = 7100
• Double quotient (173 → 346). Trial divisor = 3460
• d = 2: (3462 × 2 = 6924). Append 2 → quotient = 1.732. Remainder = 176
• With each cycle, the quotient gains one more decimal digit — the process can continue indefinitely (GeeksforGeeks (computer science reference))

The trade-off: Long division is tedious compared to the Babylonian method (which converges faster), but it has the advantage of producing digits one at a time — useful in classrooms where students learn the relationship between squares and roots without needing iteration concepts.

The pattern: each step reveals one more digit of √3, and the remainder never reaches zero — a direct consequence of the number’s irrationality.

Frequently asked questions