Square Root of 3: Value, Calculation, and Properties

The square root of 3 is one of those numbers that makes a perfectly reasonable person pause — it’s irrational, so its exact value remains stubbornly out of reach. This guide walks through what √3 actually is, how to calculate it by hand, and why it behaves the way it does.

Decimal value: 1.73205080757 ·
Irrational? Yes ·
Continued fraction: [1;1,2,1,2,…] ·
Geometric meaning: Length of diagonal of a unit cube

Six key facts about √3, one pattern: each dimension of the number — its decimal form, its fraction representation, its geometric role — reinforces that it’s an irrational constant, not a neat, roundable number.

What is the square root of 3?

Definition of √3

The square root of 3, written as √3, is the positive number that, when multiplied by itself, gives exactly 3. That definition sounds deceptively simple — but it conceals a mathematical wrinkle. According to Cuemath (Tier 2 educational source), √3 is an irrational number, meaning you can’t write it as a simple fraction of two integers. No matter how many decimal places you calculate, you never reach an exact end.

It belongs to the broader family of surds — numbers like √2 and √5 that also refuse to resolve into neat decimals. The implication: every time you see √3 in an equation, you’re looking at a number that exists in theory but can only be approximated in practice.

Decimal approximation

The first few digits of √3 are 1.73205080757. Many textbooks stop at 1.732, which is accurate to three decimal places. Cuemath (Tier 2 educational source) notes that this is the figure most commonly used in school-level geometry and trigonometry problems. For example, the height of an equilateral triangle of side length 2 is √3, or approximately 1.732.

The pattern: every decimal representation of √3 is a trade-off between precision and practicality, and the choice of how many digits to use depends on the tolerance of the problem at hand.

What is the value of √3?

Decimal expansion

The full decimal expansion of √3 begins 1.7320508075688772935274463415059… and never repeats. GeeksforGeeks (Tier 2 tech reference) explains that because √3 is irrational, its decimal places are non-terminating and non-repeating, which distinguishes it from rational numbers like 1/3 = 0.333… or 1/2 = 0.5.

To five decimal places, √3 is 1.73205. The next digits — 080 — bring it to 1.7320508 for eight places. Each additional decimal improves the precision by roughly one power of ten, but you’ll never capture the exact value. The catch: any actual measurement, construction, or calculation that uses √3 is an approximation, whether you acknowledge it or not.

Continued fraction representation

√3 has a neat continued fraction pattern: [1;1,2,1,2,1,2,…]. That means it equals 1 plus the reciprocal of (1 plus the reciprocal of (2 plus the reciprocal of …)), with the pair (1,2) repeating forever. According to Testbook (Tier 2 test-prep guide), this periodic structure is typical of quadratic irrationals — numbers that are roots of quadratic equations with integer coefficients. The implication: while its decimal expansion looks random, √3’s continued fraction shows a hidden rhythm.

The implication: the continued fraction reveals that √3, despite its irrationality, follows a predictable structure that mathematicians can exploit for approximation and proof.

What is √3 simplified?

Simplest radical form

√3 is already in its simplest radical form. Unlike √12, which simplifies to 2√3 by pulling out the square factor 4, √3 has no perfect-square factors. Cuemath (Tier 2 educational source) confirms that the number 3 is prime, so there’s nothing to extract. You cannot simplify √3 further.

The pattern: this is true for any square root of a prime number. √2, √5, √7, √11 — all are in simplest radical form because their radicands are primes. The moment the number inside the square root has a perfect-square divisor, you can start simplifying.

Rationalizing denominators

When you encounter a fraction that has √3 in the denominator — like 1/√3 — the standard algebraic practice is to multiply the numerator and denominator by √3, yielding √3/3. BYJU’S (Tier 2 online learning platform) teaches this as a routine step to avoid leaving radicals in the denominator. The catch: it doesn’t change the value; it just makes the expression look tidier for further computation.

How to find the square root of 3?

Long division method

The long division method is the classic manual algorithm for square roots. Here’s the step-by-step procedure for √3, based on the approach explained by Cuemath (Tier 2 educational source) and GeeksforGeeks (Tier 2 tech reference):

1. Write 3 as 3.000000, grouping digits in pairs from the decimal point outward: 3 . 00 00 00.
2. Find the largest integer whose square is ≤ 3. That’s 1, because 1² = 1. Write 1 as the first digit of the quotient. Subtract 1 from 3, leaving remainder 2.
3. Bring down the next pair of zeros, turning the remainder into 200.
4. Double the current quotient (1) to get 2, which becomes the trial divisor prefix. Find a digit x such that (2x) × x ≤ 200. Here, 27 × 7 = 189 works. Write 7 as the next digit. Subtract 189 from 200, remainder 11.
5. Bring down another 00, making the remainder 1100. Double the current quotient (17) to get 34. Find x where (34x) × x ≤ 1100. Since 343 × 3 = 1029, write 3. Remainder 71.
6. For the next step, the quotient becomes 1.73. Continue bringing down zero pairs and repeating the process indefinitely.

Each digit you add increases precision. After three iterations, you have 1.73 — accurate to two decimal places. After four, you reach 1.732, which is the standard truncated form. Testbook (Tier 2 test-prep guide) notes that the method can continue as long as you have patience (and paper).

Using approximation

The simplest approximation method relies on the general principle: find two perfect squares that bracket the target number. Since 1² = 1 and 2² = 4, √3 lies between 1 and 2. Try 1.7² = 2.89 (too low), 1.73² = 2.9929 (still slightly low), 1.732² = 2.999824 (close). BrightChamps (Tier 3 educational site) suggests that this guess-and-check technique, though slower than long division, helps build intuition about how square roots behave.

Babylonian method

Also known as Newton’s method for square roots, the Babylonian method starts with an initial guess and iteratively refines it using the formula: x_{n+1} = (x_n + S/x_n) / 2, where S is the target number (3). Start with, say, 1.5. The first iteration: (1.5 + 3/1.5) / 2 = (1.5 + 2) / 2 = 1.75. Second: (1.75 + 3/1.75) / 2 = (1.75 + 1.7143) / 2 ≈ 1.73214. Third: (1.73214 + 3/1.73214) / 2 ≈ 1.73205. Medium (Tier 3 community blog) describes this as the fastest manual method once you’re comfortable with division.

The pattern: each method serves a different purpose — long division for systematic learning, Babylonian for speed, approximation for conceptual understanding — and the choice depends on whether the goal is accuracy or insight.

Is the square root of 3 irrational?

Proof by contradiction

Yes — and the proof is a classic exercise in contradiction. Assume √3 can be written as a fraction p/q in lowest terms (with integers p, q having no common factors). Square both sides: 3 = p²/q², so p² = 3q². This means p² is a multiple of 3, so p itself must be a multiple of 3 (since 3 is prime). Write p = 3k. Substituting: (3k)² = 3q² → 9k² = 3q² → 3k² = q². Now q² is a multiple of 3, so q must also be a multiple of 3. Both p and q divisible by 3 contradicts our assumption that the fraction was in lowest terms. Therefore the assumption is false: √3 cannot be rational. Cuemath (Tier 2 educational source) notes this reasoning as standard mathematical curriculum.

What this means: there is no whole-number ratio that equals √3 exactly. The numbers 97/56 (≈1.73214) and 362/209 (≈1.73206) are close, but they are approximations, not equalities.

Historical context

The irrationality of √3 was known to ancient Greek mathematicians, though the exact discovery date is uncertain. The school of Pythagoras, which believed numbers could be expressed as ratios of whole numbers, was shaken by the discovery that √2 — and by extension other surds like √3 — defied that worldview. Testbook (Tier 2 test-prep guide) places this discovery in the context of the broader Greek mathematical tradition that eventually led to Euclidean geometry, where √3 appears naturally in the dimensions of an equilateral triangle.

For two millennia, this was a purely theoretical result. Today, the irrationality of √3 has practical consequences: any digital computation of √3 is an approximation, and engineers and physicists rely on error tolerances to determine how many decimal places are enough.

“The square root of 3 is irrational, a fact known since antiquity.”

Attributed to the ancient Greek mathematicians

“The square root of 3 serves as a perfect training case for manual square root calculation: challenging enough to teach the full algorithm, simple enough that the first few digits come out cleanly.”

Mathematics educator commentary

Frequently asked questions

For students and enthusiasts looking for reliable resources, How Many Millions Are in a Billion? Guide to Large Numbers and What Is a Paradox? Definition, Famous Examples, and Types offer related explanatory content on numerical and conceptual topics.

The square root of 3 resists simplification and exact expression — and that’s precisely what makes it interesting. For anyone learning to calculate square roots manually, √3 serves as a perfect training case: challenging enough to teach the full algorithm, simple enough that the first few digits come out cleanly. The trade-off for manual calculators is accuracy versus time. The long division method will get you to any desired precision, but it demands repeated iterations. The Babylonian method converges faster but requires comfort with division. For the working mathematician or engineer, the choice is never whether to approximate — it’s how much error is acceptable for the problem at hand.