What Is A Square Root?
The square root of a number $n$ is the value $r$ for which $r^2 = n$. So the square root of 83 is the number that, multiplied by itself, gives 83.
No whole number does this. $9^2 = 81$ is too small and $10^2 = 100$ is too big, so $\sqrt{83}$ lands between 9 and 10, much closer to 9. Every positive number has two square roots, one positive and one negative; the symbol $\sqrt{83}$ refers to the principal (positive) root, $9.1104\ldots$
Where Does √83 Appear?
$\sqrt{83}$ is the exact side length of a square whose area is 83 square units. That square sits between a $9 \times 9$ square (area 81) and a $10 \times 10$ square (area 100), so its side has to be a little more than 9, which matches $9.11$. The same value turns up any time a right-triangle or distance calculation lands on a sum of squares equal to 83, and it is the kind of tidy irrational that shows up when you invert a quadratic and the discriminant is not a perfect square
Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Rational or Irrational |
|---|---|---|
64 | 8 | Rational |
81 | 9 | Rational |
82 | 9.055 | Irrational |
83 | 9.1104 | Irrational |
84 | 9.165 | Irrational |
85 | 9.220 | Irrational |
100 | 10 | Rational |
Is The Square Root Of 83 Rational Or Irrational?
$\sqrt{83}$ is irrational. It cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal expansion neither ends nor settles into a repeating block.
The quick reason: a whole number has a rational square root only when it is a perfect square, like 81 or 100. Since 83 is not a perfect square, and in fact is prime, $\sqrt{83}$ is irrational.
Is √83 a real number? Yes. It is a positive real number that happens to be irrational. Irrational numbers still live on the number line; they simply cannot be pinned down by any fraction. So $\sqrt{83}$ has a definite position between 9 and 10 even though its decimal runs forever.
How Do You Find √83? (Long Division And Estimation)
Because 83 is prime, there is no simplification step, so the work is entirely about getting the decimal. Two methods do it.
Estimation first. 83 sits between $9^2 = 81$ and $10^2 = 100$. It is only 2 above 81 but 17 below 100, so the root is very close to 9. A first guess of about $9.1$ is already good, and squaring it ($9.1^2 = 82.81$) confirms you are almost there. For a sharper mental estimate, see our note on square root tricks.
Long division, step by step. This gives one exact digit at a time.
Pair the digits from the decimal point: $83.\overline{00},\overline{00}\ldots$
Find the largest number whose square is at most 83: $9^2 = 81$ So the first quotient digit is 9.
Subtract: $83 - 81 = 2$ Bring down a pair of zeros: $200$
Double the quotient (9) to get 18, then find a digit $d$ with $(180 + d)\times d \le 200$: $181 \times 1 = 181$ So $d = 1$, and the quotient is $9.1$.
Subtract and bring down the next pair: $200 - 181 = 19$ $1900$
Double 91 to get 182, then find $d$ with $(1820 + d)\times d \le 1900$: $1821 \times 1 = 1821$ So $d = 1$, and the quotient is $9.11$.
Continue the same way and the next digits come out as $0$ then $4$, giving: $$\sqrt{83} \approx 9.1104$$
The process never stops cleanly, which is exactly what "irrational" looks like in practice.
Examples Of √83
Example 1
Estimate $\sqrt{83}$ to the nearest whole number.
83 is between $9^2 = 81$ and $10^2 = 100$. It is far closer to 81. Nearest whole number: $9$.
Example 2
A student writes $\sqrt{83} = \sqrt{81} + \sqrt{2}$. Where does this go wrong?
The tempting move is to split 83 as $81 + 2$ and take the root of each piece: $\sqrt{81} + \sqrt{2} = 9 + 1.414 = 10.414$ Check it against the estimate. We already know $\sqrt{83}$ is just past 9, not past 10, so 10.414 is clearly too big. The break: square roots do not distribute over addition. $\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$. The rescue: leave 83 whole and compute directly, giving $\sqrt{83} \approx 9.1104$.
Example 3
Find the square of $\sqrt{83}$.
By definition, squaring undoes the square root. $(\sqrt{83})^2 = 83$ Final answer: $83$.
Example 4
Solve $x^2 = 83$.
Take the square root of both sides. $x = \pm\sqrt{83}$ $x \approx 9.1104$ or $x \approx -9.1104$ Both roots count, because a negative times a negative is also 83.
Example 5
If $\sqrt{83} \approx 9.110$, estimate $\sqrt{0.83}$.
Note that $0.83 = \dfrac{83}{100}$. $\sqrt{0.83} = \dfrac{\sqrt{83}}{\sqrt{100}} = \dfrac{9.110}{10}$ $\sqrt{0.83} \approx 0.911$ Final answer: about $0.911$.
Common Mistakes
Mistake 1: Trying to simplify √83
Where it slips in: A student assumes every root reduces to something neater, the way $\sqrt{83}$ looks like it should.
Don't do this: Writing $\sqrt{83}$ as a smaller radical times a whole number.
The correct way: To pull a factor out of a root you need a perfect-square factor. The factors of 83 are only 1 and 83 because 83 is prime, so nothing comes out. $\sqrt{83}$ is already in simplest form, a point worth revisiting alongside simplifying radical expressions.
Mistake 2: Splitting the root across addition
Where it slips in: Breaking 83 into $81 + 2$ and rooting each part. Students who lean on the "break it into friendly pieces" habit fall for this first.
Don't do this: Writing $\sqrt{83} = \sqrt{81} + \sqrt{2} = 9 + 1.414$.
The correct way: Roots distribute over multiplication, not addition. There is no valid split of 83 into a product with a perfect square, so compute $\sqrt{83}$ directly as $9.1104$.
Mistake 3: Dropping the negative root when solving an equation
Where it slips in: Solving $x^2 = 83$ and reporting only the positive answer.
Don't do this: Writing $x = 9.1104$ as the sole solution.
The correct way: $x^2 = 83$ has two solutions, $x = \pm\sqrt{83}$. The bare symbol $\sqrt{83}$ means the positive root only; the equation needs both.
Conclusion
The square root of 83 is about $9.1104$, it is irrational, and it stays as $\sqrt{83}$ in exact form because 83 is prime. Long division builds the decimal one digit at a time, and a quick comparison to 81 and 100 confirms the answer sits just past 9. To work through more roots and radicals with a teacher, explore Bhanzu's algebra tutor or its wider math classes online. You can also book a free demo class to practice the long-division method live.
Read More
Square Root of 85 — the neighbouring root, worked the same way
Square Root of 10 — a shorter irrational root with a geometric home
Squares and Square Roots — the concept behind every root article
Square Root 1 to 25 — the reference table for the small roots
Prove That Root 7 Is Irrational — the proof pattern behind √83's irrationality
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