Square Root 1 to 25 — Values, Chart & Examples

Square Root 1 to 25 — Values, Chart & Examples

TL;DR

The square root 1 to 25 chart gives 1\sqrt{1}1​ through 25\sqrt{25}25​, where five values (1,4,9,16,25\sqrt{1}, \sqrt{4}, \sqrt{9}, \sqrt{16}, \sqrt{25}1​,4​,9​,16​,25​) are exact whole numbers and the other twenty are irrational, rounded here to three decimals. This article gives the full table, marks rational versus irrational, and shows how to read and rebuild it.

Square Root 1 to 25 Chart

The value of the square root of a number nnn is the number that, multiplied by itself, gives nnn. From 1 to 25 only five inputs land on a clean whole number; the rest are non-terminating decimals. Here is the complete chart, with perfect squares shown exactly and every other value rounded to three decimal places.

Number Square Root Number Square Root Number Square Root
1\sqrt{1}1​ 111 (exact) 10\sqrt{10}10​ 3.162 19\sqrt{19}19​ 4.359
2\sqrt{2}2​ 1.414 11\sqrt{11}11​ 3.317 20\sqrt{20}20​ 4.472
3\sqrt{3}3​ 1.732 12\sqrt{12}12​ 3.464 21\sqrt{21}21​ 4.583
4\sqrt{4}4​ 222 (exact) 13\sqrt{13}13​ 3.606 22\sqrt{22}22​ 4.690
5\sqrt{5}5​ 2.236 14\sqrt{14}14​ 3.742 23\sqrt{23}23​ 4.796
6\sqrt{6}6​ 2.449 15\sqrt{15}15​ 3.873 24\sqrt{24}24​ 4.899
7\sqrt{7}7​ 2.646 16\sqrt{16}16​ 444 (exact) 25\sqrt{25}25​ 555 (exact)
8\sqrt{8}8​ 2.828 17\sqrt{17}17​ 4.123
9\sqrt{9}9​ 333 (exact) 18\sqrt{18}18​ 4.243

Which Square Roots From 1 to 25 Are Rational

A rational number can be written as a fraction of two integers; an irrational number cannot, and its decimal never terminates or repeats. Only the perfect squares in this range give rational roots.

Rational (perfect squares): 1, 4, 9, 16, 25\sqrt{1}, \sqrt{4}, \sqrt{9}, \sqrt{16}, \sqrt{25}1​, 4​, 9​, 16​, 25​ (five values).

Irrational (everything else): 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7}, \sqrt{8}, \sqrt{10}, \sqrt{11}, \sqrt{12}, \sqrt{13}, \sqrt{14}, \sqrt{15}, \sqrt{17}, \sqrt{18}, \sqrt{19}, \sqrt{20}, \sqrt{21}, \sqrt{22}, \sqrt{23}, \sqrt{24}2​, 3​, 5​, 6​, 7​, 8​, 10​, 11​, 12​, 13​, 14​, 15​, 17​, 18​, 19​, 20​, 21​, 22​, 23​, 24​ (twenty values).

So exactly 525\tfrac{5}{25}255​, or one in five, of the roots from 1 to 25 are rational. That fraction is worth internalising: perfect squares are rare, and the gaps between them grow as numbers climb.

Where Square Roots From 1 to 25 Appear

These small square roots show up constantly. 2≈1.414\sqrt{2} \approx 1.414 is the diagonal of a unit square, so it lives in every set square and every 45° cut a carpenter makes. 5≈2.236\sqrt{5} \approx 2.236 is the diagonal of a 1×21 \times 21×2 rectangle, and it hides inside the golden ratio 1+52\tfrac{1+\sqrt{5}}{2}21+5​​. Screen and paper sizes, standard-deviation calculations, and the distance formula in coordinate geometry all lean on roots in this range.

How to Read and Use the Table

Read each entry as a question and an answer: 7=2.646\sqrt{7} = 2.646 means "the number whose square is 7 is about 2.646." Check it and 2.6462=7.0012.646^2 = 7.0012.6462=7.001, close enough for the three-decimal rounding.

To use the chart for estimation, locate your number between two perfect squares. For 7\sqrt{7}7​, note that 4<7<94 < 7 < 94<7<9, so the answer sits between 4=2\sqrt{4}=24​=2 and 9=3\sqrt{9}=39​=3. Because 7 is closer to 9, the root leans toward 3, which matches 2.646.

The point of the chart is not to memorise twenty decimals. It is to see the structure: exact roots at 1, 4, 9, 16, 25, and a predictable climb in between that you can reconstruct with the perfect-square anchors on either side.

How to Compute Square Roots From 1 to 25

Method 1: Perfect-square recognition

For 16\sqrt{16}16​, ask which whole number times itself is 16. 4×4=164 \times 4 = 164×4=16 So 16=4\sqrt{16} = 416​=4.

The same works for 1, 4, 9, and 25. No decimals needed.

Method 2: Estimation between anchors (for non-perfect squares)

Take 12\sqrt{12}12​. Find the nearest perfect squares below and above: 999 and 161616. So 3<12<43 < \sqrt{12} < 43<12​<4. Since 121212 is closer to 999 than to 161616, estimate near 3.43.43.4. Refine: 3.42=11.563.4^2 = 11.563.42=11.56 and 3.52=12.253.5^2 = 12.253.52=12.25, so the root is between them, near 3.463.463.46. Final answer: 12≈3.464\sqrt{12} \approx 3.46412​≈3.464.

Method 3: Long division (for a precise decimal)

Take 20\sqrt{20}20​. Pair digits from the decimal point: 20.00‾00‾20.\overline{00},\overline{00}20.0000. Largest square ≤20\le 20≤20 is 16=4216 = 4^216=42, so the first digit is 444, remainder 444. Bring down 000000 to get 400400400; double the quotient (4 → 8) and find a digit ddd with 8d×d≤4008d \times d \le 4008d×d≤400; 84×4=33684 \times 4 = 33684×4=336, so next digit is 444, remainder 646464. Continue to reach 4.4724.4724.472. Final answer: 20≈4.472\sqrt{20} \approx 4.47220​≈4.472.

Common Mistakes With Square Root 1 to 25

Mistake 1: Treating the decimal as exact

Where it slips in: Writing 2=1.414\sqrt{2} = 1.4142​=1.414 with an equals sign in an exact answer.

Don't do this: State 2=1.414\sqrt{2} = 1.4142​=1.414 as if the decimal ends.

The correct way: Keep the radical for exact work, 2\sqrt{2}2​, and use ≈1.414\approx 1.414≈1.414 only when a decimal is asked for.

Mistake 2: Confusing squares with square roots

Where it slips in: Reading a "1 to 25" chart and mixing up 25=5\sqrt{25} = 525​=5 with 252=62525^2 = 625252=625.

Don't do this: Report 9=81\sqrt{9} = 819​=81.

The correct way: The square root shrinks the number back down: 9=3\sqrt{9} = 39​=3, because 32=93^2 = 932=9.

Mistake 3: Assuming every root simplifies to a fraction

Where it slips in: Trying to write 7\sqrt{7}7​ as a neat fraction.

Don't do this: Claim 7=2610\sqrt{7} = \tfrac{26}{10}7​=1026​ because 2.62.62.6 looks close.

The correct way: 7\sqrt{7}7​ is irrational; no fraction equals it exactly. The one habit that fixes this: check whether the number is a perfect square first, and if it is not, the root is irrational.

Conclusion