Math Tables
Squares, Cubes and Roots Table
Squares and cubes come up constantly in area and volume work, and the reason they are worth tabulating is that they grow unevenly. The difference between consecutive squares is 2n + 1, so the gap widens by two every step: 1, 4, 9, 16, 25 differ by 3, 5, 7 and 9. Anyone who expects the column to rise smoothly will misjudge it somewhere between here and three figures.
Roots are the awkward direction, because most of them are irrational. Of the integers from 1 to 60 only seven are perfect squares and only three are perfect cubes, so almost every square root in the table has been rounded to four decimals rather than written exactly. That rounding is fine for measurement and wrong for algebra: the square root of 2 is 1.4142 to four places, but squaring 1.4142 gives 1.99996164, not 2.
The useful habit is to keep roots symbolic as long as possible and only convert to decimals at the final step. Intermediate rounding is the most common source of drift in multi-step hand calculation, and it shows up as an answer that is right in shape and wrong in the last digit or two.
The third table covers the sequences that appear in computing and combinatorics. Powers of two are the reason storage sizes look odd, and factorials grow faster than any exponential, which is why they are the natural language for arrangements and why they become unmanageable almost immediately.
Squares, cubes, square roots and cube roots, 1 to 30
| n | n² | n³ | √n | ∛n |
|---|---|---|---|---|
| 1 | 1 | 1 | 1.0000 | 1.0000 |
| 2 | 4 | 8 | 1.4142 | 1.2599 |
| 3 | 9 | 27 | 1.7321 | 1.4422 |
| 4 | 16 | 64 | 2.0000 | 1.5874 |
| 5 | 25 | 125 | 2.2361 | 1.7100 |
| 6 | 36 | 216 | 2.4495 | 1.8171 |
| 7 | 49 | 343 | 2.6458 | 1.9129 |
| 8 | 64 | 512 | 2.8284 | 2.0000 |
| 9 | 81 | 729 | 3.0000 | 2.0801 |
| 10 | 100 | 1,000 | 3.1623 | 2.1544 |
| 11 | 121 | 1,331 | 3.3166 | 2.2240 |
| 12 | 144 | 1,728 | 3.4641 | 2.2894 |
| 13 | 169 | 2,197 | 3.6056 | 2.3513 |
| 14 | 196 | 2,744 | 3.7417 | 2.4101 |
| 15 | 225 | 3,375 | 3.8730 | 2.4662 |
| 16 | 256 | 4,096 | 4.0000 | 2.5198 |
| 17 | 289 | 4,913 | 4.1231 | 2.5713 |
| 18 | 324 | 5,832 | 4.2426 | 2.6207 |
| 19 | 361 | 6,859 | 4.3589 | 2.6684 |
| 20 | 400 | 8,000 | 4.4721 | 2.7144 |
| 21 | 441 | 9,261 | 4.5826 | 2.7589 |
| 22 | 484 | 10,648 | 4.6904 | 2.8020 |
| 23 | 529 | 12,167 | 4.7958 | 2.8439 |
| 24 | 576 | 13,824 | 4.8990 | 2.8845 |
| 25 | 625 | 15,625 | 5.0000 | 2.9240 |
| 26 | 676 | 17,576 | 5.0990 | 2.9625 |
| 27 | 729 | 19,683 | 5.1962 | 3.0000 |
| 28 | 784 | 21,952 | 5.2915 | 3.0366 |
| 29 | 841 | 24,389 | 5.3852 | 3.0723 |
| 30 | 900 | 27,000 | 5.4772 | 3.1072 |
Only four of the first thirty integers have integer square roots — 1, 4, 9, 16 and 25 — because a square has to be the product of a whole number with itself. Everything else is irrational and can only be written to a finite number of decimals.
Squares and cubes, 31 to 60
| n | n² | n³ | √n | ∛n |
|---|---|---|---|---|
| 31 | 961 | 29,791 | 5.5678 | 3.1414 |
| 32 | 1,024 | 32,768 | 5.6569 | 3.1748 |
| 33 | 1,089 | 35,937 | 5.7446 | 3.2075 |
| 34 | 1,156 | 39,304 | 5.8310 | 3.2396 |
| 35 | 1,225 | 42,875 | 5.9161 | 3.2711 |
| 36 | 1,296 | 46,656 | 6.0000 | 3.3019 |
| 37 | 1,369 | 50,653 | 6.0828 | 3.3322 |
| 38 | 1,444 | 54,872 | 6.1644 | 3.3620 |
| 39 | 1,521 | 59,319 | 6.2450 | 3.3912 |
| 40 | 1,600 | 64,000 | 6.3246 | 3.4200 |
| 41 | 1,681 | 68,921 | 6.4031 | 3.4482 |
| 42 | 1,764 | 74,088 | 6.4807 | 3.4760 |
| 43 | 1,849 | 79,507 | 6.5574 | 3.5034 |
| 44 | 1,936 | 85,184 | 6.6332 | 3.5303 |
| 45 | 2,025 | 91,125 | 6.7082 | 3.5569 |
| 46 | 2,116 | 97,336 | 6.7823 | 3.5830 |
| 47 | 2,209 | 103,823 | 6.8557 | 3.6088 |
| 48 | 2,304 | 110,592 | 6.9282 | 3.6342 |
| 49 | 2,401 | 117,649 | 7.0000 | 3.6593 |
| 50 | 2,500 | 125,000 | 7.0711 | 3.6840 |
| 51 | 2,601 | 132,651 | 7.1414 | 3.7084 |
| 52 | 2,704 | 140,608 | 7.2111 | 3.7325 |
| 53 | 2,809 | 148,877 | 7.2801 | 3.7563 |
| 54 | 2,916 | 157,464 | 7.3485 | 3.7798 |
| 55 | 3,025 | 166,375 | 7.4162 | 3.8030 |
| 56 | 3,136 | 175,616 | 7.4833 | 3.8259 |
| 57 | 3,249 | 185,193 | 7.5498 | 3.8485 |
| 58 | 3,364 | 195,112 | 7.6158 | 3.8709 |
| 59 | 3,481 | 205,379 | 7.6811 | 3.8930 |
| 60 | 3,600 | 216,000 | 7.7460 | 3.9149 |
Thirty-six and forty-nine are the squares in this range. The gap between consecutive squares grows by two each step, which is why the column accelerates rather than rising evenly.
Powers of two and ten, and factorials
| n | 2ⁿ | 10ⁿ | n! | n² |
|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 0 |
| 1 | 2 | 10 | 1 | 1 |
| 2 | 4 | 100 | 2 | 4 |
| 3 | 8 | 1,000 | 6 | 9 |
| 4 | 16 | 10,000 | 24 | 16 |
| 5 | 32 | 100,000 | 120 | 25 |
| 6 | 64 | 1,000,000 | 720 | 36 |
| 7 | 128 | 10,000,000 | 5,040 | 49 |
| 8 | 256 | 100,000,000 | 40,320 | 64 |
| 9 | 512 | 1,000,000,000 | 362,880 | 81 |
| 10 | 1,024 | 10,000,000,000 | 3,628,800 | 100 |
| 11 | 2,048 | 100,000,000,000 | 39,916,800 | 121 |
| 12 | 4,096 | 1,000,000,000,000 | 479,001,600 | 144 |
Two to the tenth is 1,024, which is why a kilobyte is 1,024 bytes in binary usage and 1,000 in decimal packaging. The gap compounds: by the thirtieth power it is over 7%.
How to Use These Tables
- Area and volume: a square of side n has area n² and a cube of side n has volume n³. A 17 inch tile covers 289 square inches.
- Going backwards: the side of a square with area A is the square root of A. A 200 square foot space is about 14.14 feet on a side.
- Avoid intermediate rounding: keep √2 as a symbol through the calculation and convert once at the end. Squaring 1.4142 loses four parts in a million.
- Estimating a root: find the nearest squares either side. √50 lies between 7 and 8, much closer to 7, and the table gives 7.0711.
Common Questions
Why is the square root of 2 not written exactly?
Because it is irrational — its decimal expansion never terminates or repeats. Any written value is a rounding. The table gives four decimals, which is 1.4142, and squaring that gives 1.99996164 rather than 2.
How many perfect squares are there below 100?
Nine: 1, 4, 9, 16, 25, 36, 49, 64 and 81. The count of perfect squares below n is the integer part of the square root of n, which is why they thin out as numbers grow.
Why are consecutive squares 2n + 1 apart?
Because (n + 1)² − n² = 2n + 1. The gap grows by two each step, so the sequence of differences is the odd numbers: 3, 5, 7, 9 and so on.
Is a cube root the same as the power of one third?
Yes. Raising a number to the power of one third is the cube root, and to the power of one half is the square root. That is how spreadsheets and calculators usually express a root.
Why does a kilobyte sometimes mean 1,024?
Because 2¹⁰ is 1,024, and binary addressing made powers of two the natural unit. Drive manufacturers use decimal powers, where a kilobyte is 1,000 bytes, which is why a drive reports less capacity than its packaging claims.
More Math Tables
Sources
- NIST Digital Library of Mathematical Functions — powers and roots — National Institute of Standards and Technology
- NIST Special Publication 811 — prefixes for binary multiples — National Institute of Standards and Technology
- Integer sequences — squares and cubes — OEIS Foundation
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