*squares up to hundreds
In order not to mindlessly square all the numbers using the formula, you need to simplify your task as much as possible with the following rules.
Rule 1 (cuts off 10 numbers)
For numbers ending in 0.
If a number ends in 0, multiplying it is no more difficult than a single-digit number. You just need to add a couple of zeros.
70 * 70 = 4900.
Marked in red in the table.
Rule 2 (cuts off 10 numbers)
For numbers ending in 5.
To square a two-digit number ending in 5, you need to multiply the first digit (x) by (x+1) and add “25” to the result.
75 * 75 = 7 * 8 = 56 … 25 = 5625.
Marked in green in the table.
Rule 3 (cuts off 8 numbers)
For numbers from 40 to 50.
XX * XX = 1500 + 100 * second digit + (10 - second digit)^2
Hard enough, right? Let's look at an example:
43 * 43 = 1500 + 100 * 3 + (10 - 3)^2 = 1500 + 300 + 49 = 1849.
In the table they are marked in light orange.
Rule 4 (cuts off 8 numbers)
For numbers from 50 to 60.
XX * XX = 2500 + 100 * second digit + (second digit)^2
It is also quite difficult to understand. Let's look at an example:
53 * 53 = 2500 + 100 * 3 + 3^2 = 2500 + 300 + 9 = 2809.
In the table they are marked in dark orange.
Rule 5 (cuts off 8 numbers)
For numbers from 90 to 100.
XX * XX = 8000+ 200 * second digit + (10 - second digit)^2
Similar to rule 3, but with different coefficients. Let's look at an example:
93 * 93 = 8000 + 200 * 3 + (10 - 3)^2 = 8000 + 600 + 49 = 8649.
In the table they are marked in dark dark orange.
Rule No. 6 (cuts off 32 numbers)
You need to memorize the squares of numbers up to 40. It sounds crazy and difficult, but in fact most people know the squares up to 20. 25, 30, 35 and 40 are amenable to formulas. And only 16 pairs of numbers remain. They can already be remembered using mnemonics (which I also want to talk about later) or by any other means. Like a multiplication table :)
Marked in blue in the table.
You can remember all the rules, or you can remember selectively; in any case, all numbers from 1 to 100 obey two formulas. The rules will help, without using these formulas, to quickly calculate more than 70% of the options. Here are the two formulas:
Formulas (24 digits left)
For numbers from 25 to 50
XX * XX = 100(XX - 25) + (50 - XX)^2
For example:
37 * 37 = 100(37 - 25) + (50 - 37)^2 = 1200 + 169 = 1369
For numbers from 50 to 100
XX * XX = 200(XX - 25) + (100 - XX)^2
For example:
67 * 67 = 200(67 - 50) + (100 - 67)^2 = 3400 + 1089 = 4489
Of course, do not forget about the usual formula for the expansion of the square of a sum (a special case of Newton’s binomial):
(a+b)^2 = a^2 + 2ab + b^2.
56^2 = 50^2 + 2*50*6 + 6*2 = 2500 + 600 + 36 = 3136.
Squaring may not be the most useful thing on the farm. You won’t immediately remember a case when you might need to square a number. But the ability to quickly operate with numbers and apply appropriate rules for each number perfectly develops the memory and “computing abilities” of your brain.
By the way, I think all readers of Habra know that 64^2 = 4096, and 32^2 = 1024.
Many squares of numbers are memorized at the associative level. For example, I easily remembered 88^2 = 7744 because of the same numbers. Each one will probably have their own characteristics.
I first found two unique formulas in the book “13 steps to mentalism,” which has little to do with mathematics. The fact is that previously (perhaps even now) unique computing abilities were one of the numbers in stage magic: a magician would tell a story about how he received superpowers and, as proof of this, instantly squares numbers up to a hundred. The book also shows methods of cube construction, methods of subtracting roots and cube roots.
If the topic of quick counting is interesting, I will write more.
Please write comments about errors and corrections in PM, thanks in advance.
Table of squares of integers from 1 to 100
1 2 = 1
| 21 2 = 441
| 41 2 = 1681
| 61 2 = 3721
| 81 2 = 6561
|
Table of squares of integers from 1 to 999 and fractions from 1.1 to 9.99.
The order of searching for fractional numbers:
For example, you want to find the square of 1.26.
Find the number 1.2 in the left vertical column, and find 6 in the top horizontal row.
The intersection of numbers 1,2 and 6 is the desired result: 1
,2
6
2
= 1,5876
Search order for integers:
Simply remove the comma and get the square of the desired integer.
Example 1 (for two-digit numbers): We need to find the square of the number 36.
Find the square of the number 3.6. This number is 12.96. This means 36 2 = 1296 (all commas removed).
Example 2 (for three-digit numbers): We need to find the square of the number 592.
We find the intersection of the numbers 5.9 and 2. This number is 35.0464. So, 592 2 = 350464.
Note:
1) the results of multiplying single-digit and double-digit numbers are in the first column (under 0).
2) to find the square of a three-digit number with a zero at the end, you just need to add two zeros to the square of a two-digit number. For example, 560 2 = 3136 00
(00 was added to 3136 and commas were removed). The results of these actions are also in the first column (under 0).
6 | ||||||||||
1,2 | 1,5876 | |||||||||
Table of squares of integers from 0 to 99.
| x 2 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 0 | 0 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 |
| 1 | 100 | 121 | 144 | 169 | 196 | 225 | 256 | 289 | 324 | 361 |
| 2 | 400 | 441 | 484 | 529 | 576 | 625 | 676 | 729 | 784 | 841 |
| 3 | 900 | 961 | 1024 | 1089 | 1156 | 1225 | 1296 | 1369 | 1444 | 1521 |
| 4 | 1600 | 1681 | 1764 | 1849 | 1936 | 2025 | 2116 | 2209 | 2304 | 2401 |
| 5 | 2500 | 2601 | 2704 | 2809 | 2916 | 3025 | 3136 | 3249 | 3364 | 3481 |
| 6 | 3600 | 3721 | 3844 | 3969 | 4096 | 4225 | 4356 | 4489 | 4624 | 4761 |
| 7 | 4900 | 5041 | 5184 | 5329 | 5476 | 5625 | 5776 | 5929 | 6084 | 6241 |
| 8 | 6400 | 6561 | 6724 | 6889 | 7056 | 7225 | 7396 | 7569 | 7744 | 7921 |
| 9 | 8100 | 8281 | 8464 | 8649 | 8836 | 9025 | 9216 | 9409 | 9604 | 9801 |
To use the table, select the number of tens vertically, the number of units horizontally, and at the intersection you will see the result. For example, 3 8 2 = 1444.
2
Table of cubes of integers from 0 to 99.
| x 3 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 0 | 0 | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 |
| 1 | 1000 | 1331 | 1728 | 2197 | 2744 | 3375 | 4096 | 4913 | 5832 | 6859 |
| 2 | 8000 | 9261 | 10648 | 12167 | 13824 | 15625 | 17576 | 19683 | 21952 | 24389 |
| 3 | 27000 | 29791 | 32768 | 35937 | 39304 | 42875 | 46656 | 50653 | 54872 | 59319 |
| 4 | 64000 | 68921 | 74088 | 79507 | 85184 | 91125 | 97336 | 103823 | 110592 | 117649 |
| 5 | 125000 | 132651 | 140608 | 148877 | 157464 | 166375 | 175616 | 185193 | 195112 | 205379 |
| 6 | 216000 | 226981 | 238328 | 250047 | 262144 | 274625 | 287496 | 300763 | 314432 | 328509 |
| 7 | 343000 | 357911 | 373248 | 389017 | 405224 | 421875 | 438976 | 456533 | 474552 | 493039 |
| 8 | 512000 | 531441 | 551368 | 571787 | 592704 | 614125 | 636056 | 658503 | 681472 | 704969 |
| 9 | 729000 | 753571 | 778688 | 804357 | 830584 | 857375 | 884736 | 912673 | 941192 | 970299 |
To use the table, select the number of tens vertically, the number of units horizontally, and at the intersection you will see the result. For example, 1 2 3 = 1728.
Form for calculating other values:
3
Table of square roots of integers from 0 to 99, rounded to the fifth decimal place.
| √ x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 0 | 0 | 1 | 1,41421 | 1,73205 | 2 | 2,23607 | 2,44949 | 2,64575 | 2,82843 | 3 |
| 1 | 3,16228 | 3,31662 | 3,4641 | 3,60555 | 3,74166 | 3,87298 | 4 | 4,12311 | 4,24264 | 4,3589 |
| 2 | 4,47214 | 4,58258 | 4,69042 | 4,79583 | 4,89898 | 5 | 5,09902 | 5,19615 | 5,2915 | 5,38516 |
| 3 | 5,47723 | 5,56776 | 5,65685 | 5,74456 | 5,83095 | 5,91608 | 6 | 6,08276 | 6,16441 | 6,245 |
| 4 | 6,32456 | 6,40312 | 6,48074 | 6,55744 | 6,63325 | 6,7082 | 6,78233 | 6,85565 | 6,9282 | 7 |
| 5 | 7,07107 | 7,14143 | 7,2111 | 7,28011 | 7,34847 | 7,4162 | 7,48331 | 7,54983 | 7,61577 | 7,68115 |
| 6 | 7,74597 | 7,81025 | 7,87401 | 7,93725 | 8 | 8,06226 | 8,12404 | 8,18535 | 8,24621 | 8,30662 |
| 7 | 8,3666 | 8,42615 | 8,48528 | 8,544 | 8,60233 | 8,66025 | 8,7178 | 8,77496 | 8,83176 | 8,88819 |
| 8 | 8,94427 | 9 | 9,05539 | 9,11043 | 9,16515 | 9,21954 | 9,27362 | 9,32738 | 9,38083 | 9,43398 |
| 9 | 9,48683 | 9,53939 | 9,59166 | 9,64365 | 9,69536 | 9,74679 | 9,79796 | 9,84886 | 9,89949 | 9,94987 |
To use the table, select the number of tens vertically, the number of units horizontally, and at the intersection you will see the result. For example, √ 1 0 ≈ 3,16228 .
Form for calculating other values:
√
Table of cube roots of integers from 0 to 99, rounded to the fifth decimal place.
| 3 √ x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 0 | 0 | 1 | 1,25992 | 1,44225 | 1,5874 | 1,70998 | 1,81712 | 1,91293 | 2 | 2,08008 |
| 1 | 2,15443 | 2,22398 | 2,28943 | 2,35133 | 2,41014 | 2,46621 | 2,51984 | 2,57128 | 2,62074 | 2,6684 |
| 2 | 2,71442 | 2,75892 | 2,80204 | 2,84387 | 2,8845 | 2,92402 | 2,9625 | 3 | 3,03659 | 3,07232 |
| 3 | 3,10723 | 3,14138 | 3,1748 | 3,20753 | 3,23961 | 3,27107 | 3,30193 | 3,33222 | 3,36198 | 3,39121 |
| 4 | 3,41995 | 3,44822 | 3,47603 | 3,5034 | 3,53035 | 3,55689 | 3,58305 | 3,60883 | 3,63424 | 3,65931 |
| 5 | 3,68403 | 3,70843 | 3,73251 | 3,75629 | 3,77976 | 3,80295 | 3,82586 | 3,8485 | 3,87088 | 3,893 |
| 6 | 3,91487 | 3,9365 | 3,95789 | 3,97906 | 4 | 4,02073 | 4,04124 | 4,06155 | 4,08166 | 4,10157 |
| 7 | 4,12129 | 4,14082 | 4,16017 | 4,17934 | 4,19834 | 4,21716 | 4,23582 | 4,25432 | 4,27266 | 4,29084 |
| 8 | 4,30887 | 4,32675 | 4,34448 | 4,36207 | 4,37952 | 4,39683 | 4,414 | 4,43105 | 4,44796 | 4,46475 |
| 9 | 4,4814 | 4,49794 | 4,51436 | 4,53065 | 4,54684 | 4,5629 | 4,57886 | 4,5947 | 4,61044 | 4,62607 |
To use the table, select the number of tens vertically, the number of units horizontally, and at the intersection you will see the result. For example, 3 √ 2 8 ≈ 3,03659 .
Form for calculating other values:
3 √
Table of values of trigonometric functions (sine, cosine, tangent, cotangent) of standard arguments.
| π |
| π |
| π |
| 2π |
| 3π |
To use the table, select the function vertically, the argument value horizontally, and at the intersection you will see the result. For example, sin 90° = 1.
Form for calculating other values:
sin cos tg ctg °
Table of inverse values of trigonometric functions (arcsine, arccosine, arctangent, arccotangent) of standard arguments in radians.
| arcf(x) | 0 | 1 | -1 | 1 / 2 | - 1 / 2 | √ 2 / 2 | - √ 2 / 2 | √ 3 / 2 | - √ 3 / 2 | √ 3 | -√ 3 | 1 / √ 3 | - 1 / √ 3 |
| arcsin( x) | 0 | π/2 | - π/2 | π/6 | - π/6 | π/4 | - π/4 | π/3 | - π/3 | - | - | 0.6155 | -0.6155 |
| arccos( x) | π/2 | 0 | π | π/3 | 2π/3 | π/4 | 3π/4 | π/6 | 5π/6 | - | - | 0,9553 | 2,1863 |
| arctg( x) | 0 | π/4 | - π/4 | 0.4636 | -0.4636 | 0.6155 | -0.6155 | 0.7137 | -0.7137 | π/3 | - π/3 | π/6 | - π/6 |
| arcctg( x) | π/2 | π/4 | 3π/4 | 1.1071 | 2.0344 | 0.9553 | 2.1863 | 0.8571 | 2.2845 | π/6 | 5π/6 | π/3 | 2π/3 |
To use the table, select the function vertically, the argument value horizontally, and at the intersection you will see the result. For example, arccos -1 = π.
Form for calculating other values (result in degrees):
arcsin arccos arctg °
Table of natural logarithms of integers from 0 to 99, rounded to the fifth decimal place.
| ln( x) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 0 | -INF | 0 | 0,69315 | 1,09861 | 1,38629 | 1,60944 | 1,79176 | 1,94591 | 2,07944 | 2,19722 |
| 1 | 2,30259 | 2,3979 | 2,48491 | 2,56495 | 2,63906 | 2,70805 | 2,77259 | 2,83321 | 2,89037 | 2,94444 |
| 2 | 2,99573 | 3,04452 | 3,09104 | 3,13549 | 3,17805 | 3,21888 | 3,2581 | 3,29584 | 3,3322 | 3,3673 |
| 3 | 3,4012 | 3,43399 | 3,46574 | 3,49651 | 3,52636 | 3,55535 | 3,58352 | 3,61092 | 3,63759 | 3,66356 |
| 4 | 3,68888 | 3,71357 | 3,73767 | 3,7612 | 3,78419 | 3,80666 | 3,82864 | 3,85015 | 3,8712 | 3,89182 |
| 5 | 3,91202 | 3,93183 | 3,95124 | 3,97029 | 3,98898 | 4,00733 | 4,02535 | 4,04305 | 4,06044 | 4,07754 |
| 6 | 4,09434 | 4,11087 | 4,12713 | 4,14313 | 4,15888 | 4,17439 | 4,18965 | 4,20469 | 4,21951 | 4,23411 |
| 7 | 4,2485 | 4,26268 | 4,27667 | 4,29046 | 4,30407 | 4,31749 | 4,33073 | 4,34381 | 4,35671 | 4,36945 |
| 8 | 4,38203 | 4,39445 | 4,40672 | 4,41884 | 4,43082 | 4,44265 | 4,45435 | 4,46591 | 4,47734 | 4,48864 |
| 9 | 4,49981 | 4,51086 | 4,52179 | 4,5326 | 4,54329 | 4,55388 | 4,56435 | 4,57471 | 4,58497 | 4,59512 |
To use the table, select the number of tens vertically, the number of units horizontally, and at the intersection you will see the result. For example, ln 4 2 = 3.73767.