Table of squares of natural numbers from 100. Search order for integers

* squares up to hundreds

In order not to mindlessly square all numbers according to 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. All you have to do is add a couple of zeros.
70 * 70 = 4900.
The table is marked in red.

Rule 2 (cuts off 10 numbers)

For numbers ending in 5.
To square a two-digit number ending in 5, multiply the first digit (x) by (x+1) and add “25” to the result.
75 * 75 = 7 * 8 = 56 … 25 = 5625.
The table is marked in green.

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 take an example:
43 * 43 = 1500 + 100 * 3 + (10 - 3)^2 = 1500 + 300 + 49 = 1849.
The table is 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's also quite difficult to understand. Let's take an example:
53 * 53 = 2500 + 100 * 3 + 3^2 = 2500 + 300 + 9 = 2809.
The table is 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 take an example:
93 * 93 = 8000 + 200 * 3 + (10 - 3)^2 = 8000 + 600 + 49 = 8649.
The table is marked in dark dark orange.

Rule #6 (cuts off 32 numbers)

It is necessary to memorize the squares of numbers up to 40. It sounds crazy and difficult, but in fact, up to 20, most people know the squares. 25, 30, 35 and 40 lend themselves to formulas. And only 16 pairs of numbers remain. They can already be memorized using mnemonics (which I also want to talk about later) or by any other means. Like a multiplication table :)
The table is marked in blue.

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, don't forget about the usual formula expansion of the square of the sum ( special case binomial Newton):
(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 in the household. You will not immediately remember the case when you may need the square of a number. But the ability to quickly operate with numbers, apply the appropriate rules for each of the numbers, perfectly develops the memory and "computing abilities" of your brain.

By the way, I think all Habra readers know that 64^2 = 4096, and 32^2 = 1024.
Many squares of numbers are remembered at the associative level. For example, I easily memorized 88^2 = 7744 because of the same numbers. Everyone will surely have their own characteristics.

Two unique formulas I first found in the book "13 steps to mentalism", which has little to do with mathematics. The fact is that earlier (perhaps even now) unique computing abilities were one of the numbers in stage magic: the magician told the bike about how he received superpowers and, as proof of this, instantly squares the numbers up to a hundred. The book also shows how to cube, how to subtract roots and cube roots.

If the topic quick count interesting - I will write more.
Please write comments about errors and corrections in PM, thanks in advance.

Table of squares of integers from 0 to 99.

To use the table, select the number of tens vertically, the number of units horizontally and you will see the result at the intersection. For example, 3 8 2 = 1444 .

Table of cubes of integers from 0 to 99.

To use the table, select the number of tens vertically, the number of units horizontally and you will see the result at the intersection. For example, 1 2 3 = 1728 .

Form for calculating other values:

Table square roots integers from 0 to 99 rounded to the fifth decimal place.

To use the table, select the number of tens vertically, the number of units horizontally and you will see the result at the intersection. 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.

To use the table, select the number of tens vertically, the number of units horizontally and you will see the result at the intersection. For example, 3 √ 2 8 ≈ 3,03659 .

Form for calculating other values:

Table of values trigonometric functions(sine, cosine, tangent, cotangent) standard arguments.

To use the table, select the function vertically, the value of the argument horizontally and at the intersection you will see the result. For example, sin 90° = 1 .

Form for calculating other values:

Table of reciprocals of trigonometric functions (arcsine, arccosine, arctangent, arccotangent) of standard arguments in radians.

To use the table, select the function vertically, the value of the argument horizontally and at the intersection you will see the result. For example, arccos -1 = π.

Form for calculating other values ​​(result in degrees):

Table of natural logarithms of integers from 0 to 99 rounded to the fifth decimal place.

To use the table, select the number of tens vertically, the number of units horizontally and you will see the result at the intersection. For example, ln 4 2 = 3.73767 .

Table of squares of integers from 1 to 100

Table of squares of integers from 1 to 999 and fractional numbers from 1.1 to 9.99.

The order of searching for fractional numbers:

For example, you want to find the square of the number 1.26.
Find the number 1.2 in the left vertical column, and find 6 in the upper horizontal row.
The intersection of the numbers 1,2 and 6 is the desired result: 1 ,2 6 2 = 1,5876

Integer search order:

Just 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. So 36 2 = 1296 (removed all commas).
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 multiplication of single-digit and double-digit numbers are in the first column (under 0).
2) to find a square 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 (add 00 to 3136 and remove commas). The results of these actions are also in the first column (under 0).