Instructions
Break down any two-digit number into its components, highlighting the number of units. In the number 96, the number of units is 6. Therefore, we can write: 96 = 90 + 6.
Square the first number: 90 * 90 = 8100.
Do the same with the second number: 6 * 6 = 36
Multiply the numbers together and double the result: 90 * 6 * 2 = 540 * 2 = 1080.
Add the results of the second, third and fourth steps: 8100 + 36 + 1080 = 9216. This is the result of squaring the number 96. After some practice, you will be able to quickly do the steps in your head, surprising your parents and classmates. Until you get the hang of it, write down the results of each step so you don't get confused.
To practice, square the number 74 and test yourself on the calculator. Sequence of actions: 74 = 70 + 4, 70 * 70 = 4900, 4 * 4 = 16, 70 * 4 * 2 = 560, 4900 + 16 + 560 = 5476.
Raise the number 81 to the second power. Your actions: 81 = 80 + 1, 80 * 80 = 6400, 1 * 1 = 1, 80 * 1 * 2 = 160, 6400 + 1 + 160 = 6561.
Multiply the number of tens by the next digit in the number line: 7 * 8 = 56.
Add the number 25 to the right: 5625 - the result of squaring the number 75.
For training, raise the number 95 to the second power. It ends in the number 5, so the sequence of actions is: 9 * 10 = 90, 9025 is the result.
Learn to square negative numbers: -95 squared equals 9025, as in step eleven. Likewise, -74 squared equals 5476, as in step six. This is due to the fact that two negative numbers always result in a positive number: -95 * -95 = 9025. Therefore, when squaring, you can simply ignore the minus sign.
Helpful advice
To keep your workout from getting boring, call a friend for help. Let him write a two-digit number, and you write the result of squaring this number. Then switch places.
Sources:
- Squaring a number
Some products are not knitted with a continuous fabric, but from individual squares. This is especially true for crocheting. In this case, it may be necessary to place a certain number of squares in width and height into the pattern so as to prevent serious deviations from the size. The need to calculate the size of the square may also arise if you are doing patchwork.
You will need
- Ruler
- Product pattern
Instructions
It is necessary to knit from individual squares strictly according to the pattern. Make it yourself, or translate it from a magazine and adjust it to what you need. If, when knitting a single piece of fabric, a craftswoman first selects threads and a hook, and only then calculates the pattern, then in this case it is necessary to do exactly the opposite.
Knit several squares according to the proposed pattern using different threads and hooks of different thicknesses. Steam them and measure the width and height. Using the pattern, measure the width and height of the intended part.
Divide the patterns into the sizes of different squares and see in which case you get a whole number. If you can't get a whole number in either case, choose an option that is slightly different.
If you need to know the squares for quilting, decide first what size the entire piece will be. For example, in order to make a patchwork bedspread, you need to know its length and width. Determine by what number both of these measures are divisible. A whole number of squares should fit both in length and width. This is especially important if the measurements are quite strict, and they cannot be increased or decreased.
Having calculated the size of the surface of the square that will be visible, do not forget that the fragments will have to be sewn together. Accordingly, seam allowances must be added to the calculated dimensions of the square. As a rule, they are the same on all sides. This will be the size of the square that you will cut from the scraps.
Helpful advice
In some cases, it is necessary to subtract the fastener allowance from the pattern dimensions.
Try to ensure that the whole number of squares fits in all parts of the pattern, including the sleeves and armholes.
Exponentiation is a common operation in mathematics. Difficulties arise when the zero degree appears. Not all numbers can be raised to this power, but for the rest several general rules apply.
Raising numbers to the zero power
Raising to the zero power is very common in algebra, although the very definition of the zero power requires additional clarification.
The definition of degree zero involves solving this simplest example. Any equation to the zeroth power is equal to one. This does not depend on whether the number is a whole number or a fraction, negative or positive. In this case, there is only one exception: the number zero itself, for which different rules apply.
That is, no matter what number you raise to the zero power, the result will only be one. Any series of numbers from 1 to infinity, integer, fractional, positive and negative, rational and irrational, when raised to the zero power, turns into one.
The only exception to this rule is zero itself.
Raising zero to a power
In mathematics, it is not customary to raise zero to the zero power. The point is that such an example is impossible. Raising zero to zero makes no sense. Any number can be raised to this power except zero itself.
In some examples there are cases where we have to deal with zero degrees. This happens when simplifying an expression with powers. In this case, the zero degree can be replaced by one and further solve the example without going beyond the rules of mathematical exercises.
Things get a little more complicated if, as a result of simplification, a variable or expression appears with variables to the zeroth power. In this case, an additional condition arises - the base of the degree must be made different from zero and then continue to solve the equation.
The exact square of any number, including zero, cannot end with the numbers 2, 3, 7 and 8, or an odd number of zeros. The second property of any squared natural number is that it is either divisible by 4, or when divided by 8 it leaves a remainder of 1.
There is also a property for dividing by 9 and by 3. The square of any natural number is either divisible by nine or, when divided by three, leaves a remainder of 1. These are the basic properties of the exact square of natural numbers. You can verify them with the help of simple proofs, as well as with the help of real examples.
Squaring zero is a difficult problem that is not taught in school. Zero multiplied by zero gives the same result, so the example itself is meaningless and rarely occurs in classical mathematics.
It's time to do a little math. Do you still remember how much it is if two are multiplied by two?
If anyone has forgotten, there will be four. It seems that everyone remembers and knows the multiplication table, however, I discovered a huge number of requests to Yandex like “multiplication table” or even “download multiplication table”(!). It is for this category of users, as well as for more advanced ones who are already interested in squares and powers, that I am posting all these tables. You can even download for your health! So:
Multiplication table
(integers from 1 to 20)
| ? | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 | 38 | 40 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | 39 | 42 | 45 | 48 | 51 | 54 | 57 | 60 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 | 52 | 56 | 60 | 64 | 68 | 72 | 76 | 80 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 | 85 | 90 | 95 | 100 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 | 78 | 84 | 90 | 96 | 102 | 108 | 114 | 120 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 | 91 | 98 | 105 | 112 | 119 | 126 | 133 | 140 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 | 104 | 112 | 120 | 128 | 136 | 144 | 152 | 160 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 | 117 | 126 | 135 | 144 | 153 | 162 | 171 | 180 |
| 10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 | 130 | 140 | 150 | 160 | 170 | 180 | 190 | 200 |
| 11 | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110 | 121 | 132 | 143 | 154 | 165 | 176 | 187 | 198 | 209 | 220 |
| 12 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 | 156 | 168 | 180 | 192 | 204 | 216 | 228 | 240 |
| 13 | 13 | 26 | 39 | 52 | 65 | 78 | 91 | 104 | 117 | 130 | 143 | 156 | 169 | 182 | 195 | 208 | 221 | 234 | 247 | 260 |
| 14 | 14 | 28 | 42 | 56 | 70 | 84 | 98 | 112 | 126 | 140 | 154 | 168 | 182 | 196 | 210 | 224 | 238 | 252 | 266 | 280 |
| 15 | 15 | 30 | 45 | 60 | 75 | 90 | 105 | 120 | 135 | 150 | 165 | 180 | 195 | 210 | 225 | 240 | 255 | 270 | 285 | 300 |
| 16 | 16 | 32 | 48 | 64 | 80 | 96 | 112 | 128 | 144 | 160 | 176 | 192 | 208 | 224 | 240 | 256 | 272 | 288 | 304 | 320 |
| 17 | 17 | 34 | 51 | 68 | 85 | 102 | 119 | 136 | 153 | 170 | 187 | 204 | 221 | 238 | 255 | 272 | 289 | 306 | 323 | 340 |
| 18 | 18 | 36 | 54 | 72 | 90 | 108 | 126 | 144 | 162 | 180 | 198 | 216 | 234 | 252 | 270 | 288 | 306 | 324 | 342 | 360 |
| 19 | 19 | 38 | 57 | 76 | 95 | 114 | 133 | 152 | 171 | 190 | 209 | 228 | 247 | 266 | 285 | 304 | 323 | 342 | 361 | 380 |
| 20 | 20 | 40 | 60 | 80 | 100 | 120 | 140 | 160 | 180 | 200 | 220 | 240 | 260 | 280 | 300 | 320 | 340 | 360 | 380 | 400 |
Table of squares
(integers from 1 to 100)
| 1 2 = 1
2 2 = 4 3 2 = 9 4 2 = 16 5 2 = 25 6 2 = 36 7 2 = 49 8 2 = 64 9 2 = 81 10 2 = 100 |
11 2 = 121
12 2 = 144 13 2 = 169 14 2 = 196 15 2 = 225 16 2 = 256 17 2 = 289 18 2 = 324 19 2 = 361 20 2 = 400 |
21 2 = 441
22 2 = 484 23 2 = 529 24 2 = 576 25 2 = 625 26 2 = 676 27 2 = 729 28 2 = 784 29 2 = 841 30 2 = 900 |
31 2 = 961
32 2 = 1024 33 2 = 1089 34 2 = 1156 35 2 = 1225 36 2 = 1296 37 2 = 1369 38 2 = 1444 39 2 = 1521 40 2 = 1600 |
41 2 = 1681
42 2 = 1764 43 2 = 1849 44 2 = 1936 45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304 49 2 = 2401 50 2 = 2500 |
| 51 2 = 2601
52 2 = 2704 53 2 = 2809 54 2 = 2916 55 2 = 3025 56 2 = 3136 57 2 = 3249 58 2 = 3364 59 2 = 3481 60 2 = 3600 |
61 2 = 3721
62 2 = 3844 63 2 = 3969 64 2 = 4096 65 2 = 4225 66 2 = 4356 67 2 = 4489 68 2 = 4624 69 2 = 4761 70 2 = 4900 |
71 2 = 5041
72 2 = 5184 73 2 = 5329 74 2 = 5476 75 2 = 5625 76 2 = 5776 77 2 = 5929 78 2 = 6084 79 2 = 6241 80 2 = 6400 |
81 2 = 6561
82 2 = 6724 83 2 = 6889 84 2 = 7056 85 2 = 7225 86 2 = 7396 87 2 = 7569 88 2 = 7744 89 2 = 7921 90 2 = 8100 |
91 2 = 8281
92 2 = 8464 93 2 = 8649 94 2 = 8836 95 2 = 9025 96 2 = 9216 97 2 = 9409 98 2 = 9604 99 2 = 9801 100 2 = 10000 |
Table of degrees
(integers from 1 to 10)
1 to the power:
2 to the power:
3 to the power:
4 to the power:
5 to the power:
6 to the power:
7 to the power:
7 10 = 282475249
8 to the power:
8 10 = 1073741824
9 to the power:
9 10 = 3486784401
10 to the power:
10 8 = 100000000
10 9 = 1000000000
*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.
