Mental arithmetic or ingenious calculation of the square number

　　

See the TV talk about 999*999 mental arithmetic method: First remove a 9, get 99, and then write a 8, then 8 front there are a few 9, after the write a few 0, and finally add 1, get 998001. Sensitive I look at it is certainly the law of mining from the rule of calculation. And it's not so complicated. Please look

999*999 = (1000-1) ^2 = 1000*1000-2*1000 + 1 = (1000-2) * 1000 + 1*1

So the simpler formula is: Write a 998 in front and write a 001 on the back. There are several numbers in front, and several in the back.

Don't believe, you calculate, 9999999*9999999 = 99999980000001

By analogy, 9997x9997 = 99940009

9988x9988 = 99760144

Did you find the formula?　99XX distance 10000 Assuming N, the final count is (99xx-n) "The first half" (NXN) "The second half of the part". The digits of the front and back parts are the same. As a result, there will be "strange" laws that "the larger the number, the better the calculation".

Similarly, you can calculate 99966x99966 = 9993201156 where 99932 = 99966-(100000-99966), 1156 = 34*34

Calculation principle: 99XX * 99xx= (10000-n) ^2 = (N-10000) ^2 = (10000-2n) x10000 + n*n = ((10000-n)-N) *10000+n*n

Notice that as long as the number of 9 is greater than or equal to the number of non 9, you can use this method to quickly calculate the square number

In this way, you can convert the square of the high number to the square of the low number. For high-level numbers, all you need is add and subtract and keep the digits equal.

For the two-digit square, the above rule still applies. For example 86*86 = [86-14]00 + 14*14 = 7396 This way, you need to be familiar with the square of the low number.

1. First, the square of the single digit 5 is very good calculation. Low two bits are always 25, high two bits are 10 digits x (10 digits +1)

For example, 45*45 = (4*5) 25 = 2025, 65*65 = (6*7) 25=4225

Calculation principle: (a*10 + 5) ^2 = a*a*100 + 100A + = A (a+1) *100 + 25

2. Second, the square of the number to be computed can be deduced by the square of the number that is easy to calculate: A*a = b*b + (a+b) (A-B)

The difference of squares of adjacent two numbers equals two numbers of sum: (a*10+b+1) ^2-(a*10+b) ^2 = (a*10+b) (a*10+b+1)

11*11 = 121, 12*12 = 144, 13*13 = 169, 14*14 = 169 + (13+14) = 196, 15*15 = 225, 16*16 = 225 + 31 = 256

As a result, the square of any number can be deduced as long as the addition and subtraction is mental arithmetic quickly.

　

(a*10+b) ^2-(a*10+d) ^2 = [((a*10+b) + (A*10+D)] * (b-d)

34*34 = 30*30 + 4*64 = 1156 or 34*34 = 35*35-(34+35) = 1225-69 = 1156

78*78 = 75*75 + 3*153 = 5625 + 459 = 6084 or 78*78 = 6400-2*158 = 6084

Close to 5 using X5 as a neighboring number, closer to 0 using X0 as a neighboring number.

Two-digit squared also has a singular kinky inventions (requires three-digit addition of fast mental arithmetic):

78 * 78 = (7*7) (8*8) + 2*7*8*10 = 4964 + 1120 = 6084 = (496+112) *10 + 4

? 67*67 = 3649 + 840 = 4489 = (364+84) *10 + 9

Calculation principle: (a*10+b) * (a*10+b) = (a*a*100 + b*b) + 20AB = [A*a][b*b] + 20AB

= 10*[10a*a+ (b*b-x)/10 + 2AB] + x

X is the single digit of the b*b.

This solves the problem of mental arithmetic or clever calculation of the two-digit square.

Square of three-digit number (high-number addition + two-digit square):

(100a+10b+c) ^2 = 100* (10a+b) ^2 + (10b+c) ^2 + (2ac-b*b) *100

764*764 = 76*76 *100 + 64*64 + (56-36) *100 = 577600 + 4096 + 2000 = 583696

In short, any mental arithmetic or clever calculation, in fact, there is a formula in the back support. This formula is simply a reorganization of the various items, making it easier to mental arithmetic or skillfully calculate the number of digits.

Mental arithmetic or ingenious calculation of the square number