Very Interesting Number Nine

[mathspassion]

Very Interesting Number Nine

Two Numbers (10 a+b) and (10 x+y)

For two digits numbers, numbers can be written in four ways like this


1. (10 a+b) and (10 x+y)

2. (10 a+b) and (10 y+x)

3. (10 b+a) and (10 x+y)

4. (10 b+a) and (10 y+x)

Multiple with each other

 (10 a+b)*(10 x+y) = 100 ax + 10 bx + 10 ay +by

 (10 a+b)*(10 y+x) = 100 ay + 10 by + 10 ax + x b

 (10 b+a)*(10 x+y) = 100 bx + 10 ax + 10 by + a y

 (10 b+a)*(10 y+x) = 100 by + 10 ay + 10 bx + ax

Subtract each one respectively

1.100 ax +10 bx + 10 a y + by - 100 a y - 10 by- 10 ax - x b = 90 ax + 9 bx -
90 a y – 9 by = 9(10 ax+bx-10 a y- by)
2.100 ax +10 bx + 10 ay + by – 100 bx – 10 ax- 10 by - a y = 90 ax - 90 bx
-9 a y – 9 by = 9(10 ax-10 bx- a y-by)
3.100 ax +10 bx + 10 a y + by – 100 by – 10 a y- 10 bx - ax = 99 ax – 99 by
= 9(11 ax -11 by)
4.100 a y + 10 by + 10 ax + x b -100 bx - 10 ax -10 by –a y = 99 ay -99bx =
9(11 a y-11 b x)
5.100 a y + 10 by + 10 a x + x b -100 by - 10 a y -10 b x –ax = 90 a y -90 by+
9 ax -9 x b = 9(10 a y-10 by +ax –x b)

For example

25 & 32, we can write them in four ways like 25, 32, 52 and 23and now multiple with each other like this

(25*32),(25*23),(52*32) and (52*23)
25*32=800
25*23=575
52*32=1664
52*23=1196
1664 – 1196 = 468 =4+6+8 =18 =1+8=9
1664 – 800 = 864 =8+6+4 =18 =1+8=9
1664 – 575 = 1089=1+0+8+9=18 =1+8=9
1196 – 800 = 396 =3+9+6 =18 =1+8=9
1196 – 575 = 621 =6+2+1=9
800 – 575 = 225 =2+2+5=9

 

Nine always Remain

Copyrighted

[Sriman Dutta]

Interesting.

[imatfaal]

You will note that your final factorization places a 9 on the outside of the brackets - ie your answer is divisible by nine. The digital root of a number that is divisible by nine is always nine (and vice versa)

 

This was known back with the ancients (some of these techniques were in Liber Abaci - by Leonardo Bonacci; much of this "new to Europe" technology was from India and Arabia). I think the ancient Indian mathematicians could well have known of this idea - they definitely knew of very closely related ideas.

[mathspassion]

sir whatever you trying to write i also know but i just try to prove hw it is .........

[Acme]

You will note that your final factorization places a 9 on the outside of the brackets - ie your answer is divisible by nine. The digital root of a number that is divisible by nine is always nine (and vice versa)

 

This was known back with the ancients (some of these techniques were in Liber Abaci - by Leonardo Bonacci; much of this "new to Europe" technology was from India and Arabia). I think the ancient Indian mathematicians could well have known of this idea - they definitely knew of very closely related ideas.

Indeed, Indian mathematicians did & do know and use the idea, though I can't attest to the exact era of its inception. See the Wiki on Vedic Squares.

In Indian mathematics, a Vedic square is a variation on a typical 9 × 9 multiplication table where the entry in each cell is the digital root of the product of the column and row headings i.e. the remainder when the product of the row and column headings is divided by 9 (with remainder 0 represented by 9). Numerous geometric patterns and symmetries can be observed in a Vedic square some of which can be found in traditional Islamic art. ...

Moreover, the idea extends to any base so that, for example, in base twelve the digital root of a number that is divisible by eleven is always eleven. Thirty-three₁₂ = 29 & 2+9=B, where B is the digit eleven.

[imatfaal]

Indeed, Indian mathematicians did & do know and use the idea, though I can't attest to the exact era of its inception. See the Wiki on Vedic Squares.

 

Moreover, the idea extends to any base so that, for example, in base twelve the digital root of a number that is divisible by eleven is always eleven. Thirty-three₁₂ = 29 & 2+9=B, where B is the digit eleven.

 

Indian maths was veiled in mystery and to an extent still is - heaven knows what we have lost.

 

On the bases - I did think that was the case; I was too lazy to check it out. Nice fact