The
Magic of Number 9
1.Finding
the Digital Roots by Casting “9”
What is Digital root?
If
we add up the digits of a number until there is only one number left
we have found what is called the digital root.
In other words, the sum of the digits of a number is called its
digital root.
Example:
For 5674, 5 + 6 + 7 + 4 = 22 and 2 + 2 =
4
»
4 is the digital root of 5674
One
use of digital roots is for divisibility tests (like 3 and 9).
This method makes it easier to calculate the digital root.
Example:
Example:
Find the digital root of 257520643
Steps:
1. 2 + 7 = 9, cross out 2 and 7.
2.4 + 3 = 9, cross out 4, 3 and 2.
3.There are no other groups of numbers adding up to 9.
4.Add up the remaining digits, 5 + 5 + 0 + 3 =
13.
5.13 is more than 9, so 1 + 3 = 4.
6.The digital root is 4.
If
there is nothing left after having cast out nines then the digital
root is 9.
2.
I do not like him, why
does he follow me?
In
the nine times table below notice that the digits of each product sum
to nine. Why does this
happen? Look at how the
digits of the product are changing each time.
I
would like to tell the class that due to some reason (Purani dushmani)
I do not like No. 9, so to get rid of him I multiply him by 5, we get
45 which is 4 + 5 = 9 then, I look skywards, roll my eyes, and say oh
oh he has come again!
Then
I say ok let me multiply him by 7.
The experience repeats.
By this time the students have caught on and want me to
multiply by 8, by 9, by 15, and so on.
3.
Inverse
Table
Write the multiplication table of 9 and
interchange the place value of every number obtained.
Observe the pattern. How
fascinating it is!
Do
you think this will work for the table 8?
Try!
4.
Snake
eats its own tail
Think
of a two digit number, say 42, then subtract the reverse of its
digits, 24, from 42
Choose
any two digits number and for each one reverse the digits and subtract
the smaller number from the larger.
Look at all the answers you get.
Do they all have a common divisor?
What do the digits sum to each time?
Some
Examples:
You
see
how fascinating and enjoying it is. In each case the difference is
divisible by 9 (i.e. the common factor is 9) and the sum of the digits
of the difference is always 9.
Do
you think this will also work for three digit number or fourdigit
number.
Try it out!
5.
Take
9 and add any number to it.
What
you have observed:
The
sum of the digits of the number added to 9 is always equal to the sum
of the digits of the result.
Take
any four digit number and try the trick.
6.
Hand
Calculator
Your
friends are amazed when you magically transform your hands into a
calculator and multiply on your fingers!
Materials: Pen
Preparation
Draw
these calculator keys on your palms with a ballpoint pen.
Presentation
Tell
your friend that she can multiply by 9 on your hands just as she would
on a regular calculator. After
she enters the numbers and pushes (=)
, just bend over the finger that is multiplied by 9.
The fingers that are standing up tell her the answer!
7.
Subtraction
Sorcery
You
ask a friend to work a subtraction problem on a calculator.
After she tell you one digit of the answer, you are able to
divulge the entire answer!
Materials
A
calculator
Paper and pencil
Finally,
ask her to tell you either the first digit or the last digit of the
answer. You are now able
to divulge the entire answer!
How to
Do It
Here
are all the possible answers when you subtract two 3digit numbers as
described.
99
198 297
396 495
594 693
792 891
(099)
Notice
that the middle digit is always 9 and that the sum of the first digit
and the last digit is 9. So
just subtract what your friend tells you from 9 to get the missing
digit.
An Exception
If
your friend tells you that the first digit or last digit is 9, her
answer will be 99.
8.
Casting
out the Nines
Casting
out the nines – by repeatedly subtracting 9 until a remainder of
less than 9 is left, or, which amounts to the same thing, dividing by
9 and noting the remainder – can be done in an oddly simple way.
The remainder when a number has been divided by 9 is the same
as the sum of the digits (or, when that sum gives a number with two
digits the sum of those digits).
As the remainder – not the number of nines – is what you
are after you can arrive at it directly.
Here are two examples:
Cast
the nines from 67 and find the remainder.
