# Properties of numbers – nine

Nine is a very nice number, especially for those, who have a hard time getting this most important of all "conquests."” mathematics - multiplication tables.

Well, you can not learn multiplication by at all 9. Why burden your memory? Enough to have 10 fingers, Place both hands on the table and raise the appropriate finger, and multiplication will complete itself, and you only have to read the result.

If e.g.. we want to multiply 9 by 3, we raise the third finger from the left side and read: the number of fingers to the left of the raised one will be tens of the product (2), and the number of fingers to the right - unity (7). if we want 7 multiply by 9, we lift the seventh finger from the left and read: 63.

Jaka szkoda – many of you will think - that it is impossible to "repack" the whole multiplication table.

Below we will also show you how to multiply on your fingers by 6, 7 i 8, a bit more complicated than the first, but still immensely simple.

Let's go back to nine. You could say, that each number consists of nine, taken the appropriate number of times and increased by the sum of the individual digits of that number.

Here are some examples:

745 = 81 • 9 + (7 + 4 + 5)

214 = 23 • 9 + (2 + 1 + 4)

84 = 8 • 9 + (8 + 4)

Any number can be written in a similar way, e.g..

68504791 = (multiple 9) + (6 + 8 + 5 + 0 + 4 + 7 + 9 + 1)

If the number is one digit with many zeros, then it is equal to the number multiplied by the number written with this number of nines, how many zeros are followed by a given digit, and even increased by the same digit; e.g:

8000 = 999 • 8 + 8
700 = 99 • 7 + 7
40= 9 • 4 + 4

Let us take a sequence of ten natural numbers

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

i pomnóżmy te liczby przez 9, pisząc iloczyny w postaci:

09, 18, 27, 36, 45, 54, 63, 72, 81, 90.

We will notice, that the first digits of these products constitute a natural sequence from 0 do 9, the second digits form a progress decreasing from 9 do 0.

A similar property can be found in any sequence of consecutive natural numbers beginning with a number ending with one. Take numbers for example:

231, 232, 233, . . 239.

When we multiply them by 9, we will get:

2079, 2088, 2097, 2106, 2115, 2124, 2133, 2142, 2151.

The last digits of these numbers constitute the natural sequence of numbers from 9 do 1, the first three digits form a series of natural numbers: 207, 208, 209 and so on.

It is easy to explain, if it weighs in, that you would multiply any integer by 9 it means the same, what to subtract this number from ten times it; e.g:

254 • 9 = 2540 – 254
7140 • 9 = 71400 – 7140

Obviously, these and similar minor observations cannot be included in the order of some extraordinary discoveries, but not everyone knows them, and they can sometimes be very useful for even the simplest numerical operations.