THE MAGIC OF NINES
Written by Espen Hænes Kristiansen, Magnus Kristiansen and Øystein Myksvoll Lande
Through the history of mathematics it has been claimed that the number nine has some mysterious properties. To Joe Public this may seem pretty absurd. Magical possibilities is something we link to David Copperfield, and not a number. There are probably several reasons why the number 9 has earned this reputation. Here we will deal with two of them. These are practical examples of what you can use the number 9 to. The examples we will work with is called "The test of nines" and "The table of nines".
"THE TABLE OF NINES"
First we will show you the table of nines (multiplication), which has this special look:
0 |
* |
9 |
0 |
11 |
* |
9 |
= |
99 |
21 |
* |
9 |
= |
189 |
||
1 |
* |
9 |
= |
9 |
12 |
* |
9 |
= |
108 |
22 |
* |
9 |
= |
198 |
|
2 |
* |
9 |
= |
18 |
13 |
* |
9 |
= |
117 |
23 |
* |
9 |
= |
207 |
|
3 |
* |
9 |
= |
27 |
14 |
* |
9 |
= |
126 |
24 |
* |
9 |
= |
216 |
|
4 |
* |
9 |
= |
36 |
15 |
* |
9 |
= |
135 |
25 |
* |
9 |
= |
225 |
|
5 |
* |
9 |
= |
45 |
16 |
* |
9 |
= |
144 |
26 |
* |
9 |
= |
234 |
|
6 |
* |
9 |
= |
54 |
17 |
* |
9 |
= |
153 |
27 |
* |
9 |
= |
243 |
|
7 |
* |
9 |
= |
63 |
18 |
* |
9 |
= |
162 |
28 |
* |
9 |
= |
252 |
|
8 |
* |
9 |
= |
72 |
19 |
* |
9 |
= |
171 |
29 |
* |
9 |
= |
261 |
|
9 |
* |
9 |
= |
81 |
20 |
* |
9 |
= |
180 |
30 |
* |
9 |
= |
270 |
|
10 |
* |
9 |
= |
90 |
When we look at this table we can see the construction of it is very simple. The last number in every number is the counting from 9-0. The next number goes from 0-9 and so on. The third number also goes from 0-9, but this time each number is used 10 times. Using this technique it is possible to find all the numbers in the table of nines without calculating them. This is what is called the "beautiful table of nines". But is this special for the number 9, what about other numbers?
Here are some other tables:
0 |
* | 2 |
= | 0 |
100 |
* | 2 |
= | 200 |
0 |
* | 5 |
= | 0 |
0 |
* | 6 |
= | 0 |
0 |
* | 7 |
= | 0 |
1 |
* | 2 |
= | 2 |
101 |
* | 2 |
= | 202 |
1 |
* | 5 |
= | 5 |
1 |
* | 6 |
= | 6 |
1 |
* | 7 |
= | 7 |
2 |
* | 2 |
= | 4 |
102 |
* | 2 |
= | 204 |
2 |
* | 5 |
= | 10 |
2 |
* | 6 |
= | 12 |
2 |
* | 7 |
= | 14 |
3 |
* | 2 |
= | 6 |
103 |
* | 2 |
= | 206 |
3 |
* | 5 |
= | 15 |
3 |
* | 6 |
= | 18 |
3 |
* | 7 |
= | 21 |
4 |
* | 2 |
= | 8 |
104 |
* | 2 |
= | 208 |
4 |
* | 5 |
= | 20 |
4 |
* | 6 |
= | 24 |
4 |
* | 7 |
= | 28 |
5 |
* | 2 |
= | 10 |
105 |
* | 2 |
= | 210 |
5 |
* | 5 |
= | 25 |
5 |
* | 6 |
= | 30 |
5 |
* | 7 |
= | 35 |
6 |
* | 2 |
= | 12 |
106 |
* | 2 |
= | 212 |
6 |
* | 5 |
= | 30 |
6 |
* | 6 |
= | 36 |
6 |
* | 7 |
= | 42 |
7 |
* | 2 |
= | 14 |
107 |
* | 2 |
= | 214 |
7 |
* | 5 |
= | 35 |
7 |
* | 6 |
= | 42 |
7 |
* | 7 |
= | 49 |
8 |
* | 2 |
= | 16 |
108 |
* | 2 |
= | 216 |
8 |
* | 5 |
= | 40 |
8 |
* | 6 |
= | 48 |
8 |
* | 7 |
= | 56 |
9 |
* | 2 |
= | 18 |
109 |
* | 2 |
= | 218 |
9 |
* | 5 |
= | 45 |
9 |
* | 6 |
= | 54 |
9 |
* | 7 |
= | 63 |
10 |
* | 2 |
= | 20 |
110 |
* | 2 |
= | 220 |
10 |
* | 5 |
= | 50 |
10 |
* | 6 |
= | 60 |
10 |
* | 7 |
= | 70 |
11 |
* | 2 |
= | 22 |
111 |
* | 2 |
= | 222 |
20 |
* | 5 |
= | 100 |
11 |
* | 6 |
= | 66 |
11 |
* | 7 |
= | 77 |
12 |
* | 2 |
= | 24 |
112 |
* | 2 |
= | 224 |
21 |
* | 5 |
= | 105 |
12 |
* | 6 |
= | 72 |
12 |
* | 7 |
= | 84 |
13 |
* | 2 |
= | 26 |
113 |
* | 2 |
= | 226 |
22 |
* | 5 |
= | 110 |
13 |
* | 6 |
= | 78 |
13 |
* | 7 |
= | 91 |
14 |
* | 2 |
= | 28 |
114 |
* | 2 |
= | 228 |
23 |
* | 5 |
= | 115 |
14 |
* | 6 |
= | 84 |
14 |
* | 7 |
= | 98 |
15 |
* | 2 |
= | 30 |
115 |
* | 2 |
= | 230 |
24 |
* | 5 |
= | 120 |
15 |
* | 6 |
= | 90 |
15 |
* | 7 |
= | 105 |
From this we understand that the table of nines is built in a special way. But still the other tables have their ways of a special construction. E.g. the table of twos. The numbers 0,2,4,6,8 makes the last line of numbers, the next is from 0-9 a. s. o. In a similar way the table of five’s is constructed.
From these results it is possible to make different conclusions. The table of nines is special, and its table is built in a very simple way. But at the same time several other numbers makes special multiplication tables. The number nine is special, but can we call it magic? In our eyes, no.
NINER TEST
This test was invented by Arabian mathematicians in the 8th century, that makes this relatively new compared to other mathematics (E.g. ancient Greece & Egypt)
The test was invented to check the result of an addition and multiplication. This made it possible to check the results without any technical instruments. Below are two examples:
Example 1 - Addition
325 + 536 = 861
We divide each number with nine and find the reminders, which are left after the dividing:
325 ÷ 9 = 36 with the reminder 1
536 ÷ 9 = 59 with the reminder 5
861 ÷ 9 = 95 with the reminder 6
If the addition is correct, should the sum of the two first reminders equal the third! We see that this is correct because: 1 + 5 = 6
Example 2 - Multiplication
11 × 12 = 132
We divide each number with nine and find the reminders, which are left after the dividing:
11 ÷ 9 = 1 with the reminder 2
12 ÷ 9 = 1 with the reminder 3
132 ÷ 9 = 14 with the reminder 6
If the multiplication is correct, should the product of the two first reminders equal the third! We see that this is correct because: 2 × 3 = 6
Above have we shown a method for checking additions and multiplications with the number nine as a basis, and like in the "table of nines" the number has some properties that seems magic, but in our opinion it’s not correct to call it magic. It’s just a practical use of the number nine!