Fun with Math: Part 3 - Squares and Differences
While reading D's replay to "Fun with Math: Part 2", I noticed he made an interesting point.
9 is the difference between 4 squared and 5 squared. 25 - 16 = 9.
While looking at that, a simple thing occurred to me. 4 + 5 = 9.
Let's take that a bit further. What combination of numbers, when added, equals 9?
Well, obviously, 4 + 5 = 9.
But so does 3 + 6, 2 + 7, 1 + 8, and 0 + 9
Let's write those pairs down like this:
9 = {0 + 9} {1 + 8} {2 + 7} {3 + 6} {4 + 5}
As D pointed out, 9 is the 5th odd number. It is not a coincidence that 9 has 5 pairs of numbers that, when added, equal it. For example, 7 is the 4th odd number. There are also only four pairs, when added, that equal 7: 0 + 7, 1 + 6, 2 + 5, and 3 + 4.
But,let's not get sidetracked. As I mentioned:
9 = {0 + 9} {1 + 8} {2 + 7} {3 + 6} {4 + 5}
Now, rather than adding the pairs, subtract the smaller of the pair from the larger of the pair and write the result above the pair like so:
9 7 5 3 1
9 = {0 + 9} {1 + 8} {2 + 7} {3 + 6} {4 + 5}
These numbers are your multipliers.
Now, we will square each half of the pair and subtract the smaller total from the larger total.
For the multiplier of 9, it will look like this:
Multiplier: 9 (9 - 0)
9 x 9 = 81
0 x 0 = 0
81 - 0 = 81
Now because the multiplier and the base number are both 9, the relation ship is a little difficult to see. Let's look at the 7 multiplier.
Multiplier: 7 (8 - 1)
8 x 8 = 64
1 x 1 = 1
64 - 1 = 63
And the 5 multiplier:
Multiplier: 5 (7 - 2)
7 x 7 = 49
2 x 2 = 4
49 - 4 = 45
The 3 multiplier:
Multiplier: 3 (6 - 3)
6 x 6 = 36
3 x 3 = 9
36 - 9 = 27
And lastly, the 1 multiplier:
Multiplier: 1 (5 - 4)
5 x 5 = 25
4 x 4 = 16
25 - 16 = 9
The trick, now becomes obvious. The difference of the squares equals the base number multiplied by the multiplier.
For example:
9 x 1 = 5^ - 4^
9 x 3 = 6^ - 3^
9 x 5 = 7^ - 2^
9 x 7 = 8^ - 1^
9 x 9 = 9^ - 0^
Wasn't that fun? Let's try another. Let's try 12.
Which even number is 12? Let's find out. 0, 2, 4, 6, 8, 10, 12. 12 is the 7th even number.
There will be 7 pairs, when added, that equal 12. We have 0 + 12, 1 + 11, 2 + 10, 3 + 9, 4 + 8, 5 + 7, and 6 + 6.
Now, we find the multipliers:
12 10 8 6 4 2 0
12 = {0 + 12} {1 + 11} {2 + 10} {3 + 9} {4 + 8} {5 + 7} {6 + 6}
What we will notice here, is this: If the base number is odd, the multiplier is odd. It the base number is even, the multiplier is even.
Multiplier: 12 (12 - 0)
12 x 12 = 144
0 x 0 = 0
144 - 0 = 144
Multiplier: 10 (11 - 1)
11 x 11 = 121
1 x 1 = 1
121 - 1 = 120
Multiplier: 8 (10 - 2)
10 x 10 = 100
2 x 2 = 4
100 - 0 = 96
Multiplier: 6 (9 - 3)
9 x 9 = 81
3 x 3 = 9
81 - 9 = 72
Multiplier: 4 (8 - 4)
8 x 8 = 64
4 x 4 = 16
64 - 16 = 48
Multiplier: 2 (7 - 5)
7 x 7 = 49
5 x 5 = 25
49 - 25 = 24
Multiplier: 0 (6 - 6)
6 x 6 = 36
6 x 6 = 36
36 - 36 = 0
The difference of the squares equals the base number multiplied by the multiplier.
12 x 0 = 6^ - 6^
12 x 2 = 7^ - 5^
12 x 4 = 8^ - 4^
12 x 6 = 9^ - 3^
12 x 8 = 10^ - 2^
12 x 10 = 11^ - 1^
12 x 12 = 12^ - 0^
Fun stuff, huh?
9 is the difference between 4 squared and 5 squared. 25 - 16 = 9.
While looking at that, a simple thing occurred to me. 4 + 5 = 9.
Let's take that a bit further. What combination of numbers, when added, equals 9?
Well, obviously, 4 + 5 = 9.
But so does 3 + 6, 2 + 7, 1 + 8, and 0 + 9
Let's write those pairs down like this:
9 = {0 + 9} {1 + 8} {2 + 7} {3 + 6} {4 + 5}
As D pointed out, 9 is the 5th odd number. It is not a coincidence that 9 has 5 pairs of numbers that, when added, equal it. For example, 7 is the 4th odd number. There are also only four pairs, when added, that equal 7: 0 + 7, 1 + 6, 2 + 5, and 3 + 4.
But,let's not get sidetracked. As I mentioned:
9 = {0 + 9} {1 + 8} {2 + 7} {3 + 6} {4 + 5}
Now, rather than adding the pairs, subtract the smaller of the pair from the larger of the pair and write the result above the pair like so:
9 7 5 3 1
9 = {0 + 9} {1 + 8} {2 + 7} {3 + 6} {4 + 5}
These numbers are your multipliers.
Now, we will square each half of the pair and subtract the smaller total from the larger total.
For the multiplier of 9, it will look like this:
Multiplier: 9 (9 - 0)
9 x 9 = 81
0 x 0 = 0
81 - 0 = 81
Now because the multiplier and the base number are both 9, the relation ship is a little difficult to see. Let's look at the 7 multiplier.
Multiplier: 7 (8 - 1)
8 x 8 = 64
1 x 1 = 1
64 - 1 = 63
And the 5 multiplier:
Multiplier: 5 (7 - 2)
7 x 7 = 49
2 x 2 = 4
49 - 4 = 45
The 3 multiplier:
Multiplier: 3 (6 - 3)
6 x 6 = 36
3 x 3 = 9
36 - 9 = 27
And lastly, the 1 multiplier:
Multiplier: 1 (5 - 4)
5 x 5 = 25
4 x 4 = 16
25 - 16 = 9
The trick, now becomes obvious. The difference of the squares equals the base number multiplied by the multiplier.
For example:
9 x 1 = 5^ - 4^
9 x 3 = 6^ - 3^
9 x 5 = 7^ - 2^
9 x 7 = 8^ - 1^
9 x 9 = 9^ - 0^
Wasn't that fun? Let's try another. Let's try 12.
Which even number is 12? Let's find out. 0, 2, 4, 6, 8, 10, 12. 12 is the 7th even number.
There will be 7 pairs, when added, that equal 12. We have 0 + 12, 1 + 11, 2 + 10, 3 + 9, 4 + 8, 5 + 7, and 6 + 6.
Now, we find the multipliers:
12 10 8 6 4 2 0
12 = {0 + 12} {1 + 11} {2 + 10} {3 + 9} {4 + 8} {5 + 7} {6 + 6}
What we will notice here, is this: If the base number is odd, the multiplier is odd. It the base number is even, the multiplier is even.
Multiplier: 12 (12 - 0)
12 x 12 = 144
0 x 0 = 0
144 - 0 = 144
Multiplier: 10 (11 - 1)
11 x 11 = 121
1 x 1 = 1
121 - 1 = 120
Multiplier: 8 (10 - 2)
10 x 10 = 100
2 x 2 = 4
100 - 0 = 96
Multiplier: 6 (9 - 3)
9 x 9 = 81
3 x 3 = 9
81 - 9 = 72
Multiplier: 4 (8 - 4)
8 x 8 = 64
4 x 4 = 16
64 - 16 = 48
Multiplier: 2 (7 - 5)
7 x 7 = 49
5 x 5 = 25
49 - 25 = 24
Multiplier: 0 (6 - 6)
6 x 6 = 36
6 x 6 = 36
36 - 36 = 0
The difference of the squares equals the base number multiplied by the multiplier.
12 x 0 = 6^ - 6^
12 x 2 = 7^ - 5^
12 x 4 = 8^ - 4^
12 x 6 = 9^ - 3^
12 x 8 = 10^ - 2^
12 x 10 = 11^ - 1^
12 x 12 = 12^ - 0^
Fun stuff, huh?