| Most numbers out there in numberland | ||
| are made up of two or more smaller numbers multiplied together. | ||
| Like say, the number 12. | ||
| If we think a bit we can think of two numbers | ||
| that we can multiply together to get 12. | ||
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Like maybe 4 x 3? |
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4 x 3 = 12 |
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| It's like the 4 and the 3 were hiding inside the 12 | ||
| and they just popped out. | ||
| We can draw that like this: | ||
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| Swell, now we have a 4 and a 3. | ||
| Could we break up the 4 or the 3 like we broke up the 12? | ||
| Well 2 x 2 = 4. | ||
| That means we can break up the 4. | ||
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| OK, how about the 3? | ||
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3 x 1 = 3 |
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| Well yeah, but: | ||
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Any number x 1 = That number |
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In fact: |
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Any number x 1 x 1 x 1 x 1 x 1 x 1 x 1 = That number |
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| So we really don't get too excited about the "1 times the number" deal. | ||
| Great, are there any other whole numbers that we can multiply together to get 3? | ||
| I don't think so. | ||
| That means there's nothing that we can do with the 3. | ||
| What about those 2's? | ||
| Are there any numbers that we can multiply together to get 2? | ||
| Only the 2 times 1 thing and we already said we don't like that. | ||
| If we can't break up the numbers any more, we're done. | ||
| Numbers that we can't break up any more are called PRIME NUMBERS. | ||
| All the numbers at the bottom of the paths in the picture are prime numbers. | ||
| If we collect them all and multiply them together, | ||
| we get the number at the top. | ||
| (Unless we made a mistake that is.) | ||
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2 x 2 x 3 = 12 |
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| Remember, since we can't break up the 2's and 3 any more | ||
| they are called prime numbers. | ||
| AND, since 2 x 2 x 3 = 12: | ||
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2, 2, and 3 are the PRIME FACTORS of 12 |
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| Let's try another one ... | ||
| Example: | ||
| Break up 24 into prime factors. | ||
| OK, 24 is an even number so it has to be equal to 2 times something. | ||
| But 2 times what? | ||
| Whatever it is will be half as big as 24. | ||
| 2 x 10 = 20 so it's bigger than 10. | ||
| 2 x 11 = 22 close. | ||
| 2 x 12 = 24 BINGO! | ||
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| We know from the last problem that 2 is a prime number | ||
| so that side of the game is done. | ||
| What about the 12? | ||
| Well 12 is an even number so it is equal to 2 times something. | ||
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12 = 2 x 6 |
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| We got another 2 so that branch is done, | ||
| what about the 6? | ||
| 6 is also an even number, | ||
| so it is equal to 2 times something. | ||
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6 = 2 x 3 |
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| This time we got a 2 and a 3. | ||
| The 2 is a prime number, what about the 3? | ||
| It's a prime number to so that part is finished. | ||
| Now gather up the numbers at the bottom of the branches. | ||
| Multiply them together and you get 24 | ||
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2 x 2 x 2 x 3 = 24 |
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| So 2, 2, 2, and 3 are the Prime Factors of 24 | ||
| There are other numbers besides 2 and 3 | ||
| that can't be broken up into smaller numbers. | ||
| 5, 7, and 11 are all prime numbers, and there are a lot more. | ||
| In fact, there are more than we could ever write down. | ||
| There is one more trick to all of this. | ||
| What if you have something like -8 to split up? | ||
| What do you do with it? | ||
| Here's the deal. | ||
| We don't like to use 1 x some number in these | ||
| because you could put as many 1 times something into any of these. | ||
| But when we have a negative number, | ||
| we use: | ||
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- 1 x (the number) |
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| So the first step when you have something like - 8 is: | ||
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- 8 = - 1 x 8 |
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| Now just leave the - 1 alone and work on the 8. | ||
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So: |
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- 1 x 2 x 2 x 2 = - 8 |
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copyright 2005 Bruce Kirkpatrick |
