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Math is like a foreign language, the symbols all stand for words and ideas. |
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| The S symbol, Greek letter capital Sigma, has a fairly involved meaning. | ||
| It tells us to sum up a bunch of terms. | ||
| It is kind of like a little computer program. | ||
| How many terms are we talkin' about? | ||
| That bit of info comes from other symbols that are above and below the S. | ||
| You might see something like: | ||
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| Maybe instead of the X you will see a Y or a k or any old letter below the S. | ||
| The important thing is that if you count from the bottom number (here a 1), | ||
| to the top number (here a 5) you get the number of terms we add together. | ||
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1, 2, 3, 4, 5 (5 terms) |
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| If we had: | ||
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| We would also have 5 terms. | ||
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103, 104, 105, 106, 107 (5 terms) |
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| Or: | ||
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| Which would be: | ||
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-4, -3, -2, -1, 0 (5 terms) |
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| WHY THIS SILLY NOTATION? | ||
| If you wanted 5 terms, why not just write "5 terms" under the S or something? | ||
| The reason is that those particular numbers: | ||
| 1, 2, 3, 4, 5 or 103, 104, 105, 106, 107 | ||
| might be needed somewhere in the problem. | ||
| Where we wrote (stuff), | ||
| we might have something like: | ||
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3X |
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| This makes the whole expression: | ||
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| Did you see that the letter under the S | ||
| matches the variable in the expression to the right? | ||
| This little group of symbols means that we have 5 terms. | ||
| Each of the terms is the number 3, raised to a power. | ||
| The power is the sequence of numbers. | ||
| This particular whole sum is: | ||
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31 + 32 + 33 + 34 + 35 = 363 |
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| Which is a whole lot different from: | ||
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| Which is: | ||
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3103 + 3104 + 3105 + 3106 + 3107 » 1.684 x 1051 |
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| Sometimes there is no variable in the statement. | ||
| We might have: | ||
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| Which is just: | ||
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4 + 4 + 4 + 4 + 4 + 4 = 24 |
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| We have a total of 6 fours here. | ||
| That's the same as 6 x 4. | ||
| so we can say that if we're "summing" a term with no variable | ||
| and the number under the S is a 1 | ||
| the answer is the number on top times the constant. | ||
| In math talk, that is: | ||
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| So: | ||
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| If we have something like: | ||
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| We get: | ||
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3(1) + 3(2) + 3(3) + 3(4) + 3(5) = 45 |
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| Remember when we did limits, and derivatives, and integrals? | ||
| The rules let us move constants to the left and deal with them later. | ||
| We can do that here too. | ||
| So: | ||
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| Also, when we worked with limits, and derivatives, and integrals | ||
| and had a bunch of terms, like: | ||
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3X2 + 4X - 5 |
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| We could work with each term by itself. | ||
| We can do that here too. | ||
| So: | ||
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| Even so, calculating this out would take a long time. | ||
| Just the first part would be: | ||
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3(12 + 22 + 32 + 42 + 52) |
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| And these things can get even more complex. | ||
| How about some shortcuts? | ||
| SURE! | ||
| If we have X to the first power: | ||
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| If we have X to the second power (X 2): | ||
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| If we have X to the third power (X 3): | ||
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| If we have some number to the X power (C X): | ||
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copyright 2005 Bruce Kirkpatrick |
