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THE MEANING OF MULTIPLICATION Mental Arithmetic Lesson 8 Section 2 The distributive property |
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In each case, we "distributed" 2 to both 100 and 5. The distributive property is the most important property of multiplication. It is a theorem that we can prove. (That is the meaning of the word theorem: it is a statement that we can prove.) |
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We expanded 24 into 20 + 4, and then "distributed" 3 to each one. Example 2. Multiply 5 × 37 mentally. Technique. Expand 37 mentally into 30 + 7. Then distribute 5:
Multiply the numbers as you read them, from left to right. The last number you say is the answer. Example 3. Multiply mentally 8 × 46. Say only, "320 + 48 = 368."
Example 4. 800 × 460 Ignore the final 0's and multiply 8 × 46. But we just saw that 8 × 46 = 368. Therefore 800 × 460 = 368,000. This is "368" with three 0's. Example 5. Multiply 6 × 7.30. (Treat problems with decimal points as dollars and cents.) Technique. Expand 7.30 mentally into 7 + .30 Then
Example 6. What is the price of five items that cost $3.25 each? Answer. Since 4 × $.25 = $1.00, then 5 × $.25 = $1.25. Say, "5 × 3.25 = 15 + 1.25 = 16.25" Example 7. Multiply 2 × 438 mentally.
The point is to say each partial sum. Say, "860 + 16 is 876." Example 8. Multiply 4 × 709.
Note: Any number times 0, or 0 times any number, is 0. Therefore, to calculate 4 × 709, simply ignore the 0 and say: "2800 + 36 = 2836." Example 9. Multiply 8,000 × 4,310. Technique. Ignore the final 0's:
Now replace the four 0's: 8,000 × 4,310 = 34,480,000 Example 10. How much is 20% of $68? Solution. 10% of $68 is $6.80. (Lesson 3.) Therefore, 20% is 2 × $6.80 = $12 + $1.60 = $13.60. Example 11. How many hours are there in one week? How many minutes are there? Solution. There are 24 hours in one day, and there are 7 days in a week. Therefore, 7 × 24 = 140 + 28 = 168 hours. Now, in each hour there are 60 minutes. To multiply 60 × 168, ignore the 0 and multiply
Now replace the 0 that we ignored: 10080. In one week, then, there are 10,080 minutes.
Two theorems A theorem is a statement that can be proved. One theorem is the distributive property of multiplication. And from that follows the order property. In algebra, distribution is an axiom. But in arithmetic we can define multiplication, which algebra cannot, and therefore we can prove it as a theorem. Here is an example of the distributive property: 3 × (20 + 4) = 3 × 20 + 3 × 4. Or, the other way around: 3 × 20 + 3 × 4 = 3 × (20 + 4). When we look at it that way, it is the theorem of Adding the Same Multiple (Euclid, V. 1): Three 20's + Three 4's = Three 24's.
Three 20's are the same multiple of 20 that Three 4's are of 4, namely the third; and Three 24's are also the third multiple of (20 + 4). To see this, look:
From this theorem we can prove the order property of multiplication (Euclid, VII. 16):
In other words, exchanging the multiplier and the multiplicand does not change the product. We will show, for example, that 3 × 24 = 24 × 3. That is, when we add 24 three times, we get the same number as when we add 3 twenty-four times: 24 + 24 + 24 = 3 + 3 + 3 + . . . + 3. We have:
At this point, please "turn" the page and do some Problems. or Continue on to Section 3: Multiplying by rounding off Introduction | Home | Table of Contents Please make a donation to keep TheMathPage online. Copyright © 2001-2007 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |
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