MULTIPLICATIONwithSQUARES

In another article (The "3M" Game) I showed you how that by the use of a special table of values (the Triangular Numbers), you could multiply two numbers by just adding and subtracting numbers appropriately chosen from that table. If you haven't read that one yet, you can go there by clicking here. Don't forget to come back though!

This time we will do much the same thing, that is, multiply two numbers without using the regular multiplication algorithm. Of course, as mentioned, we will need a table to do this work. This time it is a simple table of the square numbers. There has been much mention in my website about square numbers, so I will assume you know what that means.

Below there will be a table of squares of the numbers from 1 to 100 for your convenience in understanding this lesson. Many textbooks, especially older books, often carry tables such as this one; also they have tables of square roots, cubes, and cube roots, plus tables necessary for trigonometry. That's because in the B.C. years -- Before Calculators -- we needed some help along this line to speed things up.

### The "Squares" Algorithm Explained

This algorithm takes an unexpected turn for us. We must consider two cases for our factors. Case I is when both factors are odd numbers or both are even numbers. Case II is when we have one of each, an odd and an even. Let's take Case I first.

By way of explanation, I'll use the odd numbers 123 & 27.

1. Add the numbers: 123 + 27 = 150.

2. Take half that sum: ½ × 150 = 75

3. Subtract the numbers: 123 - 27 = 96.

4. Take half that difference: ½ × 96 = 48.

5. Look up the squares in the table of 75 and 48:

752 = 5625 and 482 = 2304

6. Subtract the two squares found in Step 5.

5625 - 2304 = 3321

That's your product for 123 × 27! (Really.)

For Case II, let's use these numbers: 38 & 145.

We will do a little algebra "trick" here before proceeding to do the algorithm described above. We mentally convert 38 to the form of (37 + 1). Now using the distributive property, we have

(37 + 1) × 145 = 37 × 145 + 145

By this means we have converted our problem back to the product of two odd numbers, plus another number. Something new from something old!

Of course, we could have done it differently if we had rewritten the 145 as (144 + 1). This would give us:

38 × (144 + 1) = 38 × 144 + 38

This time we merely have to perform the algorithm with two even numbers, then add another number to their product.

### Rationale

Now I can hear you saying, "Gee, that was a lot of work just to multiply two numbers!" and "Hey, weren't you really multiplying after all in Steps 2, 4 & 5?"

To which I say, "Well, yes and no." In Steps 2 & 4, we were perhaps multiplying a little, but we could also say that we were dividing by 2, something many people can do mentally with very little effort. (On another page I show you something called Russian Peasant Multiplication which uses this idea.) So the ease of this operation justifies its usage. In Step 5, all the multiplication to get the squares has been done earlier and by "experts"; all you do is just select the numbers from the table. So you didn't really multiply here either.

The value of this activity is the same, practically speaking, as that of the 3M Game; so see the reasons given there, if you wish. Plus it is fun sometimes just to do regular things in a different way.

Extension Activity

For students who are, or have been, in Algebra I, it would be a good challenge to prove why the algorithm works all the time.

[HINT: Let a = one factor and b = the other factor. Then proceed in a logical, algebraic way.]

 Table of Squares n n2 n n2 n n2 n n2 1 1 26 676 51 2601 76 5776 2 4 27 729 52 2704 77 5929 3 9 28 784 53 2809 78 6084 4 16 29 841 54 2916 79 6241 5 25 30 900 55 3025 80 6400 6 36 31 961 56 3136 81 6561 7 49 32 1024 57 3249 82 6724 8 64 33 1089 58 3364 83 6889 9 81 34 1156 59 3481 84 7056 10 100 35 1225 60 3600 85 7225 11 121 36 1296 61 3721 86 7396 12 144 37 1369 62 3844 87 7569 13 169 38 1444 63 3969 88 7744 14 196 39 1521 64 4096 89 7921 15 225 40 1600 65 4225 90 8100 16 256 41 1681 66 4356 91 8281 17 289 42 1764 67 4489 92 8464 18 324 43 1849 68 4624 93 8649 19 361 44 1936 69 4761 94 8836 20 400 45 2025 70 4900 95 9025 21 441 46 2116 71 5041 96 9216 22 484 47 2209 72 5184 97 9409 23 529 48 2304 73 5329 98 9604 24 576 49 2401 74 5476 99 9801 25 625 50 2500 75 5625 100 10,000

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