Everybody Needs Them
Chapter 5: Multiplication and Division: Everybody Needs Them
Popular symbols include parentheses [( )], a single dot (·) called a middle dot, the times symbol (×), or even an asterisk (*). The asterisk is used mainly on computers, adding machines, and on some calculators. Here are these signs in action:
(7)(3) 7 · 3 7 × 3 7*3
All of these examples are saying “seven times three.”
Another sign of multiplication is no sign. This situation occurs in algebra when numbers and letters appear together, such as in the term 3ab. That means 3 × a × b. We cover algebraic variables more thoroughly in Chapter 12.
Sometimes multiplication is represented as a grid, so you can see the num- bers represented as rows and columns. Figure 5-1 shows you how 3 × 10 can be represented as three rows of ten columns.
Figure 5-1:
Multiplica- tion shown as a grid of objects.
Memorizing multiplication tables:
Faster than a calculator
Sources say that the Chinese invented the multiplication table. But regard- less of who came up with it, you should commit the multiplication table to memory. It’s a must-do thing, and it’s not that hard. Here’s why nailing the times table is such a worthwhile pursuit:
✓ You probably learned it in school and may just need to review.
✓ The multiplication facts involving zero and one are easy (flip to “Easy Street: Multiplying by 0, 1, and 10” later in the chapter for the skinny on these shortcuts).
✓ It only goes up to 9 × 9.
✓ You need it to do longer multiplication and division problems.
✓ It’s faster than a calculator.
Look at that last point. It’s true. You can say “seven times seven equals forty- nine” faster than you can punch the numbers into a calculator.
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Figure 5-2 shows a classic multiplication table. It’s a 9 x 9 but includes 0 as well (so technically, it’s 10 x 10). You can also find 12 x 12 and 20 x 20 tables.
Figure 5-2:
Classic 9 x 9 multiplica-
tion table.
X 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 5 8 10 12 14 16 18
3 0 3 6 9 12 15 18 21 24 27
4 0 4 8 12 16 20 24 28 32 36
5 0 5 10 15 20 25 30 35 40 45
7 0 7 14 21 28 35 42 49 56 63
8 0 8 16 24 32 40 48 56 64 72
9 0 9 18 27 36 45 54 63 72 81 6
0 6 12 18 24 30 36 42 48 54
To use the multiplication table, find the row you want to multiply (for example, the 3 row). Read across until you come to the column you want to multiply by (such as the 6 column). The answer is where the two meet (in this example, the answer is 18). Figure 5-3 shows how to use the table to find a value.
Figure 5-3:
Finding a value in a multiplica-
tion table.
X 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9
2 0 2 4 5 8 10 12 14 16 18
3 0 3 6 9 12 15 18 21 24 27
4 0 4 8 12 16 20 24 28 32 36
5 0 5 10 15 20 25 30 35 40 45
7 0 7 14 21 28 35 42 49 56 63
8 0 8 16 24 32 40 48 56 64 72
9 0 9 18 27 36 45 54 63 72 81 6
0 6 12 18 24 30 36 42 48 54
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Chapter 5: Multiplication and Division: Everybody Needs Them
Doing Simple Multiplication Like Your Grandfather Did It
Handheld calculators didn’t become accessible to businesses and schools until the early 1970s, but of course people had to do multiplication somehow before then. That’s where the traditional multiplication (the kind you actually do with pencil and paper) in this section comes in. It can serve you when the batteries are dead and also help you gauge the reasonableness of products displayed by a handheld calculator.
It’s believed that the original calculator (a kind of abacus) was developed by the Egyptians around 2,000 BC. Other sources say it goes back farther, to ancient Mesopotamia, and others cite China. The first adding machine was developed in the 17th century, and the late 19th century saw the introduction of the first commercially developed adding machine.
Always perform every step of a multiplication problem, and make sure you do each step neatly. If the answer is wrong, you can more easily track the steps you took to arrive at it.
To solve a simple multiplication problem by hand, just follow these steps:
1. Write out the multiplication problem.
Say you want to multiply 23 by 4. Write the factors down as follows, so that the tens and ones columns in each factor line up.
2. Multiply the each column in the multiplier by the multiplicand.
You multiply 4 times 3 and 4 times 2. The same is true for longer multi- pliers, no matter how many digits it may have.
If your multiplication results in a number higher than 9, you record the ones number and carry over the tens number to the next column. After you do the multiplication for that next column, you add the carryover number to that result, carrying over again into the next column if necessary.
The product of 4 times 3 is 12. The result is higher than 10, so just write the 2 in the ones column and carry the 1.
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Then, the product of 4 times 2 is 8. Add the carried 1 and write the result, 9, in the tens column, as shown
Sometimes, though, you have a multiplier with more than one digit. In those cases, the solving process is a bit more complex.
1. Follow Steps 1 and 2 of the basic multiplication process earlier in this section.
For example, say you want to multiply 7,089,675 by 345. Write the prob- lem down as follows, so that the hundreds, tens, and ones columns in each factor line up.
Multiply each digit in the multiplicand by the ones-column digit in the multiplier.
Start with the 5 in the 345 on the bottom. You multiply 5 times 5, 5 times 7, and so on, moving from right to left until you’ve multiplied by all the digits in the multiplicand. Remember to keep your work organized by bringing each answer straight down, keeping it aligned with its appropri- ate column.
The top row in the following example shows the carryovers from this step’s multiplication:
2. Repeat Step 1 with the digit in the tens column of the multiplier, insert- ing a placeholder of 0 (zero) in the ones column of this multiplying step.
You use this zero placeholder because multiplying by 4 is really multiply- ing by 40 (because the 4 represents 4 tens, or 40, in the context of the whole number). As we note in the following section, any number times 10 shifts its decimal point one place to the right, and in the case of whole numbers, that amounts to adding a zero at the right. So adding this place- holder reminds you to account for the zero in the ones place of 40.
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Chapter 5: Multiplication and Division: Everybody Needs Them
The following shows the placeholder:
With the 0 in place you can now multiply the 4 through the multiplicand.
Write the result next to the 0 placeholder. The result follows, with the carryovers in the top row:
3. Repeat Step 2 for the remaining digits in the multiplier, adding place- holder zeroes as appropriate.
As in the previous step (multiplying by a number in the tens column), you need to add placeholder zeroes for the hundreds column, the thou- sands column, and all other columns in the multiplier. The placeholders keep things lined up and help ensure that the answer will be correct.
The following shows the placeholders for the example problem:
The following shows the example’s complete multiplication for this step, with the carryovers in the top row:
Oh, happy day! The multiplication is now over and just one final step remains.
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4. Add the partial products to get the answer.
It’s normal addition. Take the first column on the far right and add together 5 + 0 + 0. Continue with each column.
The answer is 2,445,937,875.