Math 208, Section 5.2 solutions: Why Multiplying Decimal Numbers
by 10 Is Easy
1. Using the example 10 x 23 to illustrate, explain why we move the digits in a decimal
number one place to the left when we multiply by ten.
I think it is easier to see how to multiply by ten if we write the problem as 23 x 10 , rather than
10 × 23. The answer, of course, will be the same by the commutative law. So we’ll look at
23 × 10. This means that we have 23 groups of ten objects each. By the “repeated-addition”
view of multiplication we have
+
+
+
+
+
10
10
10
10
...
10
where there are 23 copies of the number 10. Obviously the sum down the rightmost column is
0. Down the left column, there are 23 copies of the number 1, which obviously add to 23. So
our answer is 230. This is 23 with the “zero added” –or—I would rather say—the “zero
appended.” “Adding a zero” sounds like changing nothing, since any number plus zero is still
the same number.
Another way to think of this is to use base-ten blocks. If I use the small cube for 1, then a rod is
10 and 23 rods is 23 tens; which is 2 hundreds and 3 tens; we write this as 230.
2. Mary says that 10 X 3.7 = 3.70. Why might Mary think this? Explain to Mary why her
answer is not correct and why the correct answer is right. If you tell Mary a procedure, be
sure to tell Mary why the procedure makes sense.
This is another reason why “adding a zero” is not a clear instruction. Mary has thrown in a zero
at the end but this changes nothing because of the decimal point. If we use the flat in the baseten blocks as representing the number 1, then the rod is one-tenth and the small cube is onehundredth. 3.7 means 3 flats and 7 rods; 3.70 means 3 flats, 7 rods, and no cubes. In other
words, they mean the same thing. In this case we ought to say that multiplication by ten “moves
the decimal point to the right.” Here is why, in terms of the blocks. This time, let’s let a small
cube represent the number one-tenth, so a rod represents the number 1. Now, for instance, 1 x
10 means that I should have 1 group of ten objects each. If I start with one rod, then multiplying
by ten means ten rods, which we call a flat. Therefore, multiplying by 10 is really the same as
replacing rods by flats; likewise, multiplying by ten will change cubes to rods, and flats by large
cubes. So, if I start with 3 rods and 7 cubes, representing 3.7, then multiplying by ten will give
me 3 flats and 7 rods, or 37.
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3. Now that you understand why multiplying a decimal number by 10 shifts the digits on
place to the left, explain how we can deduce that multiplying by 10,000 shifts the digits 4
places to the left and multiplying by 100,000 shifts the digits 5 places to the left. How should
we think about the numbers 10,000 and 100,000 in order to make these deductions?
Because 10,000 = 10 x 10 x 10x 10 = 104, we can see that multiplying a decimal number by
10,000 is the same as multiplying that number by 10 four times. So, we shift the digits to the left
four times. Equivalently, we shift the digits 4 places to the left. Because 100,000 = 10 x 10 x 10
x 10x 10 = 105, we can see that multiplying a decimal number by 10,000 is the same as
multiplying that number by 10 five times. So, we shift the digits to the left five times.
Equivalently, we shift the digits 5 places to the left. We should think about the numbers 10,000
and 100,000 as powers of ten, 104 and 105, respectively, so that we can quickly see how many
places to move the digits to the left.
4.d) What happens to the decimal representation of a number when it is multiplied by 10?
By 100?
To multiply by 10, the decimal point moves to the right one digit. To multiply by 100, the
decimal point moves to the right two units. We can see this in the following example:
23.67 x 10 = 236.7
23.67 x 100 = 2367
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