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The last section gave a hint how numbers behave when multiplied or divided by positive powers of 10.
Table below lays it all out...
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Table: How to Multiply & Divide by Positive Powers of 10.
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When You |
By a positive power of 10
written as a power |
By a positive power of 10
written as a normal number |
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X |
Move the decimal
to the right
the same number of spaces
as the power. |
Move the decimal
to the right
the same number of spaces
as the number of zeroes. |
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% |
Move the decimal
to the left
the same number of spaces
as the power. |
Move the decimal
to the left
the same number of spaces
as the number of zeroes. |
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Note: If moving the decimal creates blank spaces, then
fill in the blank spaces with zeroes.
Example 1: 361.98 x 105 (the power of ten is written as a power)
Since we're multiplying, move to the right.
Since the 10 is to the power of 5, move the decimal 5 places.
Notice this time we have to fill in the blank spaces with zeroes.
So we get
Example 2: 17.483 x 100 (the power of ten is written as a normal number)
Since we're multiplying, move to the right.
Since there are two zeroes after the "1," move the decimal two places.
So we get:
Example 3: 3.74 / 103 (the power of ten is written as a power)
Since we're dividing, move to the left.
Since the 10 is to the power of 3, move the decimal 3 places.
Notice this time we have to fill in the blank spaces with zeroes.
So we get
Example 4: 471,000,000 / 10,000 (the power of ten is written as a normal number)
Since we're dividing, move to the left.
Since there are 4 zeroes after the "1," move the decimal 4
places.
So we get
Note:
Why Multiplying & Dividing By Powers of 10 Just Moves the Decimal
Our decimal system is based on powers of 10. As you move left through the digits in a number, each digit is worth 10 times (or one power of 10) more than the one to the right. Now take 4 x 10 = 40 as an example. The 4 is being made 10 times bigger. But "10 times more" is just one place to the left. So the 4 just moves to the left one space, which is like moving the decimal to the right one space. Now multiply 23.6 x 100 = 2360. Each digit in 23.6 (2,3, and 6) is now 100 times bigger than it used to be. Since 100 is 2 powers of 10, each digit moves two places to the left, which, again, is like moving the decimal two places to the right. When dividing by positive powers of ten, each digit is worth that power of ten times less, so the decimal moves the other way to make the number smaller.
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