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How To Write Numbers In Standard Form?

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One and forty-five ten-thousandths
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Standard form is a handy way of writing very large or very small numbers. In standard form we express the number as a number multiplied by a power of 10. Let's start with a large number: 1234.5. Move the decimal point to the left until there is only one digit (other than zero) to its left. Count how many moves it takes.

1234.5  Starting position
123.45  One move
12.345  Two moves
1.2345  Three moves

The number of moves is the same as the power of 10 we would have to multiply 1.2345 by to get back to 1234.5. In standard form 1234.5 = 1.2345 x 103.

Now let's look at a very small number? 0.000012345. We do as before but this time moving the decimal point to the right until we have one non-zero digit to its left.

0.000012345 Starting position
0.00012345 One move
0.0012345 Two moves
0.012345 Three moves
0.12345  Four moves
1.2345  Five moves

The number of moves is the same as the negative power of 10 we would have to multiply 1.2345 by to get back to 0.000012345. In standard form 0.000012345 = 1.2345 x 10-5.

In standard form the numbers are so large or so small the we often are not too worried about the exact number as long as we have it approximately. Quite often we only need the number to 2 or 3 significant figures. If we wanted our answers above to 3 significant figures they would change as follows.

1.2345 x 103   ==> 1.23 x 103 1.2345 x 10-5  ==> 1.23 x 10-5
One thing to watch out for here is rounding. If the leftmost of the numbers we knock off is 5 or greater we have to increase the last remaining digit by 1. In the following examples we want to reduce our standard form number to 2 significant digits.

1.234 x 106  ==> 1.2 x 106 (The first of the dropped digits is 3. No rounding needed.)
1.45123 x 10-7  ==> 1.5 x 10-7 (The first of the dropped digits is 5. Round the 4 up to 5.)

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