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# Powers of ten (Scientific Notation)

In so-called scientific notation , numbers are written as a product of two parts:

(i) a number between 1 and 9.9999…..

and

(ii) a power of 10.

To convert a number to scientific notation, just move the decimal point from its initial position until it is just to the right of the first nonzero digit. The number of positions you move the decimal point gives the numerical value of the power of 10 required. If the decimal point was moved to the left, the power of 10 is positive. If the decimal point was moved to the right, the power of 10 is negative.

Examples:

(Note that 4.3257 Ã— 10 3 = 4.3257 Ã— 1000 = 4325.7, demonstrating that the scientific notation form is the same numerical value as the original number.)

(Note that 5.93 Ã— 10 -3 = 5.93 Ã— 0.001 = 0.00593, demonstrating that the scientific notation form is the same numerical value as the original number.)

(Note that -7.5 Ã— 10 8 = -7.5 Ã— 100,000,000 = -750,000,000, demonstrating that the scientific notation form is the same numerical value as the original number.)

(Note that -3.6 Ã— 10 -8 = -3.6 Ã— 0.00000001 = -0.000000036, demonstrating that the scientific notation form is the same numerical value as the original number.)

Notice that the number itself can be positive or negative and the exponent on the 10 can be positive or negative, and that these two signs are quite independent of each other.

Obviously, to convert back from scientific notation to ordinary decimal numbers, you just move the decimal point the number of places left or right as indicated by the power of 10.

Examples:

(Or, using ordinary arithmetic, 5.96953 Ã— 10 4 = 5.96953 Ã— 10,000 = 59695.3.)

(Or, using ordinary arithmetic, 7.353 Ã— 10 -6 = 7.353 Ã— 0.000001 = 0.000007353.)

(Or, using ordinary arithmetic, -2.3 Ã— 10 2 = -2.3 Ã— 100 = -230.)

(Or, using ordinary arithmetic, -3.592 Ã— 10 -3 = -3.592 Ã— 0.001 = -0.003592.)

To help you remember these rules, just keep in mind that when the power of 10 is positive, the original numerical value was bigger than 1. When the power of 10 is negative, the original numerical value was a fraction, smaller than 1. The rules are not something new or mysterious. As shown in brackets following each example above, the rules just reflect simple numerical properties – they produce the result of ordinary multiplication by powers of 10.

If you have a scientific calculator capable of handling scientific notation, then entry of such numbers is straightforward. First key in the numerical part. Then press a key (typically it has an upper case ‘E’ on it) and key the exponent. You could use the above example as a test case for checking to see that you know how to enter numbers in scientific notation into your calculator.

Remark 1:

The use of scientific notation solves the problem of how to distinguish between trailing zeros in whole numbers which are significant, and those which are present only to tell us where the decimal point is located (and so are not significant digits). (To review the nature of this problem, just re-read the previous document on significant digits.)

When we use scientific notation, there can be no trailing zeros to the left of the decimal point. The only trailing zeros in a number in scientific notation are to the right of the decimal point, and therefore are automatically significant digits, according to the rules.

So, for example

32000 is written 3.2 Ã— 10 4 if none of the three zeros are significant.

32000 is written 3.20 Ã— 10 4 if just the first of the three zeros is significant, but the remaining two zeros are present only to locate the decimal point.

32000 is written 3.200 Ã— 10 4 if just the first two of the three zeros are significant, but the third zero is present only to locate the decimal point. 32000 is written

3.2000 Ã— 10 4 if all three zeros are significant.

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