Scientific Notation: What is it?

The speed of light in a vacuum is approximately 300 000 000 meters per second. The mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 91 kg. The difficulty of printing- and understanding- such large and small numbers is evident. To make our lives simpler, we use scientific notation, a shorthand notation that makes use of powers of 10.

a. Numbers larger than 1
To write a decimal number in scientific notation, we need to "factor out" the powers of 10, leaving behind a number whose value is between 1 and 10. Here's how to do it:

1) Place a decimal point after the first digit on the left. The new number you have created is between 1 and 10 (it has only one digit to the left of the decimal point). Call it "A".

For example, 13579. becomes 1.3579 That is, A=1.3579

2) Count the number of places between the new decimal point and the old one. If the original number didn't have a decimal point, it was "understood" to be after the rightmost digit. The number of places you counted is "N"

In going from 13579. to 1.3579 the decimal point was moved 4 places to the left. N=4

3) Now write the number in scientific notation: A x 10N

13579 is written as 1.3579 x 104 and read as "1.3579 times ten to the 4th power"

The following examples illustrate the procedure and the mathematics behind it:

a. 2563. = 2.563 x 1000 = 2.563 x 103 (There were 3 spaces between the old and new decimal point.)This number is read "2.563 times 10 to the third power".

b. 300 000 000 = 3.0 x 100 000 000 = 3.0 x 108 (The original decimal point was assumed to be after the rightmost zero. We don't write all of the zeros in the final number, usually just one or two. Knowing how many digits to keep is a subject of its own, which we'll touch on in the lab.)

c. 2347891.23 = 2.34789123 x 1 000 000 = 2.34789123 x 106

b. Numbers smaller than 1
If the number is smaller than one, it must be multiplied by a negative power of 10. Again, we factor out the powers of 10, leaving a number between 1 and 10. Here's the method:

1) Place a decimal point after the firstnon zero digit. The new number you have created is between 1 and 10 (it has only one digit to the left of the decimal point). Call it "A".

For example, 0.00013579 becomes 1.3579 That is, A=1.3579

2) Count the number of places between the new decimal point and the old one. The number of places you counted is "N"

In going from 0.00013579 to 1.3579 the decimal point was moved 4 places to the right. N=4

3) Now write the number in scientific notation: A x 10-N

0.00013579 is written 1.3579 x 10-4

The following examples illustrate the procedure and the mathematics behind it:

a. 0.02563 = 2.563 x 0.01 = 2.563 x 10-2 (There were 2 spaces between the old and new decimal point.)This number is read "2.563 times 10 to the negative second power".

b. 0.000 000 000 302 = 3.02 x 0.000 000 000 1 = 3.02 x 10-10

c. 0.000 234 = 2.34 x 0.000 1 = 2.34 x 10-4

 

EXERCISES TO TRY:

Write in decimal notation:

1.89 x 108

5.023 x 10-6

6.329 x 10-7

Write in scientifc notation:

625 000 000

0.000 000 345

0.000 002

ANSWERS:

189 000 000; 0.000 005 023; 0.000 000 632 9

6.25 x 108; 3.45 x 10-7; 2. x 10-6

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