\ Bryn Mawr College: Survival Skills for Problem Solving--Scientific Notation

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Scientific Notation

 
   

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Very large/very small numbers

Very large numbers can be awkward to write. For example, the approximate distance from the earth to the sun is ninety three million miles. This is commonly written as the number "93" followed by six zeros signifying that the "93" is actually 93 million miles and not 93 thousand miles or 93 miles.

Scientific notation (also called exponential notation) provides a more compact method for writing very large (or very small) numbers. In scientific notation, the distance from the earth to the sun is 9.3 x 107 miles.

Very small numbers can be as awkward to write as large numbers. A paper clip weighs a bit more than one thousandth of a pound (0.0011 LB). This would be expressed in scientific notation as 1.1 x 10-3 lb. The negative sign indicates that the decimal point is moved to the left.

Numbers are customarily written with one digit to the left of the decimal point. Numbers may be correctly represented in other ways.

Representing a number using scientific notation

The number 2,398,730,000,000 can be written in scientific notation as (the starred representation is most common):

0.239873 x 1013

****2.39873 x 1012****

23.9873 x 1011

239.873 x 1010

The number 0.003,483 can be written in scientific notation as (the starred representation is most common):

0.3483x 10-2

****3.483 x 10-3****

34.83x 10-4

348.3x 10-5

3483.x 10-6

Scientific notation and multiplication/division

Multiplication and division of large or small numbers is simplified using scientific notation. The decimal parts of the two numbers are multiplied or divided as appropriate to give the decimal part of the answer. The exponents are added together (in the case of multiplication) or subtracted (for division) and provide the exponent for the answer. The answer is adjusted so that only one digit is to the left of the decimal point in the decimal part.

Multiplying numbers written in scientific notation

To multiply 4 x 10 4 and 6 x 105:

  1. Multiply the decimal parts together:
    4 x 6 = 24

  2. Add the two exponents:
    4 + 5 = 9

  3. Construct the result:
    24 x 109

  4. Adjust the result so only one digit is to the left of the decimal point (if necessary):

    24 x 109 = 2.4 x 1010


General Rule

(a x 10x)(b x 10y) = ab x 10x+y


Dividing numbers written in scientific notation

To divide 6 x 10 5 and 4 x 104:

  1. Divide the decimal parts:
    6/4 = 1.5

  2. Subtract the two exponents:
    5 - 4 = 1

  3. Construct the result:
    1.5 x 101

  4. Adjust the result so only one digit is to the left of the decimal point (if necessary):

    24 x 109 = 1.5 x 101 No adjustment necessary


General Rule

some test text (a x 10x)/(b x 10y) = a/b x 10x-y

some test text


Scientific notation and addition/subtraction

Adding and subtracting numbers written in scientific notation is more complicated than multiplication/division. Consider adding 0.0034 and 0.021:

  0.0034

        3.4 x 10-3

+0.021

in scientific notation --->

+ 2.1 x 10-2

0.0244

 

2.44 x 10-2

 

Now, neither the decimal part or the exponential part combine together in any obvious manner (as they did with multiplication and division). When adding or subtracting numbers written in exponential notation, the numbers must first be rewritten so the exponents are identical. Then, the numbers can be added or subtracted normally

Adding/subtracting numbers written in scientific notation

To add 3.4 x 10-3 to 2.1 x 10-2:

  1. Adjust one of the numbers so that its exponent is equivalent to the otehr number. In this case, change 2.1 x 10-2 into a number which has 10-3 as it's exponential part.

    2.1 x 10-2 = 21 x 10-3

  2. Add the decimal parts together:

    21 + 3.4 = 24.4

  3. The exponential part of the result is the same as the exponential parts of the two numbers, in this case, 10-3:

    24.4 x 10-3

  4. Adjust the result so only one digit is to the left of the decimal point (if necessary):

    2.44 x 10-2

Subtraction follows the same procedure except one number is subtracted from the other in (2).

Try some problems using scientific notation!


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