Section 3: Scientific Notation

There are times when really large numbers, like 35 trillion or really small numbers like 43 ten-millionth are necessary. What do these numbers look like?

35 trillion looks like 35,000,000,000.

43 ten-millionth looks like 0.0000043

With very large numbers or very small numbers, it is not convenient to write out all of the zeros. That's why we use scientific notation. Scientific notation uses powers of 10 and exponents to write a convenient form of numbers that contain lots of zeros.

Scientific notation looks like a x 10n where 1 ≤ a < 10.

35,000,000,000 written in scientific notation would look like 3.5 x 1010.

0.0000043 written in scientific notation would look like 4.3 x 10-6.

To write a number in scientific notation, you can use the following plan:

1. Determine where to place the decimal point. The decimal point must be placed so that the number is between 1 and 10.
2. Determine the power of 10. The power of ten indicates the number of places that the decimal point was moved.

If the original number is less than one, then the exponent should be negative.

If the original number is greater than ten, then the exponent should be positive.

Example:

Convert each of the following numbers to scientific notation.

a) 245,120,000,000

b) 0.000 000 0123

Solution:

a) Convert 245,120,000,000 to scientific notation.

Determine the placement of the decimal point. The decimal should be placed between the 2 and the 4 so that the number is between 1 and 10.

2.4512

Count the number of places that the decimal point needed to be moved.
In this case, the decimal point was moved 11 places to the left.

2.45,120,000,000

Multiply by the appropriate power of 10.
The power should be positive since the original number has a value larger than 10.

2.4512 x 10¹¹

b) Convert 0.000 000 0123 to scientific notation.

Determine the placement of the decimal point. The decimal should be placed between the 1 and the 2 so that the number is between 1 and 10.

1.23

Count the number of places that the decimal point needed to be moved.
In this case, the decimal point was moved 8 places to the right.

000 000 01.23

Multiply by the appropriate power of 10.
The power should be positive since the original number has a value larger than 10.

1.23 x 10^-8

Calculations involving Scientific Notation.

Multiplication or division of expressions written in scientific notation involves the use of rules of exponents.

Multiplication.

To multiply using scientific notation, rearrange the terms so that the powers of 10 are multiplied together. Multiply the first numbers, and multiply the powers of 10. If the product of the first numbers is larger than 10, the decimal point will have to be moved and the exponent will be increased by 1.

Division.

To divide using scientific notation, divide the first numbers, and divide the powers of 10. If the quotient of the first numbers is less than 1, the decimal point will have to be moved and the exponent will be decreased by 1.

The following examples illustrate multiplication and division using scientific notation.

Example:

(5 x 10⁸) (3 x 10^-2)

Solution:

Rewrite the multiplication so that the powers of 10 are multiplied together.

(5 x 10⁸) (3 x 10-²) = (5 x 3)(10⁸ x 10^-2)

Multiply the first numbers together and multiply the powers of 10.

15 x 10⁶

Because the product of the first numbers is larger than 10, the decimal point will have to be moved and the exponent will be increased by 1.

1.5 x 10⁷

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