Scientific Notation

      Scientific Notation (exponential notation) refers to writing decimal numbers as a number multiplied by a power of 10. Now this is done in a certain way. What this boils down to is moving a decimal point, keeping track of how many places you moved it, and increasing or decreasing the power of 10 by that number of decimal point moves.

     Formally now, an exponential number has two parts, a mantissa and an exponent. The mantissa is a number whose magnitude is greater than or equal to 1 but less than 10. The exponent is a power of 10.

     Here's an example: 2.43 x 10³

           the mantissa is 2.43 the exponent is 3

    Here's another example: -5.104 x 10^-2

           the mantissa is -5.105 and the exponent is -2

Ok, so what are these numbers?

      2.43 x 10³ = 2430 and -5.104 x 10^-2 = -0.05104

     Here's the deal. The exponent is the number of decimal places to move left or right to remove the power of 10. How's that? Ok, let's consider a few examples.

Example:

      Take the number 132.45, we want to write this number in scientific notation.

      First of all this number is greater than 9.999.... so we need to rewrite it using powers of ten.

     We need to move the decimal point. Let's move it right by 1. We get 1324.5, and we're getting a number even larger than 9.999.... So let's move it left by 1. We get 13.245, we're getting closer to 9.999... but we no longer have the same number. 13.245 ≠ 132.45, but if we multiply 13.245 by 10 we'll have the same number. We have 13.245 x 10¹.

     132.45 = 13.245 x 10 (we usually don't show the 1 in the exponent)

     Well, 13.245 is still greater than 9.999... so, let's move the decimal left one more time.

      We get: 1.3245 for the mantissa and this number is in the correct range but we now need to multiply by 100 = 10² to keep the same number.

      So we have 132.45 = 1.3245 x 10²

(notice, we moved the decimal left twice and increased the exponent of ten by 2)

Example:

    consider 0.000716

     The mantissa must be greater than or equal to 1, and this number is much to small. If we move the decimal right by 1 we get 0.00716 but we need to multiply by 10^-1 to keep the same number. So if we move the decimal right by 4 we get 7.16 which is in range for the mantissa but we need to multiply by 10^-4 to keep the same number. So what we have is

       0.000716 = 7.16 x 10^-4

(we moved the decimal right 4 times and decreased the exponent of ten by 4)

    The point of scientific notation is the ability to write very large or very small numbers in a compact form.

Examples of very large numbers are:

Avogadro's Number: 6.02257 x 10²³ (number of atoms of a substance to give the number of grams equal to the atomic mass of the substance)

speed of light: 3.0 x 10⁸ meters per second

googool = 1x10¹⁰⁰

diameter of an atom = 2.5 x 10^-10 meters

Operations with numbers in exponential notation. A review of exponents would be very helpful here before proceeding with the examples.

Multiplication/Division

This is easier, so we'll do this first.

   Procedure:

       Multiply/divide the mantissas, then the powers of 10, then write the result in proper form.

Examples:

      1      8.3x10⁴ X 2.0x10⁵ = (8.3 x 2.0) X ( 10⁴ x 10⁵ ) = 16.6 x 10⁹

            now, rewriting, we get 1.66x10¹⁰

      2      (1.2 x 10^-1) / (4.0 x 10⁵) = (1.2 / 4.0) x (10^-1 / 10⁵) = 0.3 x 10^-6

            now rewriting, we get 3.0 x 10^-7

Addition/Subtraction

The exponents of the addends or subtrahends must be the same.

Procedure:  Adjust the exponents until they are the same;  as you do this you will be adjusting the mantissas as well.  Combine mantissas and rewrite the result in proper exponential form.

Example:   3.21x10³ + 8.41x10²

    we'll do this two ways.  First, we'll change the first number to an exponent of 2

       32.1x10² + 8.41x10² = 40.52x10² = 4.052x10³

now let's write the second number to have an exponent of 3

       3.21x10³ + 0.841x10³ = 4.052x10³

either method is fine.

This is another FREE Algebra PRINTABLE presented to you from the Algebra section of K12math.com