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04.23.2018: One Chapter of Math

Today's soundtrack is Orchestral Manoeuvres In The Dark: Orchestral Manoeuvres In The Dark.

I've made it to Section 4 of Basic Math & Pre-Algebra for Dummies; today, I'll be working on chapter 14, "A Perfect Ten: Condensing Numbers with Scientific Notation."

Scientific notation uses exponents. It is used to write very small or very large numbers when lots of 0s would otherwise be needed. 1 = 10 to the power of 0, 10 = 10 to the power of 1, 100 = 10 to the power of 2, 1,000 = 10 to the power of 3, and so on. Though those examples may not seem useful, 1,000,000,000,000,000,000 = 10¹⁸ - a lot easier to type and to read.

To determine what power a number is if it is a 1 followed by a bunch of 0s, just count how many 0s there are and write that number as the power of 10. For example, 1,000 has three 0s; thus, 1,000 = 10³. For a negative power of 10, do the reverse: including the 0 preceding the decimal, count the number of 0s that come before the 1. For example, 0.000001 = 10⁻⁶.

To multiply powers of ten, just add the exponents: 10⁶ x 10⁸ = 10¹⁴.

Scientific notation lets us easily work with very large or very small numbers. This can be done in one of two ways: either as a power of ten, or as a decimal equal or greater than 1 and less than 10. If I wanted to convert 230,000,000 to scientific notation, I would first convert it to a decimal: 230,000,000.0. Next, I would change it to fit within the parameters listed above: in this case, 2.3. Now I can multiply that decimal by 10 to the power of the number of places that I moved the decimal point: 8 places. So in scientific notation, 230,000,000 is 2.3 x 10⁸. If I want to write a number that's already a decimal in scientific notation - say, 0.0034 - I would again move the decimal so that I have a decimal equal or greater to 1 but less than 10, so in this case, 3.4. Then I would multiply that number to the negative power of the number of decimal places that I slid it over. Thus, 0.0034 = 3.4 x 10⁻³.

The term "Orders of Magnitude" refers to how many powers are applied to 10 multiplied by a decimal, so that we can get an estimate of just how big or small a number we are dealing with. For example, 1,000 is 10³; this is an order of magnitude of 3.

To multiply numbers that are in scientific notation, first multiply the two decimal parts, then add the exponents. For example, (2 x 10³) (3 x 10⁴) = 6 x 10⁷. If the decimal part of the solution is 10 or more, just move the decimal point one to the left and add another point to the exponent. (5 x 10³) (3 x 10⁴) = 15 x 10⁷, which comes to 1.5 x 10⁸.


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