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## Scientific Notation Worksheets

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Who invented scientific notation?

Exponents were used to handle very large numbers as far back as Archimedes (200 BC). He estimated how many grains of sand would fill the universe using a form of exponents. However, exponential notation as we use it today wasn't invented until the 1600 (by Descartes) and actual scientific notation didn't become 'scientific' until sometime in the 1930s or 1940s.

 What is scientific notation? Generally, in mathematics a notation is a convention - an agreed upon way of writing something so that the symbols and positions are understood by other mathematicians. Notations are important because they are compact ways of expressing something. New notations become popular because they are easier to use. This is the case with scientific notation too. A very large or small number in the decimal system requires too many zeros to actually write out. For instance, a million has six zeros - 1,000,000 and is moderately sized, but a million trillion would take 18 zeros (1,000,000,000,000,000,000). Because such large numbers differ by some power of 10 (each power of 10 shifts a number left or right one place) it is more convenient to use 1 x 1018 and take advantage of the fact that the exponent (18) when used with 10, means 18 zeros. This works for small numbers also if the exponent is negative. So, 0.000001 is the same as 1 x 10-6. The real advantage comes when large or small numbers have to be multiplied or divided. The rules of exponents allow us to simply add or subtract the exponents as long as they share 10 as the base. So, six million trillion times four thousandths would look like this in scientific notation: (6x 1018) X (4x 10-3 ) = (6 x 4) x (10(18-3) ) = 24 x 1015. The answer is more properly written as 2.4x 1016 because only a single one's digit is used in scientific notation. How is scientific notation used in the real world? As the name suggests, scientific notation is used throughout scientific writings. Although it came from mathematics as a compact way of writing very large numbers, scientists adopted it. In modern times, scientific exploration reaches from the realm of the very small (the mass of a proton is 1.672621637 x 10-27 kg) to the very large (the diameter of the Milky Way galaxy is approximately 9.5 x 1017 km). Scientific notation allows us to express such extreme numbers and calculate with them. A basic problem using scientific notation. Red blood cells are 7x10-6 meters wide. How many will fit, side to side, across a human hair that is 1x 10-4 meters wide? (Remember, division is done by subtracting exponents.) 1x10-4 ÷ 7x10-6 = .1429x102 which is the same as 14.29. So, you could cover a hair from one side to the other with about 14 red blood cells. An interesting fact about scientific notation. Scientific notation can't easily be displayed on a computer screen or calculator because superscripts (raised numbers) require special fonts. For this reason, 'E notation' was invented. This is why you often see 5.67E5 on a calculator or in a computer program. (The capital E is used because mathematicians already use a small 'e' to mean a particular number in mathematics.) On a calculator, 5.67E5 means the same thing as 5.67x105.