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Scientific Notation

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What is scientific Notation? Scientific notation is a way of expressing really big numbers or really small numbers. It is most often used in “scientific” calculations where the analysis must be very precise.

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Scientific notation consists of three parts: A number between 1 and 10 = coefficient The base number of 10 An exponent 5.67 x 10 5

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Changing scientific notation to standard form.

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To change scientific notation to standard form… Simply move the decimal point to the right for a positive exponent. Move the decimal point to the left for negative exponent. (Use zeros to fill in places.)

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Example 3 Given: 5.093 x 10 6 Answer: 5,093,000 (moved 6 places to the right)

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Example 4 Given: 1.976 x 10 -4 Answer: 0.0001976 (moved 4 places to the left)

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Changing standard form to scientific notation.

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To change standard form to scientific notation… Place the decimal point so that there is one non-zero digit to the left of the decimal point. Count the number of decimal places the decimal point has “moved” from the original number. This will be the exponent on the 10.

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Continued… If the original number was less than 1, and you moved the decimal to the right, then the exponent is negative. If the original number was greater than 1, and you moved the decimal to the left, then the exponent is positive.

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Example 1 Given: 289,800,000 Use: 2.898 (moved 8 places) Answer: 2.898 x 10 8

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Example 2 Given: 0.000567 Use: 5.67 (moved 4 places) Answer: 5.67 x 10 -4

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Multiplying with Exponents Multiply the base numbers together. Then add the exponents to get the power of 10. General formula: (N X 10 x ) (M X 10 y ) = (N) (M) X 10 x+y

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Example 3 Given: (3.45 X 10 7 )(6.25 X 10 5 ) Multiply bases: 3.45 X 6.25 = 21.5625 Add Exponents: 10 7 +10 5 =10 7+5 =10 12 Answer: 21.5625 X 10 12 Shift Decimal left one place. Final Answer: 2.15625 X 10 13

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Dividing with Exponents Divide the base numbers. Then subtract the exponents to get the power of 10. General formula: N X 10 x / M X 10 y = N/M X 10 x-y

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Example 4 Given:8 X 10 -3 / 2 X 10 -2 Divide bases: 8/2=4 Subtract exponents: (-3)-(-2)=-1 Answer: 4 X 10 -1

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Adding/Subtracting When adding or subtracting numbers in scientific notation, the exponents must be the same. If they are different, you must move the decimal either right or left so that they will have the same exponent.

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Moving the decimal For each move of the decimal to the right you have to subtract 1 from the exponent. For each move of the decimal to the left you have to add 1 to the exponent. It does not matter which number you decide to move the decimal on, but remember that in the end both numbers have to have the same exponent on the 10.

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Example 5 Given: 3.76 X 10 4 + 5.5 X 10 2 Shift decimal 2 places to the left for 10 4. Move:.055 X 10 2+2 Add: 3.76 X 10 4 +.055 X 10 4 Answer: 3.815 X 10 4

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Example 6 Given: 4.8 X 10 5 – 9.7 X 10 4 Shift decimal 1 places to the left for 10 5. Move:.97 X 10 (4+1) Subtract: 4.8 X 10 5 -.97 X 10 5 Answer: 3.83 X 10 5

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