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Scientific Notation (Standard Form)

Scientific Notation (Standard Form). Scientists have developed a shorter method to express very large numbers. This method is called scientific notation . Scientific Notation is based on powers of the base number 10. The mass of the earth is about 6 000 000 000 000 000 000 000 000 kg

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Scientific Notation (Standard Form)

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  1. Scientific Notation (Standard Form) Scientists have developed a shorter method to express very large numbers. This method is called scientific notation. Scientific Notation is based on powers of the base number 10. • The mass of the earth is about 6 000 000 000 000 000 000 000 000 kg • X-rays have a wavelength of about 0.000 000 095 cm A number in scientific notation is expressed as: N × 10k where: 1 ≤ N< 10 (N is a number between 1 and 10) k is an integer (k can be negative or positive) • Example: • 6 000 000 000 000 000 000 000 000 = 6 x 1024 • 0.000 000 095 = 9.5 x 10 -8

  2. 1.23000000000 11 1.23 x 10 Write the number 123,000,000,000 in scientific notation: 1. Put the decimal after the first digit and drop the zeroes. 2. To find the exponent count the number of places from the decimal to the end of the number. (count to the right) In 123,000,000,000 there are 11 places. Therefore we write 123,000,000,000 as:

  3. 0000005.78 -7 5.78 x 10 Write the number .000000578 in scientific notation: • Put the decimal after the first digit that is different from 0 • and drop the zeroes. 2. To find the exponent count the number of places from the decimal to the beginning of the number. (count to the left) In .000000578 there are 7 places. Therefore we write .000000578as:

  4. Advantages of Scientific Notation • Numbers written in the scientific notation can be used in computations with far greater ease. • This advantage was more practical before the advent of calculators and their abundance.

  5. Multiply numbers written in scientific notation • The general format for multiplying using scientific notation is as follows: (n x 10x) (m x 10y) = (n) (m) x 10x+y • First multiply the n and m numbers together and express an answer. • Secondly multiply the exponential parts together by ADDING the exponents together. • Finally multiply the two results for your final answer.

  6. Example • For example: (3 x 104) (5 x 102) • First 3 x 5 = 15 • Second (104) (102) = 104+2 = 106 • Finally 15 x 106 for the answer

  7. Divide numbers written in scientific notation • The general format for multiplying using scientific notation is as follows: (n x 10x) / (m x 10y) = (n) / (m) x 10x+y • First divide the n and m and express an answer. • Secondly multiply the exponential parts together by SUBTRACTING the exponents together. • Finally multiply the two results for your final answer.

  8. Example • For example: 8 x 10-3 / 2 x 10-2 • First 8 / 2 = 4 • Second 10-3 / 10-2 = 10 -3-( -2) = 10 -1 • Finally 4 x 10 -1 for the answer

  9. Addition and Subtraction Using Exponential Notation • The general format would be: • (N x 10x) + (M x 10x) = (N + M) x 10x or • (N x 10y) - (M x 10y) = (N-M) x 10y

  10. If the exponential equivalents do not have the same exponent then the decimal of one has to be repositioned so that it's exponent is the same as all the rest of the numbers being added or subtracted. • The reason for that is that when we add or subtract numbers we must line all the decimals up in the same position before we add or subtract columns of numbers.

  11. So for example: (2.3 x 10-2) + (3.1 x 10-3) • We recognize that the two exponents are not the same so either the exponent of the first number has to be changed to a -3 or the exponent of the second number has to be changed to a -2. • It is really arbitrary which one is changed. Let's change the first one.

  12. Example continued… 2.3 x 10-2 = 23 x 10-3 • Now both numbers will have the SAME exponent value. (23 x 10-3) + (3.1 x 10-3) = (23 + 3.1) x 10-3 = 26.1 X 10-3

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