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Exponents: Scientific Notation (page 3 of 5)

Sections: Basics, Negative exponents, Scientific notation, Engineering notation, Fractional exponents


By using exponents, we can reformat numbers. For very large or very small numbers, it is sometimes simpler to use "scientific notation" (so called, because scientists often deal with very large and very small numbers).

The format for writing a number in scientific notation is fairly simple: (first digit of the number) followed by (the decimal point) and then (all the rest of the digits of the number), times (10 to an appropriate power). The conversion is fairly simple.

  • Write 124 in scientific notation. 

    This is not a very large number, but it will work nicely for an example. To convert this to scientific notation, I first write "1.24". This is not the same number, but (1.24)(100) = 124 is, and 100 = 102. Then, in scientific notation, 124 is written as 1.24 × 102.

Actually, converting between "regular" notation and scientific notation is even simpler than I just showed, because all you really need to do is count decimal places.

  • Write in decimal notation:  3.6 × 1012

    Since the exponent on 10 is positive, I know they are looking for a LARGE number, so I'll need to move the decimal point to the right, in order to make the number LARGER. Since the exponent on 10 is "12", I'll need to move the decimal point twelve places over. First, I'll move the decimal point twelve places over. I make little loops when I count off the places, to keep track:

      3.6 _ _ _ _ _ _ _ _ _ _ _ .

    Then I fill in the loops with zeroes:   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

      3.600000000000.

    In other words, the number is 3,600,000,000,000, or 3.6 trillion

Idiomatic note: "Trillion" means a thousand billion — that is, a thousand thousand million — in American parlance; the British-English term for the American "billion" would be "a milliard", so the American "trillion" (above) would be a British "thousand milliard".

  • Write 0.000 000 000 043 6 in scientific notation.

    In scientific notation, the number part (as opposed to the ten-to-a-power part) will be "4.36". So I will count how many places the decimal point has to move to get from where it is now to where it needs to be:

      0.000000000043

    Then the power on 10 has to be –11: "eleven", because that's how many places the decimal point needs to be moved, and "negative", because I'm dealing with a SMALL number. So, in scientific notation, the number is written as 4.36 × 10–11

  • Convert 4.2 × 10–7 to decimal notation.

    Since the exponent on 10 is negative, I am looking for a small number. Since the exponent is a seven, I will be moving the decimal point seven places. Since I need to move the point to get a small number, I'll be moving it to the left. The answer is 0.000 000 42

  • Convert 0.000 000 005 78 to scientific notation.

    This is a small number, so the exponent on 10 will be negative. The first "interesting" digit in this number is the 5, so that's where the decimal point will need to go. To get from where it is to right after the 5, the decimal point will need to move nine places to the right. Then the power on 10 will be a negative 9, and the answer is 5.78 × 10–9

  • Convert 93,000,000 to scientific notation.

    This is a large number, so the exponent on 10 will be positive. The first "interesting" digit in this number is the leading 9, so that's where the decimal point will need to go. To get from where it is to right after the 9, the decimal point will need to move seven places to the left. Then the power on 10 will be a positive 7, and the answer is 9.3 × 107

 

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Just remember: However many spaces you moved the decimal, that's the power on 10. If you have a small number (smaller than 1, in absolute value), then the power is negative; if it's a large number (bigger than 1, in absolute value), then the exponent is positive.

Warning: A negative on an exponent and a negative on a number mean two very different things! For instance:

    –0.00036 = –3.6 × 10–4
      0.00036 = 3.6 × 10–4
      36,000 = 3.6 × 104
    –36,000 = –3.6 × 104

Don't confuse these!


You might be asked to multiply and divide numbers in scientific notation. I've never really seen the point of this, since, in "real life", you'd be dealing with these messy numbers by using a calculator, but here's the process.

  • Simplify and express in scientific notation: (2.6 × 105) (9.2 × 10–13)

    Since I'm multiplying, I can move things around and simplify some of this stuff easily:

      (2.6 × 105) (9.2 × 10–13)
          = (2.6)(105)(9.2)(10–13)
          = (2.6)(9.2)(105)(10–13)
          = (2.6)(9.2)(105–13)
          = (2.6)(9.2)(10–8)

    Now I have to deal with the 2.6 times 9.2, remembering to convert to scientific notation:

      2.6 × 9.2 = 23.92 = 2.392 × 10 = 2.392 × 101

    Putting it all together, I have:

      (2.6 × 105) (9.2 × 10–13)
          = (2.6)(9.2)(10–8)
          = (2.392 × 101)(10–8)
          = (2.392)(101)(10–8)
          = (2.392)(101–8)
          = 2.392 × 10–7

    Then (2.6 × 105) (9.2 × 10–13) = 2.392 × 10–7

Dividing numbers in scientific notation works about the same way.

  • Simplify and express in scientific notation:  (1.247 × 10–3) ÷ (2.9 × 10–2)

    First, I'll deal with the exponents:

      (1.247 × 10–3) ÷ (2.9 × 10–2)
          = (1.247 ÷ 2.9) (10–3 ÷ 10–2)
          = (1.247 ÷ 2.9) (10–3 × 102)
          = (1.247 ÷ 2.9) (10–1)

    Now I'll deal with the division:

      1.247 ÷ 2.9 = 0.43 = 4.3 × 10–1

    Putting it all together, I get:

      (1.247 × 10–3) ÷ (2.9 × 10–2)
          = (1.247 ÷ 2.9) (10–1)
          = (4.3 × 10–1) (10–1)
          = (4.3)(10–1)(10–1)
          = (4.3)(10–2)
          = 4.3 × 10–2

    So the answer is:  (1.247 × 10–3) ÷ (2.9 × 10–2) = 4.3 × 10–2

If you are required to do problems like these, remember that you can always check your answers in your calculator. For instance, entering "1.247 EE –3 / 2.9 EE –2" on my calculator returns "0.043", which equals 4.3 × 10–2 in scientific notation. If you have to do a lot of these problems, you may find it useful to set your calculator to display all values in scientific notation. Check your owner's manual for instructions.

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Cite this article as:

Stapel, Elizabeth. "Exponents: Scientific Notation." Purplemath. Available from
    http://www.purplemath.com/modules/exponent3.htm. Accessed
 

 

 

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