GED Mathematical Reasoning: Scientific Notation

Scientific notation is a way to make very large or very small numbers easier to work with. Going from standard form to scientific notation is a two step process

  • Step 1: Move the decimal place until you have a number between one and 10.
  • Step 2: Multiply this number by a power of ten to tell how many places you moved the decimal.

The power of ten is equal to the number of places the decimal point was moved. If the decimal point was moved to the LEFT, the power is POSITIVE. If the decimal point was moved to the RIGHT, the power is NEGATIVE.

 

Example 1

Write in standard form: 5.3 \times 10^4

 

To take the number in example one out of scientific notation and put it into standard form, first look at the power of ten. Since the power is POSITIVE, move the decimal point four places to the RIGHT.

You may need to add place-holding zeros. The result in standard form is:  53,000

algeb12

5.3 \times 10^4 = 53,000

Moving the decimal point according to the power of ten is a shortcut that makes sense because 10^4 equals 10,000. So behind the scenes, we are really just multiplying 5.3 by 10,000.

 

Example 2

Write in standard form: 1.9 \times 10^{-3}

 

To write this number in standard form, first look at the power of ten. Since the power is NEGATIVE, move the decimal point three places to the LEFT. You may need to add place-holding zeros. The result is:  0.0019

algeb13

1.9 \times 10^{-3} = 0.0019

Again, moving the decimal point according to the power of ten is a shortcut that makes sense because 10^{-3} equals ‘one over 1000’ or 0.001. So we are really just multiplying 1.9 by ‘one over 1000’ or 0.001

 

Example 3

Write using scientific notation: 34,000,000

 

To express this number in scientific notation, move the decimal point until the number is greater than or equal to one but less than ten. Remember: though you don’t see a decimal point, you may assume it is located after the right-most digit.

Moving the decimal point seven places to the left gives us a number equal to 3.4, which is greater than or equal to one but less than 10. Now we are ready to write this number in scientific notation.

We start by writing the number 3.4. Then we write the multiplication sign. Then we write our power of ten. In this case, we moved the decimal place 7 places to the left, so our power of ten will be 7.

algeb1434,000,000 = 3.4 \times 10^7

It’s important to note that when we put a number INTO scientific notation, the opposite is true – when we move the decimal point to the LEFT we use a POSITIVE power of ten.

Before we do one final example, it can be tricky to determine how many places to move the decimal point to get a number that is greater than or equal to one but less than ten. However, with practice it does get easier.

Notice that if we stopped our decimal point after only moving 6 places the value would be 34, which is greater than 10.

algeb15

Or, if we had moved the decimal point 8 places, the number would be 0.34, which is less than one.

algeb16

It’s important to stop moving the decimal point so that the resulting number is greater than or equal to one and less than 10 — meeting the criteria for scientific notation.

 

Example 4

Write using scientific notation: 0.00078

 

To express this very small number in scientific notation, move the decimal point until the number is greater than or equal to one but less than ten. Moving the decimal point four places to the right gives us a number equal to 7.8, which is greater than or equal to one but less than 10.

Now we are ready to write this number in scientific notation. We start by writing the number 7.8. Then we write the multiplication sign. Then we write our power of ten. In this case, we moved the decimal place 4 places to the right, so our power of ten will be:

Notice: when we put a number INTO scientific notation, moving the decimal point to the RIGHT yields a NEGATIVE power of ten.

algeb17

0.00078 = 7.8 \times 10^{-4}

 

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