Math Help: Exponential Notation

Help with Exponential Notation

Often, scientists and mathematicians have to work with extremely large or small measurements, like the mass of a molecule or the distance between stars. To make calculations with these types of numbers easier, they use exponential notation, which is commonly called scientific notation.

Numbers written in scientific notation include a decimal with no more than one digit to the left of the decimal point, and this decimal is multiplied by a base-ten exponent. Here are some examples of numbers written with scientific notation:

6.00 x 10^4

3.25 x 10^-2

As you can see from these examples, the construction of a number written in scientific notation includes two parts: a decimal component with a value between 1 (also called 'unity') and 10, and an exponential component, which is 10 taken to an appropriate power or exponent. In the first example, the decimal component is 6.00 and the exponential component is 10 taken to the 4th power. In the second example, the decimal number is 3.25 and the exponential component is 10 taken to the negative 2 power.

Writing Numbers Using Scientific Notation

Follow these steps to convert standard numbers into scientific notation.

1. Starting from the right, identify the non-zero digits in the original number. The non-zero number that's farthest to the left will go in the decimal's ones place. For example, if you're writing 508,000 in scientific notation, the five goes to the left of the decimal point. For 0.001034, that number would be one.

2. All digits between this number and the non-zero number with the lowest place value go to the right of the decimal point. For 508,000, the decimal would be 5.08. For 0.001034, it would be 1.034.

3. Count the number of digits between where the decimal point currently is and where it will be in the decimal you use for the scientific notation. For instance, 508,000 has five digits and 0.001034 has three.

4. This number of digits tells you what your exponent will be. If the original number is less than one, the exponent will be negative, and if it is one or higher, the exponent will be positive. For instance, the exponent will be 10^5 for 508,000 and 10^-3 for 0.001034.

5. Write your number using complete scientific notation. The number 508,000 is written as 5.08 x 10^5 and 0.001034 is expressed as 1.034 x 10^-3.

Converting Numbers from Exponential Notation

You'll also need to convert numbers that are written using scientific notation into normal numbers. For those with a negative exponent, you'll add zeros to the beginning (left side) of the decimal until the number of digits before the decimal point equals the absolute value of the exponent. Then, move the decimal point to the beginning of the number. For instance, 3.045 x 10^-7 is equal to 0.0000003045.

For numbers with positive exponents, you'll add those zeros to the end (right side) of the number, and then move the decimal point there. For 7.98 x 10^10, you would need to add eight zeros, like this: 79,800,000,000. You only add eight zeros instead of ten because there are already two digits to the right of the decimal point.