READ: Scientific Notation

READ: Scientific Notation

Brian is doing research on the Sun for science class. He finds an article that states that the average distance from Earth to the Sun is 1.496×108 kilometers. Brian is confused because 1.496 doesn't seem like a very big number, and he knows that Earth is far away from the Sun. How can Brian correctly interpret the distance from Earth to the Sun that he found in his research?

Scientific Notation: 

Very large and very small numbers are used frequently in science. Here are some examples.

• The distance between Earth and Jupiter is about 595,000,000 kilometers.
• The diameter of an insect's cell is about 0.000000000017 meters.

To make it easier to read, write, and calculate these extreme numbers, scientists use scientific notation. Scientific notation is a way of representing a very large or very small number without having to write all of the zeros at the beginning or end of the number.

When a number is written in scientific notation it is written as a product of a number that is at least 1 but less than 10 and a power of 10. Large numbers (numbers greater than 1) are written with a positive power of ten. Small numbers (numbers between 0 and 1) are written with a negative power of ten. The specific power of 10 indicates just how big or how small the number is.

Here are the same quantities from before written in scientific notation. 

• 595,000,000=5.95×108 
• 0.000000000017=1.7×10−11

Notice that the first number is very large and it has a positive exponent on the 10. The second number is very small and it has a negative exponent on the 10. Also notice that when written in scientific notation, both numbers are the product of a decimal number less than 10 and a power of 10.

Here are the steps for writing a number in scientific notation:

1. Move the decimal point so that it is to the right of the first non-zero digit of the number. The result should be a number that is at least 1 but less than 10. This will be the first part of your number in scientific notation.
2. Count how many spaces you needed to move your decimal point in step 1. The number of spaces will be your power of 10. If you moved the decimal point to the left, your exponent will be positive. If you moved your decimal point to the right, your exponent will be negative.
3. The number in scientific notation is the decimal number from step 1 multiplied by 10 to the power from step 2.

Here is an example:

Write 595,000,000 in scientific notation.

Start by finding the first non-zero digit and put a decimal point to its right. Here, the 5 at the beginning of the number is the first non-zero digit.

5.95000000 which is equal to 5.95

Notice that you don't need to write the zeros at the end of the number anymore because they are to the right of a decimal point.

Next, count how many spaces you needed to move the decimal point to get from 595,000,000 to 5.95. Remember that in 595,000,000 the decimal point is at the very end.

595,000,000 going to 5.95: Move the decimal point 8 spaces to the left.

Now, put everything together. Your number in scientific notation is 5.95 multiplied by 10 to the power of 8. Because you have a very large number and you moved the decimal point to the left in the first step, your exponent will be positive.

5.95×108

The answer is 595,000,000=5.95×108.

Sometimes you will be given a number in scientific notation and you will want to write it as a regular number not in scientific notation. To do this, just follow the steps in reverse.

Here is an example:

Write 3.24×10−5 as a number not in scientific notation.

First, look at the exponent on the 10. The exponent is -5. Because the exponent is negative, your number is a very small number less than 1 and you will be moving the decimal point to the left to get back to the original number.

Next, move the decimal point on the 3.24. You will move the decimal point 5 spaces to the left. Insert zeros into any blank spaces.

3.24 goes to 0.0000324

Notice that the result is a small number less than 1. This is exactly what you wanted since the number in scientific notation had a negative exponent.

The answer is 3.24×10−5=0.0000324.