Scientific notation is part of the language of math and science and allows us to deal with a vast array of numbers, large and small. Learn more about what it is, why it's important, and how to use it.

 

How were arithmetic problems solved before the advent of pocket calculators? In those days, there were only three methods for solving a multiplication problem that a person couldn't do in his or her head: pencil and paper, getting time on a mainframe computer, or using a slide rule. Many students would rely on a slide rule.

 

 

How does a slide rule work? First, it cannot multiply any numbers outside of a range between 1 and 9.99. So in order to multiply 187 by 26 the numbers would be rewritten so that they were something between 1 and 9.99:

 

 

Then the slide rule could be used to find:

 

 

However, this still left more work to be done to get to the final answer. Enough with the slide rule! Thank goodness for calculators. By using the slide rule, however, individuals could learn a valuable lesson about how to work with numbers of different magnitudes. This led to an appreciation for something called scientific notation.

Scientific notation is defined as a standardized way to represent any number as the product of a real number and a power of 10.

 

 

In this form, a is called the coefficient and b is the exponent.

The coefficient is defined as an integer (except zero) to the left of a decimal point plus a mantissa. A manitssa is defined as the significant digits to the right of the decimal point. Here's an example of scientific notation with the important parts of the expression labeled:

 

 

Why Scientific Notation?

Consider the following expression, which is very important to engineers and physicists:

 

 

Look at all of those zeros. This expression is the charge on a single electron. The 'C' stands for Coulomb, which is a unit of electrical charge. In this form, it is a very cumbersome number to work with. It might be easier if it was expressed as follows:

 

 

Here's another example. Sometimes the mass of the earth, in kilograms, is used to solve gravity problems. Here it is in its raw form:

 

 

Again, this is a pretty unwieldy number. It would be nicer represented as follows:

 

 

So it can be more convenient to use scientific notation. But, what are the rules for expressing a number in scientific notation, and how are the expressions found?

 

A Few Important Rules

The great thing about scientific notation is that there is a universally accepted, or standardized, form. Here are the rules:

 

 

Most scientific calculators will put any number into scientific notation, but when it is done by hand, there is a simple procedure:

With the number in its original form, start at the decimal point and count the spaces between digits, either left or right, until you get to the first non-zero digits.

45,000,000 ---> 4.5000000

Follow the example using the number 45,000,000. In this case, move the decimal point until it is after the first non-zero number. If moving to the left, go one additional space so that there is a whole number on the left and a mantissa on the right.

107

The number of spaces the decimal point was moved becomes the exponent. If it was moved right, the exponent is negative. If left, the exponent is positive. If the exponent is zero, scientific notation isn't needed.

45,000,000 = 4.5 x 107

If the number (or a result) is not quite in standard form that is because the mantissa is 10 or greater or less than 1; it can be corrected. Count left or right from the decimal point until there is just one whole number on the left. If it was moved left, then add that number to the existing exponent. If right, then subtract.

Here's what it looks like for the charge on an electron:

 

 

Counting to the right, the decimal point moves 19 digits until the first non-zero digit: in this case, 1 is isolated. That means the coefficient will be 1.62 and the exponent will be -19. This results in the form:

 

 

Here's another example using the mass of the earth:

 

 

This results in the following:

 

 

Arithmetic With Scientific Notation

Once a number is in scientific notation form, how is it used in addition, subtraction, multiplication, and division? Here are some more rules, assuming numbers in scientific notation.