Operations in Scientific Notation

Date: 08/30/2001 at 22:18:07
From: Caitlin Hannigan
Subject: Scientific Notation

How do you add, multiply, and divide in scientific notation?

Date: 08/31/2001 at 13:12:24
From: Doctor Rick
Subject: Re: Scientific Notation

Hi, Caitlin.

You can read about multiplying numbers in scientific notation in our 
Dr. Math Archives:

  Using Scientific Notation
  http://mathforum.org/dr.math/problems/dud11.1.97.html   

I'll use nearly the same example:

  (4.567839 * 10^6) * (5.493711 * 10^4)

This can be rewritten as follows:

  (4.567839 * 5.493711) * (10^6 * 10^4)

A product of powers of the same base is the same as the base to the 
sum of the powers, so we can write:

  (4.567839 * 5.493711) * (10^(6+4)

Do the multiplication and addition:

  25.094387 * 10^10

This isn't quite scientific notation because the numeric part isn't in 
the range 1 to 10. To make it so, we divide the numeric part by 10, 
and multiply the exponent part by 10 (which means add 1 to the 
exponent):

  2.5094387 * 10^11

That's the answer. Now let's do a division problem in the same way:

  (4.567839 * 10^6) / (5.493711 * 10^4)

  (4.567839 / 5.493711) * (10^6 / 10^4)

  (4.567839 / 5.493711) * 10^(6-4)

  0.831467 * 10^2

Again we need to get the numeric part into the right range. This time 
it's too small, so we multiply the numeric part by 10 and divide the 
exponent part by 10 (that is, subtract 1 from the exponent):

  8.31467 * 10^1

Are you ready for addition? I'll use the same two numbers:

  (4.567839 * 10^6) + (5.493711 * 10^4)

In order to add two numbers in scientific notation, you must first 
make the exponent parts the same. Rewrite the number with the smaller 
exponent. In this case, we want to make the second number have an 
exponent of 6, which means we must multiply the exponent part by 100 
(add 2 to the exponent). We divide the numeric part by 100 (move the 
decimal point two places left) to keep the number the same.

  (4.567839 * 10^6) + (0.05493711 * 10^6)

Now we can use the distributive property to combine the exponents:

  (4.567839 + 0.05493711) * 10^6

Do the addition.

  4.622776 * 10^6

That's the answer. Because I kept the higher of the two exponents, the 
numeric part is likely to be in the right range (1 to 10), as it is 
here. If it isn't, make the same kind of adjustment that I showed 
above.

I hope this summary helps you.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/