# Operations with Scientific Notation

The lesson below is still images taken from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. If the PowerPoint is not one of my freebies, you may also head over to my YouTube channel to see the slideshow fully animated and can pause as needed to be sure you grasp each concept before moving forward (each lesson will always be free there!).

In this lesson we go over how to compare numbers that are in scientific notation as well as how to add, subtract, multiply, and divide them. The slides begin by explaining the use of the magnitude of a number then moves on to operations. As the lesson progresses, students will gain an understanding of when to manipulate the decimal part of numbers in scientific notation as well as when to alter the exponent. The key is to work in smaller parts!

So without further ado, read through the slides below to get a feel for how to work with  numbers in scientific notation!            Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

Coming Soon!

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

• Scientific Notation
• Operations
• Objectives
• By the end of this lesson you should feel comfortable:
• Comparing numbers in scientific notation
• Adding and subtracting numbers in scientific notation
• Multiplying and dividing numbers in scientific notation
• comparisons
• It is nice to know which number is bigger in many cases.
• Observe these two:
• Two fairly large numbers with a lot of zeros. Which is larger?
• Of course you could count the places, but why not use our newest math trick?
• Convert them to scientific notation!
• Comparisons
• We know that we are going to have a decimal in both cases.
• Both numbers have “3025”:
• Remember that the decimal is always after the first digit.
• Both numbers will have “x10”:
• How many places are in the first number?
• The second?
• So now it is VERY easy to see that these numbers are almost exactly the same except for the exponents!
• Comparisons
• So for numbers with the same digits like our previous example, a higher magnitude, or multiple of 10 denoted by the exponent, is the largest number.
• Our magnitudes are the same, so what is different?
• Which is larger? 6.923 or 8.21
• 8 is clearly larger than 6, so 8.21 is our larger number!
• Thus with the same magnitude, or exponents, simply compare the decimals.
• Have you noticed that everything in scientific notation seems to be thought of in parts?
• Or math in general really!
• This idea is very important for adding or subtracting numbers in scientific notation.
• Take these two for instance:
• Our examples from before have the same exponent, which means we can add them (or subtract them)!
• But why?
• When adding numbers, we have to line them up:
• But converting takes time…And we are lazy!
• So let’s make this much faster!
• Do the zeroes really matter other than for our magnitude?
• No, we are really just adding this:
• We are careful to align our decimals, then we simply add:
• So our new number is:
• Wait!!!
• We are forgetting the form of a number in scientific notation.
• We can only have one digit to the left of the decimal place.
• So we need to move our decimal once:
• But if we moved our decimal one more place, our magnitude increased by one (or our exponent):
• So our real final answer (in the correct form) would be:
• Subtracting
• Really, it is easier to remember that if your sum has a tens place, then you will have to add one to your exponent.
• So that means when subtracting we might lose one from our exponent:
• Just like with addition, what are we really calculating here?
• So our initial answer would be:
• Technically we do have only one digit to the left of the decimal, but does it make sense?
• Moving our decimal once to the right has a significant digit!
• multiplying
• Alright then, what about multiplying? Will it be the same?
• Yes!
• ….and no.
• Remember that while there is not a rule for addition with exponents, there is with multiplying as long as we have the same base.
• We must add our exponents!
• Now that we know our magnitude, back to the same process as addition and subtraction:
• Multiplying
• First we multiply our decimals:
• Then slap on our “x10” with our new magnitude:
• Yet again, we have too many digits in front of the decimal, so we move to the left and add one to our magnitude:
• Thus our final answer would be:
• Three steps:
• Multiply the decimals.
• Move the decimal if necessary.
• Dividing
• Lastly, we should be fully equipped to understand division with numbers in scientific notation.
• Notice we have a negative exponent here.
• But it does not change the process as we now know that negative numbers are just numbers, too!
• Knowing that multiplication and division are opposites, we should be able to subtract our exponents this time:
• And then divide our decimals and slap on the magnitude:
• Move that decimal to get the right form and tada!
• conclusion
• Numbers in scientific notation can be compared in two parts:
• The number with the higher magnitude (exponent) is larger.
• For numbers with the same magnitude, the larger decimal is the larger number.
• Adding or subtracting numbers in scientific notation requires the same magnitude.
• To add or subtract numbers in scientific notation, add or subtract the decimals and keep the magnitude.
• If your sum or difference is in the wrong form, you may need to add one to or subtract one from your exponent to move the decimal place.
• Multiplying and dividing require us to add or subtract our exponents respectively.
• Then multiply or divide the decimal numbers.
• Again, it may be necessary to change the exponent to obtain the correct form.