CI - Scientific Notation (Lesson)

Scientific Notation

Introduction:

Scientific notation is a shorthand way to express large or tiny numbers. We will consider anything over 1000 to be a large number. Writing these numbers in scientific notation will help you do your calculations much quicker and easier and will help prevent mistakes in conversions from one unit to another. Like the metric system, scientific notation is based on factors of 10. A large number written in scientific notation looks like this:

1.23 x 10¹¹

The number before the x (1.23) is called the coefficient. The coefficient must be greater than 1 and less than 10. The number after the x is the base number and is always 10. The number in superscript (11) is the exponent.

Watch the following video to gain a greater understanding of the power of 10.

Part I: Writing Numbers in Scientific Notation

To write a large number in scientific notation, put a decimal after the first digit. Count the number of digits after the decimal you just wrote in. This will be the exponent. Drop any zeros so that the coefficient contains as few digits as possible.

Example: convert 123,000,000,000 into scientific notation

Step 1: Place a decimal after the first digit.

1.23000000000

Step 2: Count the digits after the decimal.

there are 11

Step 3: Drop the zeros and write in the exponent.

1.23 x 10¹¹

Writing tiny numbers in scientific notation is similar. The only difference is the decimal is moved to the left and the exponent is negative. A tiny number written in scientific notation looks like this:

4.26 x 10^-8

To write a tiny number in scientific notation, move the decimal after the first digit that is not a zero. Count the number of digits before the decimal you just wrote in. This will be the exponent as a negative. Drop any zeros before or after the decimal.

Example: convert 0.0000000426 to scientific notation

Step 1: Place a decimal after the first digit.

000000001.26

Step 2: Count the digits before the decimal.

there are 8

Step 3: Drop the zeros and write in the exponent as a negative.

1.26 x 10^-8

Part II: Adding and Subtracting Numbers in Scientific Notation

To add or subtract two numbers with exponents, the exponents must be the same. You can do this by moving the decimal one way or another to get the exponents the same. Once the exponents are the same, add (if it's an addition problem) or subtract (if it's a subtraction problem) the coefficients just as you would any regular addition problem (review the previous section about decimals if you need to). The exponent will stay the same. Make sure your answer has only one digit before the decimal - you may need to change the exponent of the answer.

Example: 1.35 x 10⁶ + 3.72 x 10⁵=??

Step 1: Make sure both exponents are the same. It's usually easier to go with the larger exponent so you don't have to change the exponent in your answer, so make both exponents 6 in this problem.

1.35 x 10⁶ + 0.372 x 10⁶ = ??

Step 2: Add the coefficients just as you would regular decimals. Remember to line up the decimals.

Image of an example equation

equation image indicator Step 3: Write your answer including the exponent, which is the same as what you started with.

1.722 x 10⁶

Part III: Multiplying and Dividing Numbers in Scientific Notation

To multiply exponents, multiply the coefficients just as you would regular decimals. Then add the exponents to each other. The exponents DO NOT have to be the same.

Example: 1.35 x 10⁶ χ 3.72 x 10⁵ = ??

Step 1: Multiply the coefficients, then move the decimal by the total number of places in the numbers you multiplied, here it would be 4.

Image of an example equation equation image indicator

50220 ―> 5.0220

Step 2: Add the exponents.

5 + 6 = 11

Step 3: Write your final answer.

5.0220 x 10¹¹

To divide exponents, divide the coefficients just as you would regular decimals, and then subtract the exponents. In some cases, you may end up with a negative exponent.

Example: 5.635 x 10³ ÷ 2.45 x 10⁶ = ??

Step 1: Divide the coefficients.

5.635 ÷ 2.45 = 2.3

Step 2: Subtract the exponents.

3 - 6 = -3

Step 3: Write your final answer.

2.3 x 10^-3

Dimensional Analysis

Introduction

Dimensional analysis is a way to convert a quantity given in one unit to an equal quantity of another unit by lining up all the known values and multiplying. It is sometimes called factor-labeling. The best way to start a factor-labeling problem is by using what you already know. In some cases, you may use more steps than a classmate to find the same answer, but it doesn't matter. Use what you know, even if the problem goes all the way across the page!

In a dimensional analysis problem, start with your given value and unit and then work toward your desired unit by writing equal values side by side. Remember you want to cancel each of the intermediate units. To cancel a unit on the top part of the problem, you have to get the unit on the bottom. Likewise, to cancel a unit that appears on the bottom part of the problem, you have to write it in on the top.

Once you have the problem written out, multiply across the top and bottom and then divide the top by the bottom.

Example: How many seconds are there in 3 years?

Step 1: Start with the value and unit you are given, there may or may not be a number at the bottom

Step 2: Start writing in all the values you know, making sure you can cancel top and bottom. Since you have years on top now, you need to put years on the bottom in the next segment. Keep going, canceling units as you go, until you end up with the unit you want (in this case seconds) on the top.

Step 3: Multiply all values across the top. Write in scientific notation if it's a large number. Write units on your answer.

3 x 365 x 24 x 60 x 60 = 9.46 x 10⁷ seconds

Step 4: Multiply all the values across the bottom. Write in scientific notation if it's a large number. Write units on your answer if there are any. In this case, everything was canceled, so there are no units.

1 x 1 x 1 x 1 = 1

Step 5: Divide the top number by the bottom number. Remember to include units.

Step 6: Review your answer to see if it makes sense. 9.46 x 10⁷ is a really big number (97,600,000). Does it make sense for there to be a lot of seconds in three years? YES! If you had gotten a tiny number, then you would need to go back and check for mistakes.

In lots of APES problems, you will need to convert both the top and bottom units. Don't panic! Just convert the top one first and then the bottom one.

Example: 50 miles per hour = ? feet per second

Step 1: Start with the value and units you are given. In this case, there is a unit on top and on the bottom.

Step 2: Convert miles to feet first

Step 3: Continue the problem by converting hours to seconds.

Step 4: Multiply across the top and bottom. Divide the top by the bottom. Be sure to include units on each step. Use scientific notation for large numbers.

Self-Practice: Dimensional Analysis

Be sure to read and study the content on this page then complete the Dimensional Analysis Review and Practice Sheet. Although this assignment will not be graded, it is highly encouraged that you complete it before completing the graded assignment.

Click here to download:

Dimensional Analysis Review and Practice Sheet Links to an external site.

Significant Figures Notes Links to an external site.