Scientific Notation
Scientific notation combines the order of operations and powers of 10 to compress long numbers. In doing so, we end up with numbers that are much easier to understand and deal with.
The purpose of this page is to help you apply what you’ve learned so far about exponents in a new application. You’ll come away with a better understanding of both scientific notation and of exponents.
Start with the first tab below. Read the notes. The lesson tab has a PowerPoint and lesson guide that you can use to help you learn, or in a classroom. Try the assignment, then watch the attached video to help summarize your learning.
If you’re a teacher and would like to have an organized, easy to use and reference zip file of all of the materials found on this page, and materials not published on this site, consider downloading the Exponents Packet.
Scientific Notation
Number Unit
Writing Really Big or Really Small Numbers
in a
Concise Fashion
In science, especially when measuring things, we get some long numbers. Sometimes they’re really small, like someone’s chances of passing a math test without paying attention, and sometimes really big, like how far away Jupiter is from Earth.
There is some conceptual understanding of exponents, division, decimals and multiplication required to understand scientific notation. Hopefully, by this time you’re solid in those areas. Otherwise, the only hang-up is making sure to follow the rules of scientific notation. Unlike the “rules” of exponents, these really are just rules. They’re completely made up, but serve a purpose. They provide us a way to communicate in a clear and concise manner with each other in written form.
When we look at some piece of information about the size of the sun it will be written in scientific notation. This is useful because the authors will follow the same rules we, the readers, will follow when reading it.
Here’s how it goes: Scientific notation always has a leading number (ones-value) followed by a decimal place and then however many decimals are practical or required (called significant figures). (Note that for Cambridge IGCSE, unless otherwise specified there will be three significant figures.)
This number followed by a decimal will be multiplied by a power of ten. An example is:
3.14159 ×
1015
If you read this number you will see it is huge because ten to the fifteenth power will have 15 zeros. That is 1,000,000,000,000,000.
By contrast, consider the number below:
3.14159 × 10-15
This is 3.14159 ÷ 1,000,000,000,000,000. Writing this number out would be:
0.0000000000000031415159
That is a zero followed by a decimal place and then fourteen zeros, then the 314159.
Hopefully you see a short cut for writing these numbers. If you were asked to write 7.26 × 103 power, you would recognize you’re multiplying 7.26 by 1,000. That would be 7,260. Do you see the decimal is “moved” three places to the right? This is because the number is 1,000 times larger than 7.26.
The negative exponents move the decimal the other direction, to the left. This is because we are dividing by a power of ten. For example, 7.26 × 10-3. This is 0.00726.
Try these for practice.
Rewrite the following numbers in Standard Form (write them out is what that means here in the US. In England, Standard Form means what we call Scientific Notation. What we call Standard Form, they call, “the normal way.” Seriously.)
4.52 × 105, 9.0909 × 10-7
Why are the following numbers not written in scientific notation:
32.4 × 103
0.23 × 103
1.45 × 23
Write the following in scientific notation:
a) 0.000523 b) 12.003 × 105 c) 1,203,123
Now the tricky stuff with scientific notation comes into play when you have to do arithmetic with them. Remember, exponents are repeated multiplication of the same exact number. Sometimes that makes life easy, like in the case of the following examples:
2.45 × 105 + 7.3 × 105
This is 9.55 × 105, because the units ( of × 105) are the same. If you read to a 6 year old, what is 2 “ten to the fifths” plus seven “ten to the fifths” equal to, they’d say 9 “ten to the fifths.”
Now what will make this trickier is if we had this instead:
2.45 × 105 + 7.6 × 105
This would give us:
10.05 × 105
This number is no longer in scientific notation. We need to rewrite this as:
1.005 × 106
Multiplication can be trickier. Consider how you would calculate the following:
105 × 105
That would be 1010. Now consider that fact as it applies to the following number:
(3.2 × 105) (4.7 × 105)
Try it on your own before reading further. Go ahead, you can figure this out!
The way I would tackle this is by separating the decimal expressions from the powers of ten. The product of 3.2 and 4.7 is 15.04. The product of 105 and itself is 1010. That gives us:
(3.2 × 105) (4.7 × 105) = 15.01 × 1010
This is not in scientific notation because 15.01 needs to be 1.501. To do that we divide it by 10. See 15.01 ÷ 10 = 1.501. However, that changes the value of the number. To keep it the same quantity, we must then multiply the 1010 by 10, to balance out the division by 10. Our end result is:
(3.2 × 105) (4.7 × 105) = 1.501 × 1011
Now let us consider division.
(3.2 × 105) ÷ (4.7 × 105)
Take a moment and figure out what the powers of ten will become.
Now 3.2 divided by 4.7 is 3.2 is 0.681, rounded to the thousandths place. Hopefully you go that the powers of ten reduce to just 1. Remember 100 = 1. This gives us:
(3.2 × 105) ÷ (4.7 × 105) = 0.681 × 100
This answer needs to be converted back to scientific notation. The number 0.681 needs to be multiplied by 10, giving us 6.81. To balance that we must then divide the 100 by 10. Remember that negative exponents are division, so this would also be 100 × 10-1. Our final answer, written in scientific notation is:
(3.2 × 105) ÷ (4.7 × 105) = 6.81 × 10-1
The last thing we will discuss is these numbers written in scientific notation being raised to a power. For example:
(8.881 × 10-6)3
Now this is a great way to reinforce how you are supposed these types of expressions. Consider the expression below:
(ab6)3
Here the base is the number “a times b to the sixth power.” The exponent for that number is three. So it would be written as:
(ab6)(ab6)(ab6)
That gives us:
a3b18
The point is that the base is the entire number grouped together by the parenthesis. Looking at our number written in scientific notation and writing it out would give us:
(8.881 × 10-6)3 = (8.881 × 10-6) (8.881 × 10-6) (8.881 × 10-6)
Now this is very ugly, so instead we could write:
(8.881 × 10-6)3 = 8.813 × 10-6 × 3
Performing this calculation we get:
700.46 × 10-18
That’s not in scientific notation. Dividing the 700.46 by 100, and multiplying the 10-18 by 100, we get:
7.0046 × 10-16
Summary:
In summary, scientific notation is just a way to write big ugly numbers, or super small, but equally as long to write, numbers in a fashion that is easy and does not take up a lot of space. The convention is that we have a single digit number followed by a decimal, then multiplied by a power of ten.
The larger the power of ten, the larger the number. The smaller the power of ten (negative numbers are smaller), the smaller the number.
To make this learning solid, please try the following problems.
Practice Problems
Write the following numbers in standard form (not scientific notation):
1. 8.23 × 10-12 2. 1.023 × 106
Write the following numbers in scientific notation:
1. 51.08 × 10-12 2. 0.0025 × 107
3.
(3.21 × 107)
× (4.998 × 103) 4. (3.21 × 107)
+ (4.998 × 103)
5. [(3.21 × 107) × (4.998 ×
103)]3
6. Without a calculator, perform the following calculations.
a. 6.23×10−3+4.2×10−4
b. (3.9×105)(1.2×103)
c. (1.1×10−3)2
Select Practice Problems Review
6. Without a calculator, perform the following calculations.
a. 6.23×10−3+4.2×10−4
We need these to be like terms. You can change either term, the -3 into a -4, or the exponent of -4 into -3 for the second number.
6.23×10−3+.42×10−3=6.65×10−3
b. (3.9×105)(1.2×103) Here we need to multiply the coefficients and then the powers of ten. Make sure you write your final answer in scientific notation. Keep in mind, when you see a problem like this on a test, you will not be allowed a calculator!
(3.9×105)(1.2×103)=4.68×108
c. (1.1×10−3)2 Here we need to keep in mind that there are two bases, 1.1 and 10-3.
(1.1×10−3)2=1.12×10−6
1.12×10−6=1.21×10−6
Teaching Scientific Notation
Note: Students will be asked, on Cambridge IGCSE exams, to calculate, without a calculator, things like and write their answer in scientific notation. There are easy ways to handle this type of problem if students learn that scientific notation uses exponents as a way to write long numbers concisely.
Big Idea
Scientific notation is just arithmetic. Using what has been learned before about how when terms are like we can multiply and sometimes even add before dealing with exponents, the problems presented with scientific notation are easily accessible.
Key Knowledge
It is convention that a number written in scientific notation has a single digit, followed by a decimal point and a decimal expansion, which is multiplied by a power of 10.
Pro-Tip
(for students)
When dealing with addition of numbers in scientific notation, it is best to rewrite the expressions so they are like terms.
To quickly and accurately remember how to manipulate the numbers in scientific notation, remember that 10 to the power of a positive number is large, where 10 to the power of a negative number is small (because of division). This can help remind students which way to “move,” the decimal point.
To download a copy of the PowerPoint, Click Here.
Lesson Guide
|
Time minutes |
Notes |
Slide # |
|
|
Hidden Slide: Teacher Notes |
1 |
|
5 |
Bellwork |
2 |
|
15 |
Homework Review … review homework through discussion. |
3 |
|
5 |
Introduction of the topic. Challenge them to find something they didn’t know or understand about Scientific Notation in today’s lesson. |
4 – 5 |
|
5 |
Just to make sure all students are on-board with how Scientific Notation works. |
6 |
|
5 |
Have students work through translating the numbers into standard notation. Help them understand that 3.81 × 105 literally means 3.81 × 10,000. That’s why the “decimal moves right.” |
7 |
|
5 |
Students that understand these as numbers should get this right away. Have them help each other. |
8 |
|
5 |
Articulate the point that was subtle in slide 7 … that we are multiplying numbers by powers of 10. |
9 |
|
10 – 15 |
Have students change the numbers from scientific notation into like terms, by using properties of exponents. |
10 – 11 |
|
5 |
Closure … make sure students address the things learned in slides 10 and 11. |
12 |
|
|
Homework |
13 |
