A 1.4 – Algebra Extension: Scientific Notation and Relative Size
Chapter 1, Section A4
Algebra Topics – Scientific Notation
Elementary Education – Connections to Base Ten Decimals and Expressing Relative Size
Base Ten Place Values
In a previous section, we looked at the powers of ten. We saw how the negative powers of ten can be written as decimals or fractions. We focused on the fraction form in that section. Now, let’s look at the decimal form and what that can tell us about base ten place values. We can then use the decimal form to write very large and very small numbers in scientific notation, which will allow us to quickly see their relative size. We’ll also see how you can share with your own students how to think of large and small objects, from pennies to elephants to planets.
We previous wrote this table of powers of ten.
Powers of Ten
| 10-4 | 0.0001=  |
| 10-3 | 0.001=  |
| 10-2 | 0.01 =  |
| 10-1 | 0.1 =  |
| 100 | 1 |
| 101 | 10 |
| 102 | 100 |
| 103 | 1,000 |
| 104 | 10,000 |
| 105 | 100,000 |
| 106 | 1,000,000 |
Writing the table horizontally instead, shows us how this relates to place value. The names of each place value are now included.
Place Values
| 106 | 105 | 104 | 103 | 102 | 101 | 100 | 10-1 | 10-2 | 10-3 | 10-4 |
| 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 | 0.1= |
0.01 = |
0.001 = |
0.0001 = |
| Millions | Hundred thousands | Ten thousands | Thousands | Hundreds | Tens | Ones | Tenths | Hundredths | Thousandths | Ten Thousandths |
You can see how the fractions give the place values below 1 their names. Hundredths is 1/100, thousandths is 1/1000, and so on.
To name a number, we say the whole number part based on where it starts on this table, “and” where the decimal is, and then we say the decimal part based on where the number ends. For example, the number 408.5 starts in the hundreds place and ends in the tenths place, so we say, “Four hundred eight and 5 tenths.”
Example 1 How would you say the number 1,342.351?
This number starts in the thousands place and ends in the thousandths place, so we say, “One thousand three hundred forty two, and three hundred fifty one thousandths. Now, actually, most of the time what we actually say is “One thousand three hundred forty two, point three five one,” probably because it’s easier and faster. But let’s practice this the long way, so that we can get a better idea of size and place value.
If you’re having trouble seeing where the number starts and ends, imagining lining the number up on the place value chart, above, with the decimal point between 100 and 10-1
| 1 | 3 | 4 | 2 | . | 3 | 5 | 1 | ||||
| 106 | 105 | 104 | 103 | 102 | 101 | 100 | 10-1 | 10-2 | 10-3 | ||
| 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 | 0.1= |
0.01 = |
0.001 =  |
0.0001 = |
|
| Millions | Hundred thousands | Ten thousands | Thousands | Hundreds | Tens | Ones | Tenths | Hundredths | Thousandths | Ten Thousandths |
Multiplying by Positive or Negative Powers of Ten
What happens when we multiply a whole number by a positive power of ten? What about when we multiply a decimal by a positive power of ten?
4 × 101 = 4 × 10 = 40
4 × 102 = 4 × 100 = 400
4 × 103 = 4 × 1,000 = 4,000
4 × 104 = 4 × 10,000 = 40,000
You might notice a pattern, that the number of zeros increases each time. Look also at how the power of ten, the exponent, relates to the number of zeros. For example, how many zeros does 4 × 104 = 40,000 have? It has 4 zeros, the same as the exponent.
What happens when we multiply a decimal number by a power of ten?
4.237 × 101 = 4.237 × 10 = 42.37
4.237 × 102 = 4.237 × 100 = 423.7
4.237 × 103 = 4.237 × 1,000 = 4,237
4.237 × 104 = 4.237 × 10,000 = 42,370
Now you might notice that the decimal place moves to the right each time. How is where the decimal place ends up related to the exponent? For example, for 4.237 × 102 = 423.7, how many places to the right did the decimal place move?
The decimal point moves to the right the same amount as the power of ten, the exponent.
These two patterns are actually based on the same principle. With the whole number, the decimal place is also moving to the right as we multiply by ten. For example, for 4 × 103 , the decimal place is directly after the 4; we can write 4.0 or even 4.00 or 4.000, because the number of zeros after the decimal place does not change the number. 4 × 103 = 4.0 × 103 = 4,000.0 the decimal has moved 3 places to the right.
Example 2 Predict the answers to 12 × 107 and 12.381 × 107 without using a calculator.
12 × 107 will have 7 zeros following the 12, so 12 × 107 = 120,000,000
12.381 × 107 will have the decimal place moved to the right 7 places. We can move right 3 places, then add on 4 zeros to get a total of 7 places. 12.381 × 107 = 123,810,000
Notice that these two numbers are roughly the same size — both start in the hundred millions place.
What happens when we multiply a whole number or decimal by a negative power of ten?
4 × 10–1 = 
= 0.4
4 × 102 = 
= 0.04
4 × 103 = 
= 0.004
4 × 104 = 
= 0.0004
Now the decimal moves to the left the same number of places as the exponent.
Example 3 Predict the answers to 12 × 10–7 and 12.381 × 10–7 without using a calculator.
For each of these, we need to move the decimal to the left 7 places. For 12 x 10-7, after we move the decimal two places to the left, we will have 0.12, so now we need to move 5 more place left, by adding in 5 zeros. 12 × 10-7 = 0.0000012
Similarly, 12.381 × 10-7 = 0.0000012381
Again, notice that these two decimals are roughly the same size. Sometimes people think that 0.0000012381 is much larger than 0.0000012, but it is not — the 381 part is very, very small. We will talk more about comparing sizes of decimals in the next section.
Scientific Notation
A number written in scientific notation has the form a × 10n where a is a number greater than or equal to 1 and less than 10. For example, 2.43 × 103 is written in scientific notation, while 14.85 × 103 is not, since 14.85 is greater than 10. Why do we care that a is less than 10? Because then we can compare the sizes of things by just looking at the the power of ten.
For example, in 2020, the population of New York City (the five boroughs) was 8.8 million, or 8.8 × 106, according to the 2020 census. By comparison, the population of Los Angeles was 3.8 × 106, also a large city because it has the same magnitude (power of ten) as New York City, but still less than half of New York’s population (half of 8.8 is 4.4, and 3.8 is less than that).
We can also compare the sizes of very small things using scientific notation. For example, according to the U.S. mint, a penny is 1.905 x 101 cm in diameter, while a quarter is 2.426 × 101 cm — very similar in size — and an atom is about 1.0 × 10-7 cm — much, much smaller.
Example 4 Use scientific notation to compare 4,200,000,000 and 1,502,000,000,000. Which number is larger?
To write these numbers in scientific notation, we write just the first digit, followed by a decimal point and any other digits, leaving off the final zeros. Then figure out the power of ten you would need to move the decimal point to the end of the number. 4,200,000,000 = 4.2 × 109 because you would need to move the decimal point over to the right 9 times to get it to the end of the number. 1,502,000,000,000 = 1.502 × 1012 because you would need to move the decimal 12 times to the right to get 1,502,000,000,000. The order of magnitude for 1.502 × 1012 is 12, which is the trillions place, while the order of magnitude for 4.2 × 109 is 9, which is the billions place.
We can also use scientific notation to compare sizes of decimals.
Example 5 Use scientific notation to compare 0.000814 and 0.014. Which number is larger?
Again, we write just the first digit, followed by the decimal point and any other digits. Now we leave off the initial zeros. 0.000814 = 8.14 × 10-4 because we would have to move left four decimal places to get our original number. 0.014 = 1.4 × 10-2. This is a much larger number, since the 1 is in the hundredths place (10-2 = 1/100), as opposed to 8.14 × 10-4, where the 8 is in the ten thousandths place, a much smaller place value.
Calculator note
Your calculator will use scientific notation if a number gets too large to display. For example, if you multiply 800,000,000 × 1,00,000,000 on a calculator (including the one on your phone), it will probably display 8E16. This doesn’t mean that there is an error. Instead, it means that the answer is 8 × 1016. You can also easily get the answer by hand, using scientific notation. 800,000,000 = 8 × 108, and 1,00,000,000 = 1 × 108 so 800,000,000 × 1,00,000,000 = (8 × 108) × (1 × 108) = (8 × 1) × (108 × 108) = 8 × 1016 , because we add the exponents.
Unless otherwise specified, images in this section are by Kathleen Offenholley and licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.