Consider the following pairs of numbers written in scientific notation. Evaluate each number and write it in standard form and determine which of the two numbers is larger.

Scientific notation	Standard form	Which is larger?
1.2\times 10^{-2} \text{ and } 1.2 \times 10^2	0.012 \text{ and } 120	1.2 \times 10^2
2.3\times 10^2 \text{ and } 3.4 \times 10^2
4.8\times 10^{-5} \text{ and } 2.3 \times 10^5
8\times 10^4 \text{ and } 6 \times 10^8
2.3\times 10^{-3} \text{ and } 3.4 \times 10^{-3}
4.8\times 10^{-5} \text{ and } 2 \times 10^{-2}

If the numbers in scientific notation have the same decimal number, what helps us to determine which is larger?
If the numbers in scientific notation have the same exponent, what helps us to determine which is larger?
How do negative exponents and positive exponents on the power of 10 in scientific notation change the number in standard form?

You can compare two numbers written in scientific notation by looking at their powers of 10. We can compare numbers in scientific notation by looking at their powers. When ordering numbers in scientific notation, there are a few strategies we can utilize.

The number with the greater power of 10 will be the greater number.
If two numbers have the same power of 10, then compare the decimal numbers to determine the greater number.
Numbers in scientific notation with negative exponents will always be smaller than numbers in scientific notation with positive exponents.

We can use these strategies when asked to order numbers in scientific notation in either ascending or descending order. Listing numbers in ascending order means listing them from least to greatest. Listing numbers in descending order means listing them from greatest to least.

Examples

Example 1

Which of the numbers is larger:

2.7 \times 10^{4} \enspace \text{or} \enspace 3.4 \times 10^{3}

Worked Solution

Create a strategy

The powers are different values so we can compare the powers to determine which number is larger.

Apply the idea

\displaystyle 10^{4}	\displaystyle >	\displaystyle 10^{3}	Compare the powers of 10
\displaystyle 4	\displaystyle >	\displaystyle 3	The exponent 4 is greater than the exponent 3
\displaystyle 2.7\times 10^{4}	\displaystyle >	\displaystyle 3.4\times10^{3}

So, 2.7 \times 10^{4} is the larger number.

Example 2

Order the numbers from least to greatest:

7.23 \times 10^{7}, \, \, 7.1 \times 10^{6}, \, \,5.6 \times 10^{7}

Worked Solution

Create a strategy

We can start by comparing the powers of 10, then by comparing the decimal numbers.

Apply the idea

The powers of 10 of the three numbers are 7, 6, and 7.

\displaystyle 10^{7}	\displaystyle >	\displaystyle 10^{6}	Compare the powers of 10
\displaystyle 7	\displaystyle >	\displaystyle 6	The exponent 7 is greater than the exponent 6
\displaystyle 7.23 \times 10^{7} \text{and } 5.6 \times 10^{7}	\displaystyle >	\displaystyle 7.1 \times 10^{6}	So, 7.1 \times 10^{6} is the smallest number.

The remaining two numbers have an equal power of 10^{7}, so we will compare the decimal values.

\displaystyle 7.23	\displaystyle >	\displaystyle 5.6	Compare the place value farthest to the left.
\displaystyle 7	\displaystyle >	\displaystyle 5	In the ones place 7 is greater than 5
\displaystyle 7.23 \times 10^{7}	\displaystyle >	\displaystyle 5.6 \times 10^{7}	So, 7.23 \times 10^{7} is the largest number.

And that means 5.6 \times 10^{7} is the middle number. So from least to greatest we have:

7.1 \times 10^{6},\, 5.6 \times 10^{7}, and\, 7.23 \times 10^{7}.

Idea summary

When ordering and comparing numbers in scientific notation, we can do the following:

If the powers of 10 are different, the larger number is the one with the larger exponent.
If the powers of 10 are equal, compare the decimal numbers. The larger number is the one with the larger decimal number.

Outcomes

7.NS.1

The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

7.NS.1d

Compare and order no more than four numbers greater than 0 written in scientific notation. Ordering may be in ascending or descending order.*

1.07 Compare and order numbers in scientific notation

Ideas

Compare and order numbers in scientific notation

Exploration