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- Topic Cluster: Pure Mathematics
- Topic: Scientific Notation and Orders of Magnitude
- Objective: Given two values written in scientific notation, determine which is greater and defend your answer
- Content:
- Level: 2

It can be difficult to determine what is meant by numbers with scientific notation.

Which number is greater? $$ 1.23 \times 10^9 \text{ or } 5.67 \times 10^8 $$ Some students will say that \( 5.67 \times 10^8 \) must be greater because \( 5.67 > 1.23 \). But look carefully! The values have different orders of magnitude, and that turns out to be much more important. In fact $$ 1.23 \times 10^9 = 12.3 \times 10^8 > 5.67 \times 10^8 ! $$

If two values are in proper and correct scientific notation, and they have different orders of magnitude, then the value with the greater order of magnitude will always be the greater value, regardless of the significant figures.

Determine which value is greater.

- \( 4.58 \times 10^9 \text{ vs. } 1.45 \times 10^{8} \)
- \( 9.87 \times 10^{6} \text{ vs. } 8.80 \times 10^{5} \)
- \( 4.56 \times 10^{8} \text{ vs. } 2.23 \times 10^{-6} \)
- \( 3.33 \times 10^{-6} \text{ vs. } 1.23 \times 10^{-7} \)
- \( 4.83 \times 10^{-10} \text{ vs. } 2.45 \times 10^{-9} \)
- \( 6.67 \times 10^{24} \text{ vs. } 8.78 \times 10^{23} \)
- \( 9.99 \times 10^{-15} \text{ vs. } 1.00 \times 10^{-14} \)

If two numbers are not in proper scientific notation and you are trying to figre out which one is greater, the best way to proceed is to put them into proper scientific notation, then follow the same steps as above!

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