1 Lesson 5.2.4 Comparing Numbers in Scientific Notation Comparing Numbers in Scientific Notation.

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Lesson 5.2.4Lesson 5.2.4

Comparing Numbers in

Scientific Notation

Comparing Numbers in

Scientific Notation

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Lesson

5.2.4Comparing Numbers in Scientific NotationComparing Numbers in Scientific Notation

California Standard:Number Sense 1.1Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10), compare rational numbers in general.

What it means for you:You’ll learn how to tell which is the larger of any two numbers written in scientific notation.

Key words:• scientific notation• exponent• coefficient• power

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Comparing Numbers in Scientific NotationComparing Numbers in Scientific NotationLesson

5.2.4

When you look at two numbers, you can usually tell straightaway which is larger.

But once you know what part of the number to look at first, it becomes much more straightforward.

9.5 > 3.2

If the two numbers are in scientific notation, you might need to think a bit harder.

9.5 × 10–3

3.2 × 103

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Comparing Numbers in Scientific NotationComparing Numbers in Scientific Notation

Look at the Exponent First, Then the Other Factor

Lesson

5.2.4

The problem with comparing numbers written in scientific notation is that each number has two parts to look at.

There’s the number between 1 and 10... 2.89 × 106

... and there’s the power of 10. 2.89 × 106

The first thing to look at is the power. The number with the greater exponent is the larger number.

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Example 1

Solution follows…

Lesson

5.2.4

Which of the following numbers is larger? 4.23 × 108 or 7.91 × 106

Solution

8 > 6, so 4.23 × 108 is the larger number.

The exponent in 4.23 × 108 is 8.

The exponent in 7.91 × 106 is 6.

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Comparing Numbers in Scientific NotationComparing Numbers in Scientific NotationLesson

5.2.4

If two numbers have the same power of 10, then you need to look at the other factor — the number between 1 and 10.

The number with the greater factor is the larger number.

3.93 × 105 6.41 × 105

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Example 2

Solution follows…

Lesson

5.2.4

Which of the following numbers is larger? 4.23 × 108 or 7.91 × 108

Solution

7.91 > 4.23, so 7.91 × 108 is the larger number.

The exponents of the power of 10 in these two numbers are the same, so you need to compare the other factors.

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Guided Practice

Solution follows…

Lesson

5.2.4

In Exercises 1–6, say which of each pair of numbers is greater.

1. 9.1 × 106, 8.2 × 109 2. 4.61 × 105, 1.05 × 1010

3. 3.21 × 103, 6.8 × 103 4. 8.4 × 108, 5.75 × 107

5. 6.033 × 1012, 2.46 × 1012 6. 2.6 × 104, 2.09 × 104

8.2 × 109 1.05 × 1010

6.8 × 103 8.4 × 108

6.033 × 1012 2.6 × 104

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Guided Practice

Solution follows…

Lesson

5.2.4

In Exercises 7 and 8, write out each set of numbers in order from smallest to largest:

7. 8.34 × 1010 7.1 × 109 5.71 × 1010 9.64 × 109

8. 3.8 × 105 3.09 × 106 3.41 × 105 4.12 × 105

7.1 × 109

3.47 × 105

, 9.64 × 109 , 5.71 × 1010 , 8.34 × 1010

, 3.8 × 105, 4.12 × 105, 3.09 × 106

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Be Careful with Negative Exponents

Lesson

5.2.4

You need to take care when you compare numbers in scientific notation that have negative powers.

The number with the greater exponent is still larger — but remember that negative numbers can look like they’re getting bigger, when they’re actually getting smaller.

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Example 3

Solution follows…

Lesson

5.2.4

Which of the following numbers is larger? 4.23 × 10–5 or 7.91 × 10–7

Solution

–5 > –7, so 4.23 × 10–5 is the larger number.

The exponent in 4.23 × 10–5 is –5.

The exponent in 7.91 × 10–7 is –7.

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Example 4

Solution follows…

Lesson

5.2.4

If the negative exponents are the same, then the number with the greater coefficient is still larger.

Which of the following numbers is larger? 4.23 × 10–9 or 7.91 × 10–9

Solution

7.91 > 4.23, so 7.91 × 10–9 is the larger number.

The exponents in these two numbers are the same, so you need to compare the coefficients.

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Guided Practice

Solution follows…

Lesson

5.2.4


9. 1.4 × 10–4, 2.3 × 10–6 10. 5.0 × 10–6, 4.8 × 10–6

11. 7.42 × 10–33, 3.89 × 10–23 12. 1.57 × 10–4, 9.31 × 10–5

13. 6.04 × 10–86, 6.2 × 10–86 14. 9.99 × 10–40, 1.45 × 10–17

1.4 × 10–4 5.0 × 10–6

3.89 × 10–23 1.57 × 10–4

6.2 × 10–86 1.45 × 10–17

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Guided Practice

Solution follows…

Lesson

5.2.4

In Exercises 15 and 16, write out each set of numbers in order from smallest to largest:

15. 4.97 × 10–8 4.52 × 10–7 3.08 × 10–8 3.18 × 10–7

16. 6.4 × 10–15 6.04 × 10–13 6.44 × 10–13 6.14 × 10–15

3.08 × 10–8

6.14 × 10–15, 6.4 × 10–15 , 6.04 × 10–13 , 6.44 × 10–13

, 4.97 × 10–8, 3.18 × 10–7, 4.52 × 10–7

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Independent Practice

Solution follows…

Lesson

5.2.4


1. 4.25 × 1018, 3.85 × 1019 2. 9.16 × 10–12, 6.4 × 10–10

3. 2.051 × 107, 1.19 × 104 4. 8.04 × 10–9, 7.96 × 10–9

5. 5.22 × 1045, 7.01 × 1045 6. 6.861 × 10–22, 4.0 × 10–21

7. 7.89 × 1011, 7.9 × 1011 8. 3.642 × 10–30, 1.886 × 10–28

3.85 × 1019 6.4 × 10–10

2.051 × 107 8.04 × 10–9

7.01 × 1045 4.0 × 10–21

7.9 × 1011 1.886 × 10–28

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Solution follows…

Lesson

5.2.4

This table shows the mass of one atom for five chemical elements. Use it to answer Exercises 9–11.

9. Which is the heaviest element?

10. Which element is lighter, silver or titanium?

11. List all five elements in order from lightest to heaviest.

Element Mass of atom (kg)

Titanium 7.95 × 10–26

Lead 3.44 × 10–25

Silver 1.79 × 10–25

Lithium 1.15 × 10–26

Hydrogen 1.674 × 10–27

Lead

Titanium

Hydrogen, Lithium, Titanium, Silver, Lead

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Solution follows…

Lesson

5.2.4

This table shows the approximate distance from Earth of six stars. Use the table to answer Exercises 12–14.

Name of star Approx. distance from Earth (miles)

Bellatrix 1.4 × 1015

Sirius 5.04 × 1013

Barnard’s star 3.50 × 1013

Castor A 3.1 × 1014

Peacock 1.1 × 1015

Deneb 1.9 × 1016

12. Which of these stars is nearest to Earth?

13. Which of these stars is furthest from Earth?

14. How many of these stars are nearer than Castor A?

Barnard’s star

Deneb

Two

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Solution follows…

Lesson

5.2.4

This table shows the approximate distance from Earth of six stars. Use the table to answer Exercises 15–16.

Name of star Approx. distance from Earth (miles)

Bellatrix 1.4 × 1015

Sirius 5.04 × 1013

Barnard’s star 3.50 × 1013

Castor A 3.1 × 1014

Peacock 1.1 × 1015

Deneb 1.9 × 1016

15. Which is closer to Earth, Peacock or Sirius?

16. List all six stars in order from nearest to furthest.

Sirius

Barnard’s star, Sirius, Castor A,Peacock, Bellatrix, Deneb

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Round UpRound Up

Lesson

5.2.4

If you have two numbers written in scientific notation and you want to know which is larger, look at the two parts of the numbers.

First compare their exponents, and then, if necessary, compare their other factors. And don’t forget to watch out for negative exponents.

1 Lesson 5.2.4 Comparing Numbers in Scientific Notation Comparing Numbers in Scientific Notation.

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