COMPARING AND ORDERING NUMBERS IN SCIENTIFIC NOTATION

Example 1 :

Compare 5.62 x 10⁶ and 7.39 x 10⁵.

Solution :

First, notice the exponents of 10.

They are 6 and 5.

Here, the exponents are different.

We know that the number with the greater exponent is greater in value.

Because 6 is greater than 5,

5.62 x 10⁶ is greater

Therefore,

5.62 x 10⁶  >  7.39 x 10⁵

Example 2 :

Compare 4.29 x 10^-3 and 5.38 x 10^-3.

Solution :

First, notice the exponents of 10.

That is -3.

Here, the exponents are same.

Because the exponents are the same, we have to compare the decimal numbers to determine the greater number.

Comparing 4.29 and 5.38, clearly 5.38 is the greater number.

Therefore,

4.29 x 10^-3  <  5.38 x 10^-3

Example 3 :

Order the following list of numbers from least to greatest.

1.2 x 10^-1, 8.2 x 10⁴, 6.2 x 10⁵, 2.4 x 10⁵, 1 x 10^-1, 9.9 x 10^-4

Solution :

Step 1 :

List the numbers in order by powers of 10.

9.9 x 10^-4, 1.2 x 10^-1, 1 x 10^-1, 8.2 x 10⁴, 6.2 x 10⁵, 2.4 x 10⁵

Step 2 :

Order the numbers that have the same powers of 10.

9.9 x 10^-4, 1 x 10^-1, 1.2 x 10^-1, 8.2 x 10⁴, 2.4 x 10⁵, 6.2 x 10⁵

Example 4 :

Order the following list of numbers from least to greatest.

5.2 x 10^-3, 3 x 10¹⁴, 4 x 10^-3, 2 x 10^-12, 4.5 x 10³⁰, 4.5 x 10¹⁴

Solution :

Step 1 :

List the numbers in order by powers of 10.

2 x 10^-12, 5.2 x 10^-3, 4 x 10^-3, 3 x 10¹⁴, 4.5 x 10¹⁴, 4.5 x 10³⁰ 

Step 2 :

Order the numbers that have the same powers of 10.

2 x 10^-12, 4 x 10^-3, 5.2 x 10^-3, 3 x 10¹⁴, 4.5 x 10¹⁴, 4.5 x 10³⁰

Example 5 :

Explain how to find the product 0.8 × 10³ . Compare this product with 10³

Solution :

= 0.8 × 10³

To find the product of this, we observe the number we have at the power of 10. Since we have 3, we have to move the decimal 3 digits to the left.

= 0.008

10³ = 10 x 10 x 10

= 1000

Comapring these two, we get 1000 is greater which is 10³ is greater.

Example 6 :

Jake found that the quotient 9.236 ÷ 10⁴ is 0.0009236. If Jake’s result is correct, justify his answer. If not, give the correct decimal.

Solution :

= 9.236 ÷ 10⁴

Since we have 4 at the exponent, we have to move the decimal to the left of 4 digits.

= 0.0009236

Example 7 :

To find the quotient 62.5 ÷ 10⁵ , Gina first identified the number of zeros in the power of 10. When written as a decimal, 10⁵ has five zeros. Describe how Gina can complete the problem.

Solution :

In 105, since we have power 5 we have to move the decimal 5 digits to the left.

62.5 ÷ 10⁵ = 0.000625

Example 8 :

The mass of the Sun is about 2 × 10²⁷ metric tons, or 2 × 10³⁰ kilograms. How many kilograms are in one metric ton?

Solution :

Mass of sun = 2 × 10²⁷ metric tons

Mass of sun in kilograms = 2 × 10³⁰ kilograms

2 × 10²⁷ metric tons = 2 × 10³⁰ kilograms

1 metric ton = (2 × 10³⁰) / (2 × 10²⁷)

= 10^30-27

= 10³

So, one metric ton is 10³ kilograms.

Example 9 :

In computer technology, a kilobyte is 2¹⁰ bytes in size. A gigabyte is 2³⁰ bytes in size. The size of a terabyte is the product of the size of a kilobyte and the size of a gigabyte. What is the size of a terabyte?

Solution :

The size of terabyte = size of kilobyte x size of gigabyte

= 2¹⁰ x 2³⁰

= 2^{10 + 30}

= 2⁴⁰

Example 9 :

Compare the two numbers to find which is greater. Explain how you can compare them without writing them in standard notation first. 4.5 × 10⁶ or 2.1 × 10⁸

Comparing 210 and 4.5, 210 × 10⁶ is greater. That is, 2.1 × 10⁸ is greater.

Example 10 :

Place the numbers in order from least to greatest.

0.24, 4 × 10^-2, 0.042, 2 × 10^-4, 0.004

A) 2 × 10^-4, 4 × 10^-2, 0.004, 0.042, 0.24

B) 0.004, 2 ×10^-4, 0.042, 4 × 10^-2, 0.24

C) 0.004, 2 × 10^-4, 4 × 10^-2, 0.042, 0.24

D) 2 × 10^-4, 0.004, 4 × 10^-2, 0.042, 0.24

Solution :

0.24, 4 × 10^-2, 0.042, 2 × 10^-4, 0.004

0.24, 0.042, 0.004 are already decimals.

= 4 × 10^-2 

Moving the decimal 2 digits to the left .

= 0.04

= 2 × 10^-4

Moving the decimal 4 digits to the left.

= 0.0002

Comparing the decimals, writing from least to greatest.

0.0002, 0.004, 0.04, 0.042, 0.24

2 × 10^-4, 0.004, 4 × 10^-2, 0.042, 0.24

So, option D is correct.

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