If we were just given those numbers in a list it would be difficult to know which ones are larger or smaller than others. It is helpful to convert the numbers you are comparing to be in the same form.

The following number line shows all of the same numbers, converted to decimal form. It is much easier to see that 2.83\lt 3.14 than to know that \sqrt{8} \lt \pi.

A number line from -5 to 5. The following are plotted: -0.5, 0.25, 0.75,2.83, and 3.14

We can order real numbers in ascending and descending order.

Ascending order means arranging numbers from the smallest to the largest. On a number line, numbers are automatically arranged in ascending order.

Descending order is the opposite, where numbers are arranged from the largest to the smallest. The descending order is just the reverse of the order on a number line.

Examples

Example 1

For each of the following pairs of numbers, select the number with the smaller value.

4\pi

Worked Solution

Create a strategy

Find an approximate value for 4\pi and compare this to 12.

Apply the idea

Since \pi \approx 3.14 we know that \pi is greater than 3. So 4 \pi must be greater than 4 \cdot 3=12.

12 is smaller than 4 \pi, so the correct answer is option A.

\pi^2

3\sqrt{8}

Worked Solution

Create a strategy

Approximate the value of \pi^{2} and compare it to an approximation of the square root of 8 multiplied by 3.

Apply the idea

We know that \pi\approx 3.14 so \pi^{2} will be greater than 9 since 3^{2}=9 and 3.14^{2} must be larger.

To estimate the value of \sqrt{8}:

\displaystyle \sqrt{4}\lt \sqrt{8}	\displaystyle <	\displaystyle \sqrt{9}	Find the closest perfect squares
\displaystyle 2\lt \sqrt{8}	\displaystyle <	\displaystyle 3	Evaluate the square roots

This means \sqrt{8} will be less than 3. Multiplying a number less than 3 by 3 will result in a number less than 9.

Since we know \pi^{2} must be greater than 9, the correct answer is Option B.

Reflect and check

We can check our answer with a calculator:

\displaystyle \pi^2	\displaystyle \approx	\displaystyle 9.869604401089359\ldots
\displaystyle 3\sqrt{8}	\displaystyle \approx	\displaystyle 8.48528137423857\ldots

-\sqrt{75}

-\dfrac{17}{2}

Worked Solution

Create a strategy

Estimate the value of the square root by identifying the closest perfect square and evaluating their roots. Convert the fraction to decimal.

Apply the idea

Estimate the value of \sqrt{75}. The approximate value would then be multiplied by the negative coefficient.

\displaystyle \sqrt{64}\lt \sqrt{75}	\displaystyle <	\displaystyle \sqrt{81}	Find the closest perfect squares
\displaystyle 8\lt \sqrt{75}	\displaystyle <	\displaystyle 9	Evaluate the square roots

See if \sqrt{75} is closer to 8 or 9.

\displaystyle 81-75	\displaystyle =	\displaystyle 6	Subtract the two largest values
\displaystyle 75-64	\displaystyle =	\displaystyle 11	Subtract the two smallest values

This shows that \sqrt{75} is closer to 9.

Therefore, -\sqrt{75} must be somewhere between -8.5 and -9.

Converting -\dfrac{17}{2} to decimal, we get -8.5.

So, -\sqrt{75} \lt -8.5.

The correct answer is Option A.

Example 2

Compare the numbers -\dfrac{4}{3}, \dfrac{\pi}{3}, -1.25, \sqrt{3}, and 130\% and arrange them in ascending order.

Worked Solution

Create a strategy

Convert the numbers into decimal form to compare and arrange them from least to greatest.

Apply the idea

\displaystyle -\dfrac{4}{3}

\displaystyle =

\displaystyle -1.\overline{3}

Convert to decimal

The value of \pi is approximately 3.14, so:

\displaystyle \dfrac{\pi}{3}	\displaystyle \approx	\displaystyle 3.14\div 3	Divide by 3
	\displaystyle \approx	\displaystyle 1.05	Evaluate the division

To estimate \sqrt{3}:

\displaystyle 1 \lt 3	\displaystyle <	\displaystyle 4	Identify closest perfect squares
\displaystyle \sqrt{1} \lt \sqrt{3}	\displaystyle <	\displaystyle \sqrt{4}	Square root the numbers
\displaystyle 1 \lt \sqrt{3}	\displaystyle <	\displaystyle 2	Evaluate the square roots of the perfect squares

Determining whether \sqrt{3} is closer to 1 or 2:

\displaystyle 4-3	\displaystyle =	\displaystyle 1	The difference between the two largest squares
\displaystyle 3-1	\displaystyle =	\displaystyle 2	The difference between the two smallest squares

\sqrt{3} is closer to 2, so it is somewhere between 1.5 and 2.

1.5 \lt \sqrt{3} \lt 2

Convert 130\% to decimal: \dfrac{130\%}{100}=1.3

The list from smallest to largest is: -\dfrac{4}{3},\,-1.25,\,\dfrac{\pi}{3},\,130\%,\,\sqrt{3}.

Reflect and check

We can also plot the numbers in a numberline which shows us our numbers from smallest to largest.

A number line from -2 to 2. The following are plotted: -4/3, -1.25, pi/3, 130%, and square root of 3.

Idea summary

In comparing and ordering real numbers, it is always helpful to convert all numbers you are comparing to the same form. Usually decimal form is most appropriate, especially when irrational numbers are involved.

Outcomes

8.NS.1

The student will compare and order real numbers and determine the relationships between real numbers.

8.NS.1b

Use rational approximations (to the nearest hundredth) of irrational numbers to compare, order, and locate values on a number line. Radicals may include both positive and negative square roots of values from 0 to 400 yielding an irrational number.

8.NS.1c

Use multiple strategies (e.g., benchmarks, number line, equivalency) to compare and order no more than five real numbers expressed as integers, fractions (proper or improper), decimals, mixed numbers, percents, numbers written in scientific notation, radicals, and Ï€. Radicals may include both positive and negative square roots of values from 0 to 400. Ordering may be in ascending or descending order. Justify solutions orally, in writing or with a model.

1.03 Compare and order real numbers

Ideas

Compare and order real numbers

Examples

Example 1

Example 2

Outcomes

8.NS.1

8.NS.1b

8.NS.1c

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