B. Number · Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents

Comparing and Ordering Real Numbers

B1.2 A1About 40 minutes

Learning goals

I can compare rational and irrational numbers written in different forms.
I can order a mixed list of fractions, decimals, square roots and π on a number line.
I can compare negative real numbers.
I can find a rational or irrational number between two given numbers.

Words to know

benchmark
A friendly number you know well, used to compare or estimate. √9 = 3 and √16 = 4 are benchmarks for √10.

opposite
A number the same distance from 0 on the other side, like 2.5 and −2.5.

inequality symbols
< means "is less than" and > means "is greater than".

Remember from Grade 7

In Grade 7 you compared and ordered rational numbers, like −2.3 and −2.25, on number lines. Now the lists include irrational numbers too, like √10 and π.

A real problem

Kai wants to make a square backyard rink with an area of 40 m². Each side would be √40 m long. The boards he has are 6.5 m long. Is one board long enough for a side?

We need to compare √40 and 6.5. They are in different forms, so let's put them in the same form.

A square backyard ice rink is surrounded by low wooden boards in a snowy yard, with a shovel leaning on one side. — A backyard rink like Kai's needs boards along every side. To know if one board is long enough, compare the side length, √40 m, with the board length, 6.5 m. AI-generated image

Strategies for comparing real numbers

Change everything to decimals, going to 2 or 3 decimal places (more if numbers are very close). Use a calculator for square roots and π if you need to.
Use benchmarks. √40 is between √36 = 6 and √49 = 7.
Square both (for positive numbers). 6.5² = 42.25, which is more than 40, so √40 < 6.5.
Draw a number line. The number farther right is greater.
For negatives, think about opposites. The bigger a number's distance from 0, the smaller its negative. √5 > 2, so −√5 < −2.

Back to Kai: √40 ≈ 6.32 m, and 6.32 < 6.5. Yes, one board is long enough.

Each number on the line has its square above it. For positive numbers, the bigger square goes with the bigger number, so √40 ≈ 6.32 < 6.5. One of Kai's boards is long enough for a side.

Worked example

Example 1: Positive numbers in different forms

Order from least to greatest: √3, 1.7, 7/4, 1.73

Change to decimals: √3 ≈ 1.732 05, 1.7 = 1.700, 7/4 = 1.750, 1.73 = 1.730.
Compare: 1.700 < 1.730 < 1.732 05 < 1.750.
Notice 1.73 is a little less than √3, because √3 = 1.732 05…

1.7, 1.73, √3, 7/4

Worked example

Example 2: Negative numbers

Order from least to greatest: −√10, −3.2, −π, −3 1/8

Change to decimals: −√10 ≈ −3.162, −3.2 = −3.200, −π ≈ −3.142, −3 1/8 = −3.125.
For negatives, the one farthest from 0 is least. Distances from 0: 3.200, 3.162, 3.142, 3.125.
So the order from least to greatest is −3.200, −3.162, −3.142, −3.125.

−3.2, −√10, −π, −3 1/8

The four numbers from Example 2 on a number line from −3.3 to −3.0. −3.2 is farthest left, so it is least. −3 1/8 is closest to 0, so it is greatest.

Worked example

Example 3: Find numbers between

Find a rational number and an irrational number between √2 and √3.

√2 ≈ 1.414 and √3 ≈ 1.732.
Rational: try 1.5. Check by squaring: 1.5² = 2.25, which is between 2 and 3. ✓
Irrational: √2.5 is between √2 and √3, because 2.5 is between 2 and 3. 2.5 is not the square of any fraction, so √2.5 ≈ 1.581 is irrational. (Another choice: π/2 ≈ 1.571.)

For example, 1.5 (rational) and √2.5 (irrational).

Think like a mathematician

Before you reach for a calculator, estimate. If you know √10 is a bit more than 3 and π is a bit more than 3, you know you'll need more decimal places to compare them. Then use a calculator to check, and ask: "Does my order make sense on a number line?"

Just for fun

Riddle: what did √40 say to 6.5?

"Let's square up and see who's bigger." (6.5² = 42.25 and (√40)² = 40, so 6.5 wins.)

Hands-on activity

Human number line

You need: painter's tape or a long piece of string, 8 sticky notes or paper scraps, a calculator, family members or stuffed animals to hold the cards.

Make a number line on the floor from −4 to 4 with tape or string. Mark the integers about one big step apart.
Write these numbers on cards: √5, −√5, π, −3.1, 2.25, 9/4, −1 1/2, √0.36.
Estimate where each one goes before using a calculator. Place the cards (or have people stand on the spots).
Check with a calculator and adjust.

Answers, from least to greatest: −3.1, −√5 (≈ −2.236), −1 1/2, √0.36 (= 0.6), √5 (≈ 2.236), 2.25 and 9/4 (the same spot, since 9/4 = 2.25), π (≈ 3.14). Careful: √5 is just a tiny bit less than 2.25.

Math talk: "Which two cards were hardest to place? Why?"

Your turn

Which is greater: √30 or 5.5?

Which is greater: π or 3 1/7?

Order from least to greatest: 2.2, √5, 2 1/3, 2.23

Which is greater: −√2 or −1.4?

Order from least to greatest: −0.6, −2/3, −0.66

Which of these is between 3 and 4: √8, √12, √17?

Without a calculator, decide whether √40 is greater or less than 6.3.

More practice: warm-up

Which is greater: 0.7 or 0.65?

Which is greater: −2 or −5?

Which is greater: √16 or 3.9?

Which is greater: 3/4 or 0.8?

Which is greater: π or 3?

More practice: on your own

Order from greatest to least: √50, 7.1, 7 1/20

Order from least to greatest: −1.75, −√3, −1 2/3, −1.7

A square rink has an area of 40 m². Boards are 6.5 m long. Is one board long enough for one side? Explain.

Find a rational number between √2 and √3.

Find an irrational number between 5 and 6.

Challenge

Order from least to greatest: π, √10, 3.15, 22/7, 3.14

How many whole numbers n make √n greater than 4 and less than 5?

Ali says −√20 > −4 because √20 > 4. Is he right? Explain.

Check your understanding

Quiz

Which number is greatest?

Which number is between 6 and 7?

Which statement is true?

Which list is in order from least to greatest?

7.8² = 60.84. What does this tell you?

Which rational number is between √2 and √3?

For parents and teachers

Big idea: B1.2 asks students to compare and order real numbers, rational and irrational, "separately and in combination". Students choose a strategy: decimals, benchmarks, squaring, or a number line.

Common difficulties:

Negative numbers: thinking −3.2 > −3.1 because 3.2 > 3.1. Use a number line or temperature (−20°C is colder than −10°C).
Not using enough decimal places when numbers are close, like π and 22/7.
Comparing square roots with other numbers by looking only at the number under the root sign. √17 > √16 works (bigger number under the root, bigger root), but to compare √17 with 4.1 you must estimate √17 or square 4.1.

How to help: Make the human number line together, and talk about winter temperatures for negatives. Ask "How do you know?" for each comparison. Choosing and explaining a strategy builds A1 skills.