Plot Compare Order

What You're Learning

In this section you will learn how to plot numbers on a number line, compare them using inequality symbols, and order a mixed set of rational and irrational numbers from least to greatest (or greatest to least).

The Number Line

Key Ideas

  • A number line is a straight line where every point corresponds to a real number.

  • Numbers increase as you move right and decrease as you move left.

  • Zero sits in the middle; negatives are on the left, positives on the right.

What Can Be Plotted?

Any real number — integers, fractions, decimals, and even irrational numbers like 2\sqrt{2} — has exactly one point on the number line.

Comparing Numbers Using Symbols

The Three Symbols

  • > means greater than

  • < means less than

  • == means equal to

How to Compare

  1. Convert both numbers to the same form (usually decimals).

  2. Look at the number line — the number farther right is greater.

  3. Write the symbol that makes the statement true.

Example: Compare 34\frac{3}{4} and 0.80.8

  • 34=0.75\frac{3}{4} = 0.75

  • 0.75 < 0.8

  • So \frac{3}{4} < 0.8 ✓

Comparing Decimals

Line Up the Decimal Points

  1. Write both numbers so they have the same number of decimal places (add trailing zeros if needed).

  2. Compare digit by digit from left to right.

Example: Compare 0.3050.305 and 0.320.32

  • Rewrite 0.320.32 as 0.3200.320

  • Compare: 0.3050.305 vs 0.3200.320

  • First two digits match (0.3), third digit 0 < 2

  • 0.305 < 0.32 ✓

Estimating Square Roots for Comparison

When a number is under a radical, estimate its decimal value so you can compare it to other numbers.

A. Find the Two Perfect Squares It Falls Between

Example: Estimate 10\sqrt{10}

  • 32=93^2 = 9 and 42=164^2 = 16

  • 9 < 10 < 16, so 3 < \sqrt{10} < 4

B. Narrow It Down

  • 10 is closer to 9 than to 16, so 10\sqrt{10} is closer to 3.

  • 10≈3.16\sqrt{10} \approx 3.16

C. Use the Estimate to Compare

Compare 10\sqrt{10} and 3.53.5

  • 10≈3.16\sqrt{10} \approx 3.16

  • 3.16 < 3.5

  • \sqrt{10} < 3.5 ✓

Plotting Numbers on a Number Line

Steps

  1. Convert every number to a decimal (approximate if irrational).

  2. Choose a scale that fits all values (e.g., mark every 0.5 or every 1).

  3. Place a dot at each value's position.

  4. Label each dot with the original form of the number.

Ordering a List of Numbers

Steps

  1. Convert every number to decimal form.

  2. Sort the decimals from smallest to largest (or largest to smallest).

  3. Rewrite the list using the original forms.

Example: Order from least to greatest: 5,  2.5,  94\sqrt{5},\; 2.5,\; \frac{9}{4}

  • 5≈2.236\sqrt{5} \approx 2.236

  • 2.5=2.52.5 = 2.5

  • 94=2.25\frac{9}{4} = 2.25

  • Sorted: 2.236,  2.25,  2.52.236,\; 2.25,\; 2.5

  • Answer: 5,  94,  2.5\sqrt{5},\; \frac{9}{4},\; 2.5 ✓

Common Mistakes

  • Forgetting to convert before comparing — always turn fractions and roots into decimals first.

  • Ignoring negative signs — -3 < -1 (farther left on the number line = smaller).

  • Rounding too early — keep enough decimal places to see the difference.

  • Mixing up > and < — the symbol's "mouth" always opens toward the bigger number.

Quick Tips

  • Convert everything to decimals to make comparisons easy.

  • On a number line, right = bigger.

  • For square roots, find the two perfect squares the radicand sits between.

  • When ordering, always write back the original form — don't leave decimals if the problem gave fractions or roots.