Locating and Comparing Irrational Numbers

Lesson 2 of 2: Rational Approximations

In this lesson:

• Place irrationals exactly on a number line
• Compare two irrationals, and estimate expressions like

What You Will Be Able To Do

By the end of this lesson, you should be able to:

1. Locate an irrational number on a number line using its bounds
2. Compare two irrationals, refining when their intervals overlap
3. Estimate expressions involving irrationals using

Where Exactly Is the Square Root of 2?

You can squeeze any as tightly as you want.

So: on a line from 0 to 4, where exactly does sit?

And is bigger or smaller than ?

Placing the Square Root of 2

From last lesson: .

So on a 0-to-4 line, lands a bit past .

Plotting a point — a skill you already own from grade 6.

The Number Line Holds Every Real Number

• Every real number — rational or not — is one exact point
• The bound pins it to within a hundredth
• That's finer than any tick mark you could draw by hand

Four Irrationals on One Line

Each is labeled with its name and its rational bound.

Zooming In Shows More Precision

Same numbers, finer scale — placement precision matches your bounds.

Compare: Square Root of 8 vs. Pi

• : , , so

Since , we get .

Predict: Square Root of 10 vs. Pi

Round to one decimal:

They look equal. Can we conclude ? Predict before advancing.

Refine Until the Intervals Separate

, , so .

The Strategy for Comparing Two Irrationals

1. Approximate both to the same precision
2. If the intervals don't overlap — you can compare
3. If they do overlap — add a digit and try again

Your Turn: Order Four Numbers

Order from least to greatest:

Approximate each, then arrange them. Show your bounds.

From Locating Points to Computing Values

You replaced with to plot it.

Now replace it inside an expression and compute:

Estimate Pi Squared by Substituting

, so:

So — close to 10, but a little less.

Estimate Two More Irrational Expressions

Substitute the approximation, then do the arithmetic.

When No Approximation Is Needed

Squaring a square root returns the original number.

Spot this before reaching for .

Quick Check: Pi Squared or 10?

You found .

So which is greater — or ? How do you know?

Estimate These Three On Your Own

Estimate each value:

3. Area of a circle with radius cm:

Use . Watch for any exact simplification.

Key Takeaways From This Lesson

✓ Every irrational is one exact point — locate by its bound
✓ When comparing, refine until intervals stop overlapping
✓ Estimate expressions by substituting, then computing

Watch out: 's exact value is — write
Watch out: one decimal isn't always enough

Where We Go From Here Next

Soon, the Pythagorean theorem will hand you side lengths like .

These tools turn those exact lengths into measurements you can use.