Trapping Irrationals Between Rational Bounds

Lesson 1 of 2: Rational Approximations

In this lesson:

• Pin down as tightly as you want, no calculator
• See why each step buys one more correct digit

What You Will Be Able To Do

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

1. Bound an irrational number between two consecutive integers
2. Narrow the interval one decimal place at a time by squaring
3. Explain why the process gets closer forever but never lands exactly

How Big Is the Square Root of 2?

We know is irrational — its decimal never ends.

But about how big is it?

Without a calculator — is closer to 1, or closer to 2?

Squaring Catches It Between 1 and 2

• and
• Since , we know

We just trapped between two whole numbers.

The Test Is Squaring and Comparing

To check whether a candidate is below or above :

• If , then — too small
• If , then — too big

Searching the Tenths by Squaring

Searching the Hundredths the Same Way

From the last step:

• — too small
• — too big

So .

Each Round Shrinks the Interval Tenfold

It Never Lands Exactly on 2

Push one more step:

• — too small
• — too big

The squares close in on 2 — but never equal 2.

The Three-Step Routine for Any Root

1. Bounds — find the consecutive values it sits between
2. Test — square the next candidate and compare to
3. Decide — keep the sub-interval that brackets the root

Quick Check: Between Which Tenths?

is between 1 and 2.

Test and by squaring.

Which two tenths trap ? Think before advancing.

Guided: Square Root of 3 to Hundredths

We found:

• — too small
• — square it: is it over or under 3?

So is between and .

When the Root Is Exact: Perfect Squares

Some radicands don't need squeezing at all:

• exactly, because
• The interval collapses to a single point

First check: is a perfect square?

Spot the Error in This Claim

A student writes:

Check it:

Is equal to ? What symbol should replace the ?

Your Turn: Square Root of 7

Approximate to the hundredths place.

1. Which integers bound it?
2. Which tenths?
3. Which hundredths?

Show your squaring at each step.

Key Takeaways From This Lesson

✓ Trap any between rational bounds by squaring and comparing
✓ Each round adds one digit and shrinks the interval tenfold
✓ Perfect squares are exact; others narrow forever

Watch out: , never — write
Watch out: more digits means better, never worse

Where We Go From Here Next

You can now pin any to any precision.

Next: place these numbers exactly on a number line, and settle which of two irrationals is bigger.