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Compare and Order Irrational Numbers

Students use rational approximations to compare irrational numbers to other irrationals and to rationals, placing them in order on a number line and using inequality symbols. They evaluate expressions such as √50 vs. 7 or π vs. 22/7 by computing approximations to sufficient precision.

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Compare and Order Irrational Numbers

Grade 8 · The Number System · question 1 of 3

0 of 20

20 rungs. A star at the top.

Which compares √10 and 2 correctly?

How it is taught

0:00 of 3:14

To compare a square root with a decimal, square the decimal.

If the square is smaller than the inside number, the root is larger.

Read it: Square roots between whole numbers

√20 is the number that times itself makes 20. 4 × 4 = 16 and 5 × 5 = 25, and 20 is between them, so √20 is between 4 and 5.

  1. Find the squares just below and just above the number.
  2. The root is between the numbers that make those squares.
  3. It sits nearer the whole number whose square is nearer.

A square root undoes squaring: 4 × 4 = 16, so √16 = 4, and 5 × 5 = 25, so √25 = 5.

20 is between 16 and 25, so √20 is between √16 and √25: between 4 and 5.

20 is not a square, so √20 is irrational: its decimal never ends and never repeats.

√n is between the whole numbers whose squares are just below and just above n.

Halving the number: 20 ÷ 2 = 10.

A root times itself makes the number: 4 × 4 = 16 and 5 × 5 = 25, so √20 is between 4 and 5.

√99 is between ___ and 10 — 9. 9 × 9 = 81 and 10 × 10 = 100, so √99 is between 9 and 10.

Order √2, 1.5, √3 from least to greatest

Some numbers, like √2 and √3, can't be written as nice fractions. So how do we put them in order?

Let's order three numbers from least to greatest: square root of two, one point five, and square root of three.

We'll use a number line from one to two, since all three numbers live in here.

One point five is easy — it sits exactly halfway between one and two.

Now √2. We know 1.4 squared is 1.96, just under two. So √2 is just over 1.4 — about 1.41.

We plot √2 at about 1.41 — a tiny bit to the left of 1.5.

For √3, try 1.7: 1.7 squared is 2.89, close to three. So √3 is about 1.73.

How do we order numbers like √2 and √3?Order from least to greatest:√21.5√3121.51.4² = 1.96 → √2 ≈ 1.41(just under 2, so √2 is just over 1.4)√2≈ 1.411.7² = 2.89 → √3 ≈ 1.73(just under 3, so √3 is just over 1.7)√3≈ 1.73√2 < 1.5 < √31.41 < 1.5 < 1.73ORDER√2 < 1.5< √3least to greatest
Grade 8 · Use rational approximations to place numbers on a number line

We plot √3 at about 1.73 — to the right of 1.5.

Reading from left to right: √2, then 1.5, then √3.

So √2 is less than 1.5, which is less than √3. Approximate, then compare — that's how we order irrationals!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
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Divide within 100

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MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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