04 Irrational Numbers on the Number Line

How do we mark irrational numbers on the number line? #

It’s pretty easy to mark natural number on the number line. While fractions and decimals are slightly higher to be marked, once you know the process (dividing spaces between natural numbers), they become pretty easy to mark on the number line as well.

Can we do the same for irrational numbers? Irrational numbers when estimated as decimals keep going on and on so how do we mark them? Do we keep dividing the space between numbers until eventually we get the irrational number? That would go on forever.

So we cannot exactly mark irrational number on a number line, like with rational. Instead we can estimate where it could possibly be.

We first estimate the decimal expansion of the irrational number upto a certain point and then we mark it approximately on the number line.

Let’s try it our with a few numbers -> √2, √3, √5, ∛25

√2:

First we assume where √2 lies:

(√2)² 1 < 2 < 4
√1 < √2 <√4
1 < √2 < 2

We start with 1.5² and see if it close to 2.

1.5² = 2.25 (Too high)
1.4² = 1.97 (Too low)
1.45² = 2.1 (A little high)
1.44² = 2.07 (A little high)
1.414² = 1.999

Let’s go with this value.

We so the same for √3 and √5, giving us 1.732 and 2.236 as estimates.

Let’s do the same for ∛25:

(∛25)³
8 < 25 < 27
∛8 < ∛25 < ∛27
2 < ∛25 < 3

So ∛25 lies between 2 and 3. Let’s try out 2.8 to see what we get.

2.8³ = 21.952 (Too low)
2.9³ = 24.389 (Pretty close, but still low)
2.95³ = 25.672 (A little high)
2.93³ = 25.153 (A little high)
2.925³ = 25.025

Let’s go with this!

Let’s now mark these numbers on the number line.

√2 = 1.414
√3 = 1.732
√5 = 2.236
∛25 = 2.925