03 Irrational Numbers

Introduction to Irrational Numbers #

Let us start from the very start. Have you heard about the Pythagorean Theorem? The exact time and person who first proved this theorem is contested. We are not discussing the different proofs. Here, we are concerned only about the discovery of irrational numbers by a Pythagorean follower around the 5th century BC.

As you have already learnt in geometry, the Pythagorean Theorem states that,” The sum of the areas of the squares on the legs (a and b) of a right angle triangle equals the area of the square on the hypotenuse (c).” In other words, a² + b² = c².

Hippassus, a Pythagorean follower, was supposedly working with his teacher’s theorem to find the length of the diagonal of a unit square.

He used a² + b² = c² and got c = √2. No matter how much he tried, he could not write this number (√2) as a ratio of two integers. Up until that point of time, all the numbers that had been in use (rational numbers) were the numbers that could be written this way. It was believed that every information in the world could be expressed by using these rational numbers.

√2 was something different. Hippassus went on to prove that this number cannot be written as the ratio of two integers, so it cannot be a rational number.

Let’s not go into how he came to this conclusion, here. Instead, let us try to write √2 as a ratio of two integers. Who knows, we might be able to prove him wrong!

Remember what we learn about square roots? √2 is the side length of a square with an area of 2 units.

We know,

Area of a square (A) = length²

or, 2 = length x length

What rational number gives 2 when multiplied with itself? Let us try and substitute some numbers here.

Let’s start with 3/2.

3/2 x 3/2 = 9/4 = 2.25. This is a bit greater than 2.

So, we need a number less than 3/2. What about 7/5?

7/5 x 7/5 = 49/25 = 1.96. This is a bit smaller than 2.

We need a number more than 7/5. What about 17/12?

17/12 x 17/12 = 289/144 = 2.000694444. This is again a bit more than 2.

We could go on and on but still we will not be able to find a rational number that could be used in place of √2. So, here we go! √2 must be an irrational number. It has a location on the number line but it can not be found on a number line by dividing the segment from 0 to 1 into b equal parts and going a of those parts away from 0 (if a and b are whole numbers).

The square root of any whole number is either a whole number or an irrational number.

√4 = 2 is a whole number and this can be written as 2/1.

But √5 is an irrational number. Some other examples of irrational numbers are: -√3, √10, √155, √5/2, pi, ∛7, etc.

Approximation and Decimal Expansion #

In the previous lesson, we learned about irrational numbers. They cannot be written in the form a/b (where a and b are integers and b is not equal to zero). Also, their location in the number line can not be found by dividing the segment from 0 to 1 into b equal parts and going a of those parts away from 0 (if a and b are whole numbers) like we used to do for rational numbers. But, we can still approximate irrational numbers as rational numbers.

Let us stick with the first ever irrational number to be talked about; √2. In order to approximate the value of √2 to a rational number, we need to first take reference of whole number squares. The square of 1 is 1. The square of 2 is 4. As the square of √2 is 2, the value of √2 must lie between 1 and 2 (1 < √2 < 2). Now what? Let’s use some values between 1 and 2 to get a more exact idea.
1.3² = 1.69.
This is less than the square of √2. Let’s try a greater number.
1.4² = 1.96.
This is close to the square of √2. What if we use a slightly bigger number?
1.5² = 2.25.
This is greater than the square of √2.

The value of √2 should lie between 1.4 and 1.5 (1.4 < √2 < 1.5).

If we need an approximation up to hundredths, we can continue the process by using some values between 1.4 and 1.5.

1.43² = 2.044.
This is greater than the square of √2. Let’s try a smaller number.

1.42² = 2.016.
This is quite close to the square of √2. What if we use a slightly smaller number?

1.41² = 1.988.
This is smaller than the square of √2.

The value of √2 should lie between 1.41 and 1.42 (1.41 < √2 < 1.42).

This can be continued indefinitely further, as we could continue in case of repeating decimals. The precision of approximation required determines the number of decimal digits we look into.

√2 ≈ 1.41421356237…

This is a non-repeating non-terminating decimal. All irrational numbers have non-repeating non-terminating decimal representation.

We can approximate any given irrational number in the same manner. Let us try approximating ∛23 up to two decimal digits. In order to approximate the value of ∛23 to a rational number, we need to first take reference of whole number cubes.

1³= 1, 2³= 8, 3³= 27.

We stop here. As the cube of ∛23 is 23, the value of ∛23 must lie between 2 and 3 (2 < ∛23 < 3). Now what? Let’s use some values between 2 and 3 to get a more exact idea.

2.5³= 15.62.
We need to use a number greater than 2.5.

2.7³= 19.68.
This is still much too lower than the cube of ∛23.

2.9³= 24.38.
This is greater than the cube of ∛23. Let us use a smaller number.

2.8³= 21.95.
We can see that ∛23 lies between 2.8 and 2.9. (2.8 < ∛23 < 2.9)

2.83³= 22.665
2.84³= 22.906 2.85³= 23.149.

We can see that ∛23 lies between 2.84 and 2.85. (2.84 < ∛23 < 2.85)