Have you ever wondered how to create an irrational number? There are infinite irrational numbers between any two rational numbers, yet they are difficult to find.
No list enumerates all the irrational numbers. There are more irrational numbers than rational numbers. It is crazy to even think about listing all of them! But we can use some of their properties to discover them. We can also take the help of prime numbers to do this.
Following are the lists of irrational numbers:
- List 1 – The Square Root of Primes: √2, √3, √5, √7, √11, √13, √17, √19 …
- List 2 – Logarithms of primes with prime base: log23, log25, log27, log35, log37 …
- List 3 – Sum of Rational and Irrational: 3 + √2, 4 + √7 …
- List 4 – Product of Rational and Irrational: 4π, 6√3 …
- List 5 – Infinite continued fraction: , ,
- List 6 – Special Numbers: Pi, Euler’s number, Golden Ratio
These lists are not exclusive but do provide a way to create irrational numbers.
How many irrational numbers are there between any two rational numbers (for example 1 and 100)?
We cannot list all the irrational numbers between two rational numbers (as they are infinite). However, we know that 1229 irrational numbers between 1-100 are square roots of prime. These are listed below:
√2, √3, √5, √7, √11, √13 … √9949, √9967, and √9973.
Now we can create infinite irrationals using these and the multiplication rule.
Irrational Number – Definition
Any real number that is not rational is irrational.
Rational numbers are of the form a / b ( a, b integers, b ≠ 0 ). They are quotient by definition. So by definition, irrational (= not rational) numbers cannot be quotients of two integers. Irrational numbers have the following properties:
- Endless digits after the decimal
- Irrational number: digits never end, like 1.252252225…
- Rational number: digits in a number end, like 1.25. It is easily converted to a quotient = 125/100
- The digits after decimal have no repeating pattern
- Irrational number: pattern does not repeat, like 1.252252225…
- Rational number: repeating pattern, even endless, like in 1.3252525…, is converted to a quotient = 1312/99
The above properties help identify if a number is irrational but not discover new irrational numbers.
Prime Square Roots
We can use prime numbers to find irrational numbers. For example, √5 is an irrational number. We can prove that the square root of any prime number is irrational. So √2, √3, √5, √7, √11, √13, √17, √19 … are all irrational numbers.
Logarithms of Primes
The logarithm of a prime number with a prime base, like log35 or log72, is irrational. See the proof below:
Let us assume log35 = x/y
where x, y are integers and y ≠ 0
log35 = x/y gives:
3x/y = 5
(3x/y)y = 5y
3x = 5y
3 and 5 are prime numbers. x and y are integers. So the above equation is not balanced. Our assumption has led us to a contradiction. Therefore the assumption:
log35 = x/y = rational number is false.
∴ log35 is an irrational number.
✩ Known Irrationals
Sum of Rational & Irrational
Adding a rational number to an irrational number is an easy way to create new irrational numbers. See the lists of numbers created using this method:
- List A: 1 + √2, 2 + √2, 3 + √2, …..
- List B: 1 + π, 2 + π, 2 + π, ….
- List C: 1 + log35, 2 + log35, 3 + log35, …
You can go on creating irrational numbers endlessly.
Here is proof that such a sum is always an irrational number.
Product of Rational Irrational
What works for the sum of a rational and an irrational number, works for their product also. This provides yet another method to create examples of irrational numbers. See the lists of such numbers below:
- List A: 2√2, 3√2, 4√2, …
- List B: 2π, 3π, 4π, …
- List C: 2log35, 3log35, 4log35, …
✩ Irrational Result of Operations
Following operations between rational and irrational numbers result in an irrational number. Whatever the order of operations, the outcome is always an irrational number.
Infinite Continued Fraction
This is one of the better ways to represent irrational numbers. It takes the form:
For irrational numbers, we can limit ai, bi to be integers (in generic definition, these are any complex numbers). Let us see how they can be used for irrational numbers. Take the example of the square root of a prime number p. We have the formula:
, which can be expanded to:
Some more examples of this formula are:
Representation of Euler’s number e, using continued fraction:
In the case of Euler’s number bi = 1 and ai are: a0 = 2, a1 = 1, a2 = 2, a3 = 1, a4 = 1, a5 = 4, a6 = 1, a7 = 1….
All irrational numbers can be represented in this form though it is challenging to do so.
π – A Ratio that is not Rational
The number Pi originated from geometry. It is the ratio of the circumference and the diameter of a circle. It remains constant, independent of the size of the circle.
If π is the ratio of the circumference and diameter, it must be a rational number. Wrong! Suppose the circumference of a circle is 30 units, then π would be 30 / 2r, where r is its radius. Why is it not a rational number in this case? If the circumference of a circle is rational, the radius is irrational. If you make the radius rational, the circumference is irrational. So this ratio always involves an irrational number.
There are many proofs for π being an irrational number. These proofs involve quite a bit of maths!
The Number e – sum of infinite Quotients
The number e is a recent discovery compared to Pi. Jacob Bernoulli was trying to compute a continuously compounded interest growth in the 17th century. In a nutshell, he was evaluating (1 + 1/n)n, as n grows to infinity. The graph below plots of values of this expression along the y-axis and n along the x-axis. You can see that as x increases, the blue line approaches the number 2.718 e.
Graph of Bernoulli Expression – Euler’s Number
Later Euler calculated this number. Euler used the following formula, an endless summation, to calculate the value of e up to 18 digits.
e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5!….
Euler also found that e could be represented as a continuous, infinite fraction and proved that it is an irrational number. Here is proof that e is an irrational number