Is It Irrational?
Here we look at whether a square root is irrational ... or not!
Rational Numbers
A "Rational" Number can be written as a "Ratio", or fraction.
Example: 1.5 is rational, because it can be written as the ratio 3/2
Example: 7 is rational, because it can be written as the ratio 7/1
Example 0.317 is rational, because it can be written as the ratio 317/1000
But some numbers cannot be written as a ratio!
They are called irrational (meaning "not rational" instead of "crazy!")
The Square Root of 2
The square root of 2 is irrational. How do I know? Let me explain ...
Squaring a Rational Number
First, let us see what happens when you square a rational number:
If the rational number is a/b, then that will become a^{2}/b^{2} when squared.
Example: (3/4)^{2} = 3^{2}/4^{2}
Notice that the exponent is 2, which is an even number.
But to do this properly we should really break the numbers down into their prime factors (any whole number can be made by multiplying prime numbers together):
Example (3/4)^{2} = ( 3/(2×2) )^{2} = 3^{2}/2^{4}
Notice that the exponents are still even numbers. The 3 has an exponent of 2 (3^{2}) and the 2 has an exponent of 4 (2^{4}).
In some cases we may need to simplify the fraction:
Example: (16/90)^{2}
Firstly: 16 = 2×2×2×2 = 2^{4}, and 90 = 2×3×3×5 = 2×3^{2}×5
So: (16/90)^{2} | = [ (2^{4})/(2×3^{2}×5) ]^{2} | |
= [ 2^{3}/(3^{2}×5) ]^{2} | ||
= 2^{6}/(3^{4}×5^{2}) |
But one thing becomes obvious: every exponent is an even number!
So we can see that when you square a rational number, the result will be made up of prime numbers whose exponents are all even numbers.
When you square a rational number, each prime factor has an even exponent.
Back to 2
Now, let us look at the number 2: could this have come about by squaring a rational number?
As a fraction, 2 is 2/1
Which is 2^{1}/1^{1} ,and that has odd exponents!
We could write 1 as 1^{2} (so it has an even exponent), and then we have:
2 = 2^{1}/1^{2}
Even better would be to simplify it to 2^{1}, but either way:
There is an odd exponent (in this case 1)
You could even try things like 2 = 4/2 = 2^{2}/2^{1}, but you still cannot get rid of an odd exponent
So it could not have been made by squaring a rational number!
This means that the value that was squared to make 2 (ie the square root of 2) cannot be a rational number.
In other words, the square root of 2 is irrational.
Try Some More Numbers
How about 3? 3 is 3/1 = 3^{1}
But the 3 has an exponent of 1, so 3 could not have been made by squaring a rational number, either.
The square root of 3 is irrational
How about 4? 4 is 4/1 = 2^{2}
Yes! The exponent is an even number! So 4 can be made by squaring a rational number.
The square root of 4 is rational
Conclusion
This is a neat way of discovering if the square root of a number is irrational or not. This idea can also be extended to cube roots, etc.
It also shows you that there must be irrational numbers (such as the square root of two), in case you ever doubted it!