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Why fractions are not enough numbers
The is no fraction which square is 2.
The natural numbers are intuitive to us. We all can count, and most of us can count until we get bored.
If you have ever overdrawn your bank account, you know how painfully real negative natural numbers are. So there you have the whole numbers.
If you want to share a whole thing equally among several people, then you know what a fraction is. And if you look a bit closer, you realize there are a lot of fractions. But are there enough? The answer is no.
Here is a very elegant, and easily to understand proof why there is no fraction p which is equal to two if squared. It only requires the knowledge of fractions, of even and odd numbers, and multiplication and division.
This proof is taken from the book “Principles of Mathematical Analysis” by Walter Rudin.
Theorem: There is no fraction p of whole numbers for which p*p = 2 is true.
Proof: Let us say p were such a fraction. So p*p = 2.
Since p is a fraction, we can write p as a fraction of two natural numbers: p = m/n.
A fraction can always be written as a fraction of two whole numbers where not both are even. For example: 16/14 can be written as 8/7. 2 can be written as 2/1.
So we can assume that not both m and n are even. This is vital for this proof!
So far, we have (m*m)/(n*n) = (m/n)*(m/n) = p*p = 2. We can do this, because this is how fractions work.
Example: 25/4 = (5*5)/(2*2) = (5/2)*(5/2).
Now we multiply the left and right side of the equation with (n*n):
(n*n)*(m*m)/(n*n) = 2 * (n*n)
We can do this, because if you have two numbers that are equal, they remain equal if you multiply them both with the same number.
We can simplify the left side:
m*m = 2*n*n
This is the same as writing 1/2 instead of 2/4.
So, m*m is divisible by two, and therefore it is an even number. Therefore, m is even.
Why? Any whole number can be written as the product of prime numbers. For example, 8 is equal to 2*2*2, 21 is equal to 3*7. These are called prime factors.
So if we take the number 12, which is equal to 3*4, then 144 = 12*12 = (3*4)*(3*4)=3*3*4*4. So if the square of a number has a prime factor, then the number itself has that prime factor. So if m*m is divisible by two, then m is also divisible by two, and therefore even.
Now, since we assumed at the start of this proof that not both m and n are even, n should be odd. We are going to prove that this is not possible.
Since m is even, it is divisible by two, and we can write m as 2*k, where k is another whole number. So we have
m*m = (2*k)*(2*k) = 2*2*(k*k) = 4*(k*k)
We plug this into the equation from before
4*(k*k) = m*m = 2*(n*n)
We can divide both sides by 2:
2*(k*k) = n*n
So n is even, for the same reason that m is even. This is a contradiction to our assumption that not both m and n are even. So there can be no fraction which square is equal to two! This completes the proof.
Study on the future of food by the UK GO for Science
You can find the entire study here: http://www.bis.gov.uk/foresight/our-work/projects/current-projects/global-food-and-farming-futures/reports-and-publications
Here is an interesting quote from the executive summary:
“Hunger remains widespread. 925 million people experience hunger: they lack access to sufficient
of the major macronutrients (carbohydrates, fats and protein). Perhaps another billion are thought
to suffer from ‘hidden hunger’, in which important micronutrients (such as vitamins and minerals)
are missing from their diet, with consequent risks of physical and mental impairment. In contrast,
a billion people are substantially over-consuming, spawning a new public health epidemic involving
chronic conditions such as type 2 diabetes and cardiovascular disease. Much of the responsibility for
these three billion people having suboptimal diets lies within the global food system.”
So despite having enough money to buy food, and access to information about what constitutes
a healthy diet, some people are just as bad off as the 925 million that face hunger.