Zero: The Biography of a Dangerous Idea
Today I started reading about Zero in the book Zero: The Biography of a Dangerous Idea. The part of the book I read was about how ancient people were scared of zero.
One of the main points it talked about was dividing by 0. A fundamental rule of ancient mathematics was that x/y*y=x. 0 ruined this equation. x/0*0 should be x, right? Wrong. This sparked a paradox in the ancient world. They calculated x/0*0, and it was 0, yet their fundamental rule said it would be x. An example of this is the proof that Winston Churchill is a carrot. Behold:
Mathmatical Proof that Winston Churchill is a Carrot
Don't believe me? Take a look at this:
a = 1
b = 1
b^2 = ab (eq. 1)
Since a equals itself, it is obvious that
a^2 = a^2 (eq. 2)
Subtract equation 1 from equation 2. This yields
a^2 – b^2 = a^2 – ab (eq. 3)
We can factor both sides of the equation; a^2 – ab equals a(a – b). Likewise, a^2 – b^2 equals (a + b)(a – b). Substituting into equation 3 we get
(a + b)(a – b) = a(a – b) (eq. 4)
So far, so good. Now divide both sides of the equation by (a – b) and we get
a + b = a (eq. 5)
Subtract a from both sides and we get
b = 0 (eq. 6)
But we set b to 1 at the very beginning of this proof, so this means that
1 = 0 (eq. 7)
Going further, we know that Winston Churchill has one head. But one equals zero by equation 7, so that means that Winston has no head. Likewise, Churchill has zero leafy tops, therefore he has one leafy top. Multiplying both sides of equation 7 by 2, we see that
2 = 0 (eq. 8)
Churchill has two legs, therefore he has no legs. Churchill has two arms, therefore he has no arms. Now multiply equation 7 by Churchill's waist size in inches. This mans that
(Winston's waist size) = 0 (eq. 9)
This means that Winston Churchill tapers to a point. Now what color is Winston Churchill? Take any beam of light that comes from him and select a photon. Multiply equation 7 by the wavelength, and we see that
(Winston's photon's wavelength) = 0 (eq. 10)
But multiplying equation 7 by 640 nanometers, we see that
640 = 0 (eq. 11)
Combining equations 10 and 11, we see that
(Winston's photon's wavelength) = 640 nanometers
This means that this photon – or any other photon that comes from Mr. Churchill – is orange. Therefore Winston Churchill is a bright shade of orange.
To sum up, we have proved, mathematically, that Winston Churchill has no arms, no legs; instead of a head, he has a leafy top; he tapers to a point; and he is bright orange. Clearly, Winston Churchill is a carrot.
As you can see, the problem with this proof is that you divide by zero when you divide by (a-b). This confused the ancient people, because then they could prove that 0 equals any number in the world.
Another thing that confused ancient people is multiplying by 0. Imagine multiplying as stretching a rubber band. If you multiply a number by two, the rubber band is stretched to two times its original length. If you multiply by one half, then the rubber band is relaxed to half its original length. Then what happens when you multiply by 0? The rubber band breaks. The number line has collapsed. This was a scary idea to ancient people, because when they thought of zero, they thought of the void. This scared them, because if the world started as a void, which they believed it did, then surely it would end in a void, as they were also scared of infinity. That made them scared, because they didn't want to die.
Zero, the counterpart to infinity, the middle of everything, was invented by the Babylonians, banned by the Greeks, worshiped by the Hindus, and used to fend off heretics by the Church. This controversial story of a simple number shows how much impact this had on ancient mathematics.
One of the main points it talked about was dividing by 0. A fundamental rule of ancient mathematics was that x/y*y=x. 0 ruined this equation. x/0*0 should be x, right? Wrong. This sparked a paradox in the ancient world. They calculated x/0*0, and it was 0, yet their fundamental rule said it would be x. An example of this is the proof that Winston Churchill is a carrot. Behold:
Mathmatical Proof that Winston Churchill is a Carrot
Don't believe me? Take a look at this:
a = 1
b = 1
b^2 = ab (eq. 1)
Since a equals itself, it is obvious that
a^2 = a^2 (eq. 2)
Subtract equation 1 from equation 2. This yields
a^2 – b^2 = a^2 – ab (eq. 3)
We can factor both sides of the equation; a^2 – ab equals a(a – b). Likewise, a^2 – b^2 equals (a + b)(a – b). Substituting into equation 3 we get
(a + b)(a – b) = a(a – b) (eq. 4)
So far, so good. Now divide both sides of the equation by (a – b) and we get
a + b = a (eq. 5)
Subtract a from both sides and we get
b = 0 (eq. 6)
But we set b to 1 at the very beginning of this proof, so this means that
1 = 0 (eq. 7)
Going further, we know that Winston Churchill has one head. But one equals zero by equation 7, so that means that Winston has no head. Likewise, Churchill has zero leafy tops, therefore he has one leafy top. Multiplying both sides of equation 7 by 2, we see that
2 = 0 (eq. 8)
Churchill has two legs, therefore he has no legs. Churchill has two arms, therefore he has no arms. Now multiply equation 7 by Churchill's waist size in inches. This mans that
(Winston's waist size) = 0 (eq. 9)
This means that Winston Churchill tapers to a point. Now what color is Winston Churchill? Take any beam of light that comes from him and select a photon. Multiply equation 7 by the wavelength, and we see that
(Winston's photon's wavelength) = 0 (eq. 10)
But multiplying equation 7 by 640 nanometers, we see that
640 = 0 (eq. 11)
Combining equations 10 and 11, we see that
(Winston's photon's wavelength) = 640 nanometers
This means that this photon – or any other photon that comes from Mr. Churchill – is orange. Therefore Winston Churchill is a bright shade of orange.
To sum up, we have proved, mathematically, that Winston Churchill has no arms, no legs; instead of a head, he has a leafy top; he tapers to a point; and he is bright orange. Clearly, Winston Churchill is a carrot.
As you can see, the problem with this proof is that you divide by zero when you divide by (a-b). This confused the ancient people, because then they could prove that 0 equals any number in the world.
Another thing that confused ancient people is multiplying by 0. Imagine multiplying as stretching a rubber band. If you multiply a number by two, the rubber band is stretched to two times its original length. If you multiply by one half, then the rubber band is relaxed to half its original length. Then what happens when you multiply by 0? The rubber band breaks. The number line has collapsed. This was a scary idea to ancient people, because when they thought of zero, they thought of the void. This scared them, because if the world started as a void, which they believed it did, then surely it would end in a void, as they were also scared of infinity. That made them scared, because they didn't want to die.
Zero, the counterpart to infinity, the middle of everything, was invented by the Babylonians, banned by the Greeks, worshiped by the Hindus, and used to fend off heretics by the Church. This controversial story of a simple number shows how much impact this had on ancient mathematics.
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