# Pop Video: Proving That 1 = 2 Using Basic Algebra

In this video we will present an apparent algebraic proof that 1 is equal to 2, and then examine it line by line to discover a logical error in the working out.

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### Video Transcript

In this video, weβre gonna be using some basic algebra to prove that one is equal to two. Well, maybe.

First of all letβs define two variables, π and π. And weβll say that the value of π is equal to the value of π. Now that gives us the equation π is equal to π. Now letβs take that equation and multiply both sides by π. So π times π is equal to π times π or simply ππ is equal to ππ. But wait! ππ just means π times π or π squared, so we can say π squared is equal to π times π or ππ.

And now letβs subtract π squared from each side of that equation. So weβve done the same thing to both sides of that equation, so itβs still an equation. And π squared minus π squared is equal to ππ minus π squared. Now when weβve got a square thing minus another square thing, we call this the difference of two squares. And we can factorize that expression like this: π squared minus π squared is equal to π minus π times π plus π.

So letβs just check that: positive π times positive π is positive π squared; positive π times positive π is positive ππ; negative π times positive π is negative ππ; and negative π times positive π is negative π squared. Now ππ means π times π, and ππ means π times π. But multiplication is commutative, and that means that it doesnβt matter in which order you multiply things together in. So π times π gives you the same result as π times π, and that means that I can re-express ππ as ππ.

So our expression now reads π squared plus ππ minus ππ minus π squared. And ππ takeaway ππ is nothing. And in multiplying out π minus π times π plus π, Iβve proved that these two things are in fact equal. And that means I can write out the left-hand side of my equation like this. So Iβve now got π minus π times π plus π is equal to ππ minus π squared.

So Iβve factorized the left-hand side of our equation. But over on the right, we can see that both terms have a common factor of π as well, so I can factorize that side too. So weβve now got π minus π times π plus π is equal to π times π minus π. Now youβll notice weβve got π minus π on both sides of our equation, so if I divide both sides by π minus π, I will have π minus π in the numerator and the denominator on both sides and I can cancel them out.

π minus π divided by π minus π on the left-hand side is one and likewise on the right-hand side. So on the left-hand side, Iβve got one times π plus π over one, which is just π plus π. And on the-right hand side, Iβve got π times one over one, which is just π. And that leaves me with π plus π is equal to π.

Now remember, back at the beginning we said that π was equal to π, and that means I can replace π with π in my equation, which gives us π plus π is equal to π. Well π plus π is just equal to two π. So now two π is equal to π. Now I can Divide both sides by π; and on the left, π divided by π is one; and on the right, likewise, π divided by π is one. And that leaves us with two times one over one on the left-hand side, which is just two, and one divided by one on the right-hand side, which is just one.

So there we have it! Two is equal to one, or, indeed, one is equal to two. Well there it is; weβve broken maths! Not much point in carrying on! Okay, pause the video, have a look through line by line, and see if you can find any errors in our logic.

Well, obviously one isnβt equal to two. If it was, I think weβd have heard it on the news by now. So letβs go through our working out line by line and see if we can work out whatβs happened. Well defining two variables π and π and letting them be equal, thereβs no problem with that. And itβs perfectly okay to multiply both sides of our equation by the same thing π. And yes, π squared is equal to π times π, so thatβs correct.

Also subtracting the same thing from each side of our equation does keep it equal, so thatβs alright too. Now we looked at the difference of two squares factorization in some detail earlier on, and thatβs perfectly okay as well. And factorizing out the common factor of π on the right-hand side, thatβs perfectly okay.

Now in moving from step six to step seven, we divided both sides of the equation by π minus π. Well this might normally be okay, but we did say at the beginning that π was equal to π. So π minus π is equal to something minus itself; thatβs zero. So we are dividing both sides of our equation by zero. Now that is a problem. For example, whatβs one divided by zero? Well it doesnβt matter how many times you add zero to itself, youβre never going to reach one.

Anything divided by zero is undefined. So when you divide things in equations, you really must check that what youβre dividing by is not equal to zero. Okay letβs look at it a slightly different way. In line six, weβve got π minus π times π plus π, but π minus π is zero. So the left-hand side really means zero times π plus π, and π minus π is zero on the right-hand side as well so that becomes π times zero.

So weβre saying that zero times something is equal to something times zero. Well zero times something is zero, and something times zero is zero, so here weβre saying that zero is equal to zero. And yup, that is true. But that doesnβt mean that the things that weβre multiplying by zero are also equal. So moving from line six to seven, we donβt know that π plus π is equal to π. We only know that zero times π plus π is equal to zero times π. And that means that all of this is wrong; we havenβt proved that one is equal to two, weβve proved that zero times one is equal to zero times two. Hooray! It looks like that the world of maths is safe after all.