A proof that 1=0

 

Jill: Imagine a number line.

Bill: Okay. 

Jill: So, at one point, there's zero, and then, what's to the right of it?

Bill: 1, 2, 3, and on to infinity.

Jill: What about to the left of it?

Bill: -1, -2, -3, and so on.

Jill: Are there any other numbers?

Bill: What do you mean?

Jill: Like, 1/2, say.

Bill: Sure.

Jill: Where's that?

Bill: Halfway between 0 and 1.

Jill: And what about 0.9?  Is it there?

Bill: Yes, just before the 1.

Jill: And π? 

Bill: That's a little bit beyond 3.

Jill: Okay, great.  How does adding work here?

Bill: The same way it works elsewhere, I suspect.

Jill: Like, say I said to you, add 3+4.  How would that work?

Bill: It would be 7.

Jill: But, I mean, on your number line, how would it work?  You've got 2 points, each of them to the right of the zero, one at 3, and one at 4.  They're 1 apart from each other.  What do you do with them?

Bill: Well, you start at 3, and then you count 4 steps to the right: 4, 5, 6, 7.

Jill: Could you add 6+π? 

Bill: Sure, why not?

Jill: But how?  Surely you can't count π steps to the right.

Bill: Well, maybe not, but you could start at π and count 6 steps to the right.  You'd be somewhere around 9.14159....

Jill: So there isn't an exact answer?

Bill: There's an exact answer, a single point, but I can't say what it is.

Jill: Why not?

Bill: Because the decimal expansion of π is infinitely long.  I could be here all day reciting digits and never get to the end of it. 

Jill: Could you add π+6.5?  

Bill: Sure.  It's just the same principle.  It's just a half a unit past π+6.  So, 9.64159....

Jill: What about π + 6 1/3?

Bill: Uh huh.  It's the same thing.  You start at π, and then you put a line segment 6 1/3 units long at the end of it....

Jill: Wait - line segments?

Bill: Yes.

Jill: I thought numbers were points on the line.  Now, all of the sudden, you're talking about line segments.

Bill: Yes, well, when you add two numbers, what you're really doing is taking the line segments that go from zero up to that point and stacking one right after the other one.  So, when you add 3 + 4, you take the line segment from 0 to 3, and the line segment from 0 to 4, and you put one right after the other, to make a larger line segment from 0 to 7.  Another way to put it is "length."  You're taking something of length 3 and something of length 4 and putting them together, and then seeing how far you are from zero.  

Jill: Okay, I get it.  In fact, one way of thinking of a number line is just: "How far you are from zero."  So, every point on that line is labeled with a number, which is just how far that point is from zero.

Bill: Yes!  In fact, yet another way of thinking about addition here is: You start at zero.  Then you go up to point 3.  Now you make a new number line, superimposed on that first number line, but shifted over, so that the "zero" of this number line is at the 3 of the previous number line.  Now: where is 4, on this new number line?  Well, it's at the point that is 7 away from zero on the original number line.  Do you see what I mean?

Jill: Oh!  Yeah, wow!  I see! 

Bill: And it's the same with π + 6 1/3, or any of the other addition problems.  One word for putting a line segment at the end of another line segment is "concatenation".

Jill: Ah, okay, got it.  And when, earlier, you said that you couldn't count π up from 6, but you could count 6 up from π....

Bill: That's because addition is commutative.  In other words, you can switch around the terms and still get the same answer.  2+3 = 3+2.  It's like taking those line segments, those little sticks, and switching them around.  You're still going to get to the same point. 

Jill: So what you're saying is that adding any two numbers, a and b, is equivalent to determining the length of a line segment constructed by concatenating line segments of lengths a and b.

Bill: Yup.

Jill: And what about more than two numbers?  Like, adding a and b and c is equivalent to concatenating line segments of lengths a and b and c?

Bill: You got it.

Jill: And a sum of n numbers is equivalent to the length of a concatenation of n line segments of the lengths of those numbers?

Bill: Yes.

Jill: Okay.  Let's focus on the region starting at 0 and ending at 1.

Bill: Okay.

Jill: What if you add 1/2 plus 1/2?

Bill: You get 1.

Jill: And if you add 1/3 + 1/3 + 1/3?

Bill: 1.

Jill: And 1/4 + 1/4 + 1/4 + 1/4?

Bill: 1.  In fact, if you have lengths 1/n, and you have n of them, it's always going to add up to 1.

Jill: Always?

Bill: Always.

Jill: So, if I had little sticks of length 1/1000000, and I had a million of them, and I stacked them up, all each in a row, I would eventually get to 1?

Bill: Yup.

Jill: I have a question.

Bill: Shoot.

Jill: What is a circle?

Bill: A circle is the locus of points equidistant from a single point in 2 dimensions.

Jill: What does locus mean?

Bill: It's the set of points defined by a rule.  So in this case, the rule is that the points have to all be the same distance from one point.  That point is the center of the circle, and that distance is the radius.

Jill: Okay.  And what is a line?

Bill: Well, mathematicians use the term "line" a little differently than most people do.  For a mathematician, a line is infinitely long in both directions.  So, like, the number line we were discussing earlier is a line, in the true geometrical sense.  It goes to infinity in the positive direction and the negative direction.  That's why we use the term "line segment" to refer to just a little portion of that line. 

Jill: Can you state that a little more formally?

Bill: Sure.  A line is the locus of all points in one dimension.

Jill: And the definition of a line segment?

Bill: A line segment is the locus of contiguous points in one dimension beginning with and including one point, and ending with and including another point.

Jill: So, back to our number line, formally describe the line segment from 0 to 1.

Bill: It's a line segment, which is the locus of points beginning with and including 0, and ending with and including 1.

Jill: So when you say that 1/2 + 1/2 = 1, what you are saying is....

Bill: ...that if there exists a line segment beginning with and including 0, of length 1/2, and another line segment is transposed so that it begins at the end of the first line segment, and it is also at length 1/2, then these together will form a new line segment beginning at 0 and ending at 1.

Jill: And the same goes for 1/3 + 1/3 + 1/3.  We're just stacking these line segments.

Bill: Yes: If there is a line segment of length 1/3, beginning at 0, and then there is another line segment of length 1/3, transposed so that it begins at the end of the first, and another of length 1/3 transposed so that it begins at the end of the second, we will again have constructed a line segment of length 1.

Jill: And in general....

Bill: If we have n line segments, each of length 1/n, the first one beginning at point zero, and each subsequent one beginning at the endpoint of the previous line segment, we will have constructed a line segment of length 1.

Jill: So this line segment of length 1 - it is a line segment.

Bill: Obviously.

Jill: Which is a locus of points.

Bill: Yup.

Jill: How many points?

Bill: An infinite number of points.

Jill: Is it just some of the points between zero and 1, or all of them?

Bill: All of them.

Jill: So, all the decimals between 0 and 1....

Bill: Yup.

Jill: All the fractions between 0 and 1....

Bill: Yup.

Jill: All the irrational numbers between 0 and 1.

Bill: Yes.  All of them.

Jill: And it is a locus of points.

Bill: Yes.

Jill: That's what it is.

Bill: Yes. 

Jill: Is it anything other than a locus of points?

Bill: No.

Jill: So all of the points between 0 and 1 - that's what this line segment is.

Bill: Yes.  It is constructed out of these points.

Jill: So, if you took all of the points between 0 and 1, inclusive, and put them together, you'd get this line segment.  Because that's what it is.

Bill: Yes. 

Jill: So when you said, earlier, that adding n numbers is equivalent to finding the length of a concatenation of n line segments of the lengths of those n numbers, what you meant was that adding numbers is equivalent to finding the length of a concatenation of n loci of points of those lengths, because that's what a line segment is - a locus of points - right?

Bill: Right.

Jill: And what is a locus, again?

Bill: A set determined by a rule.

Jill: So a line segment is a set of points.

Bill: A set of points determined by a rule, yes.

Jill: How big is a point?

Bill: Mathematicians say that a point has zero size.

Jill: Zero size?

Bill: Yes.

Jill: So we're taking these things that have zero size and adding them up and getting 1.

Bill: Yesss.... the locus of points between and including 0 and 1 forms a line segment of length 1.

Jill: What is 0+0?

Bill: 0.

Jill: What is 0+0+0?

Bill: 0.

Jill: And 0+0+0+0?

Bill: Still 0.

Jill: And what if I had an infinite series of zeroes and added them together?  What would that equal?

Bill: Still 0.

Jill: But you just said that we have constructed a line segment of length 1 out of an infinite number of points, each of length 0.

Bill: Well, yes.

Jill: Let me ask you this: What if we had the locus of points greater than or equal to zero, but strictly less than 1?  Would that form a line segment?

Bill: Yes.

Jill: And what would be the length of that line segment?

Bill: 1.

Jill: But this has one less point than the previously discussed line segment.  It doesn't have the point 1.  

Bill: Right.

Jill: But they're the same length?

Bill: Yes.  Because you're just subtracting the length of that one point, which is zero.  So they're the same length.

Jill: So if we talked about the line segment strictly greater than 0 and strictly less than 1, would that also have length 1?

Bill: Yes.

Jill: Exactly 1, or just really close to 1?

Bill: Exactly 1. 

Jill: So you're saying that a line segment of length 1 is constructed out of a locus of points, each of length 0?

Bill: Yes.  An infinite number of points.

Jill: But we've already discussed the fact that adding an infinite series of zeros still equals zero.

Bill: Yes.

Jill: So length 1 equals an infinite sum of lengths zero.

Bill: Yes.

Jill: Which is zero. 

Bill: Yes.

Jill: And as we've already discussed, geometrical concatenation of lengths is equivalent to addition of numbers.

Bill: Yes. 

Jill: So if I say that an infinite concatenation of lengths is 1, and I say that that same infinite concatenation of lengths is zero, that's equivalent to saying that a sum of an infinite series of numbers is 1, and that that same sum equals zero.

Bill: Yes.

Jill: And if one value equals a second, and that second value equals a third, then the first and the third must be equal to each other.

Bill: True.

Jill: So if an infinite sum of numbers equals 1, and it also equals 0, then 1 must equal 0.

Bill: Unfortunately, that would seem to be the case. 

Jill: So 1 = 0.

Bill: [sighs.] Yes.