When math just doesn't make sense

I was browsing through my system logs when suddenly, boredom washed over me like a waxer on the McKenzie building's marble floors. Naturally, I fired up fortune and played with it for half an hour. After finding a funny WC Fields quote or two to add to my signature file, this popped up:

(1) X=Y                         ; Given
(2) X^2=XY                      ; Multiply both sides by X
(3) X^2-Y^2=XY-Y^2              ; Subtract Y^2 from both sides
(4) (X+Y)(X-Y)=Y(X-Y)           ; Factor
(5) X+Y=Y                       ; Cancel out (X-Y) term
(6) 2Y=Y                        ; Substitute X for Y, by equation 1
(7) 2=1                         ; Divide both sides by Y
                -- "Omni", proof that 2 equals 1

What a wonderful puzzle! I have seen a couple proofs like this before, but had never been able to sort it out to my satisfaction. Work on it for a while yourself, then read the last paragraph of my blog for today for the solution.

This week is devoted to packing everything into tiny boxes in preparation for our drive down to college on Monday the 5th. Charlotte's done fairly well in sorting and cleaning her belongings. I, on the other hand, still have piles of books, polo shirts, and unidentifiable Korean candies scattered around my room in a roughly horseshoe-shaped gauntlet. I've talked myself out of bringing any novels to read. This pains me greatly, since I just received many new ones for my birthday three weeks ago and haven't yet had the opportunity to pore over them. The necessity of having enough room in the van and the realization that I won't have any time to read them at school anyway have sadly impressed themselves into my mental packing list even as a parent might unknowingly impress his shoe into your stealthy playdoh earthworms while taking a basket of laundry upstairs to sort. *sigh*

Whether we'll actually be able to fit into the van is becoming more and more of a concern. I'm trying to cut back the things I'm bringing. Now that I have my slim LCD monitor, I won't be needing my computer desk -- I'll just set up shop on the desk in the room. If my roommates also have desktops, I'll move it onto my bed somehow. Charlotte's friend Sarah Grace is coming with us, though I don't anticipate her bringing as much as Charlotte is. Charlotte has a HUUUUUUGE pile of ... stuff in HALF of the foyer. She says that's not everything either. We're also bringing something down for a friend who stayed at PCC for the summer.

Speaking of PCC, they made the following announcement today about hurricane Katrina. I'd link to it, but our school's website is an amorphous blob of jelly that for some reason has unpredictably non-persistent content.

Monday, August 29, 8:30 a.m.
Hurricane Katrina is currently making landfall in the southeast Louisiana/Mississippi area. Pensacola is far out on the edge of the storm and is experiencing only tropical storm winds. At the present time, local channel 3 television is reporting winds in our area of 20-30 miles per hour, which is not unusual in stormy weather. Channel 3 also reports that the weather we are presently experiencing is the worst that we will have, though they anticipate it will continue for several hours. At the college campus, we have had one-half inch of rain since midnight. We are expecting gusty winds until later in the day, and students will remain in their shelter areas until the winds calm significantly.

Hopefully, I'll get to talk to some of my friends who are still down there tomorrow sometime. Another friend is supposed to fly to Pensacola on Friday via New Orleans, so I suppose he'll have to have a flight or two rerouted.

In a little over three hours, Dad and I will go to the Sunrise Grill for our Tuesday morning Bible Study with some of the other men from Bethany Chapel. We've been reading through C. H. Mackintosh's Short Papers, and today's article is The Law and the Gospel. The passage under study is the familiar story of the Good Samaritan, found in Luke 10:25-35. Mackintosh applies the story in a rather novel way. He points out that Jesus tells the lawyer who approached him that the end of the law is to love the Lord your God with all your heart, and your neighbor also. Mackintosh reiterates the point found in scripture that the law was given to condemn man in his own righteous, and that it is the grace and love of Christ alone that can save us from sin and death. Here are a few choice quotations:

The law makes no provision for imperfect obedience, however sincere. It makes no allowance for infirmity. Its one brief, pointed inquiry is, “Have you continued in all things?” If you say No (and who can say otherwise?) it can only curse you. Why? Because it is perfect. Were it to pass over a single transgression, it would not be a perfect law. Its very perfection insures the condemnation of the transgressor. “As many as are of works of law (that is, as many as work on the principle, stand on the ground, occupy the platform of works of law) are under the curse,” and cannot possibly be anything else. This establishes the point unanswerably. The law can only prove to be a ministration of death and condemnation to the sinner, simply because he is a sinner and “the law is holy, and just, and good.” It is no use for a man to say, “I am not looking to the law for life or justification, but merely as a rule and for sanctification.” As a rule for what? For the sanctification of what? If you say, “for my old nature,” the answer is, so far from being “a rule of life,” it is “a ministration of death;” and so far from sanctifying the flesh, it condemns it, root and branch. If, on the other hand, you say it is for the new nature, then is your mistake equally obvious, since the apostle expressly declares that “the law is not made for a righteous man” (1 Tim. 1: 9).

...

The law and the sinner are complete opposites — wholly irreconcilable. I must get a new nature, stand upon new ground, be in the new creation, before I can delight in the law of God. “I delight in the law of God after the inward man” (Rom. 7: 22). But how do I get this “inward man,” this new nature? How do I get into the new creation? Not by works of law of any shape or description, but by faith of Jesus Christ. I become united to Christ in the power of a new and endless life, upon which the law has no claim. I died in Christ. Hence the law has no further demand on me. If a man is in prison for murder and dies there, the law is done with him, inasmuch as the life in which the crime was committed is gone. Thus it is with the sinner who believes in Jesus. God sees him to be dead. His old man is crucified. The sentence of the law has been put into execution upon him in the Person of Christ. Had it been executed upon himself, it would have been death eternal, but having been executed upon Christ, His death is of infinite, divine and eternal effectiveness. Moreover, having the power of eternal life in Himself, He rose, as a Conqueror from the tomb after having met every claim. And wonderful to declare, the believer, having died in Him, now lives in Him forever. Christ is his life; Christ is his righteousness; Christ is his rule of life; Christ is his model; Christ is his hope; Christ is his all and in all (Rom. 6, 7; Gal. 2: 20-21; Gal. 3, Gal. 4; Eph. 2: 4-6; Col. 2: 10-15).

You can see the full article and all of Mackintosh's writings on this page. Just search for the title of the article.

Wow! You made it to the end of the post! The Math Forum @ Drexel has a page here on precisely this problematic proof. I admit that I didn't come up with anything better than the error occurred between the 4th and 5th steps. The problem -- and this is the problem with all such apparently impossible proofs -- is that we unwittingly divide by zero. I didn't accept the answer until I worked out the problem several more times myself. Now, I wonder how often I have made this mistake in my math courses. There were so many times when I'd work a problem through and get gibberish, that I wonder if some of that could have been prevented by simple preventative error checking. Perhaps this problem and its solution should be presented to all algebra students as soon as possible in order to avoid simple errors later in their mathematical studies!