Posted: 1/28/2009 8:24:09 AM EDT
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I've always sort of been curious at why 99*101 is not equal to 100^2. Obviously, it isn't, but I couldn't get my head around why.
While thinking about it today, I noticed that the difference will always follow a simple relationship: (x-y) * (x+y) = (x^2) - (y^2) This is probably a very simple algebraic relationship, but I don't remember the necessary rules involved. What's the proof for that, or what's a good board to ask that on? |
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I believe XKCD has one of the best math forums on the internet.
I can't get it to load at the moment, but if you post there, you'll get a lot of really good answers; whereas, you may get just a few answers here. http://xkcd.com/ at the moment, i can't get it to load, but it's really good. you should check it out sometime. edit: it seems they're back up and running: http://forums.xkcd.com/viewforum.php?f=17 |
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I've always sort of been curious at why 99*101 is not equal to 100. Obviously, it isn't, but I couldn't get my head around why. While thinking about it today, I noticed that the difference will always follow a simple relationship: (x-y) * (x+y) = x - (y^2) This is probably a very simple algebraic relationship, but I don't remember the necessary rules involved. What's the proof for that, or what's a good board to ask that on? Your equation is incorrect. Using your example with x = 100 and y =1: LHS (100-1)*(100+1) =9999 RHS 100 - 1^2 = 99 |
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I've always sort of been curious at why 99*101 is not equal to 100. Obviously, it isn't, but I couldn't get my head around why. While thinking about it today, I noticed that the difference will always follow a simple relationship: (x-y) * (x+y) = x - (y^2) This is probably a very simple algebraic relationship, but I don't remember the necessary rules involved. What's the proof for that, or what's a good board to ask that on? that does not equal x-y^2. 
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I've always sort of been curious at why 99*101 is not equal to 100. Obviously, it isn't, but I couldn't get my head around why. While thinking about it today, I noticed that the difference will always follow a simple relationship: (x-y) * (x+y) = x - (y^2) This is probably a very simple algebraic relationship, but I don't remember the necessary rules involved. What's the proof for that, or what's a good board to ask that on? Um... dude... (x-y) * (x+y) = x^2 - xy - y^2 And the reason 99*101 is not 100 is because you are MULTIPLYING. i.e., adding up 99 + 99 one hundred one times. Is this a serious question? Please tell me this is not a college student I'm responding to... my 13 yr old can do this in her sleep... |
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I've always sort of been curious at why 99*101 is not equal to 100. Obviously, it isn't, but I couldn't get my head around why. While thinking about it today, I noticed that the difference will always follow a simple relationship: (x-y) * (x+y) = x - (y^2) This is probably a very simple algebraic relationship, but I don't remember the necessary rules involved. What's the proof for that, or what's a good board to ask that on? Um... dude... (x-y) * (x+y) = x^2 - xy - y^2 And the reason 99*101 is not 100 is because you are MULTIPLYING. i.e., adding up 99 + 99 one hundred one times. Is this a serious question? Please tell me this is not a college student I'm responding to... my 13 yr old can do this in her sleep... no. (x-y)*(x+y) = x^2-y^2 
good grief. I hope you're not tutoring her in Algebra 1 and 2 
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I've always sort of been curious at why 99*101 is not equal to 100. Obviously, it isn't, but I couldn't get my head around why. While thinking about it today, I noticed that the difference will always follow a simple relationship: (x-y) * (x+y) = x - (y^2) This is probably a very simple algebraic relationship, but I don't remember the necessary rules involved. What's the proof for that, or what's a good board to ask that on? Um... dude... (x-y) * (x+y) = x^2 - xy - y^2 And the reason 99*101 is not 100 is because you are MULTIPLYING. i.e., adding up 99 + 99 one hundred one times. Is this a serious question? Please tell me this is not a college student I'm responding to... my 13 yr old can do this in her sleep... (x-y)(x+y)=x^2-y^2 |
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Wrong, I've been out of college for 8 years.
Please change the thread title to "Help a college kid cheat on his exam". I'm genuinely just curious. And, yes, I meant 100^2, not just 100. Sorry... it just sounded dorky enough that I made some assumptions. |
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Wrong, I've been out of college for 8 years.
Please change the thread title to "Help a college kid cheat on his exam". I'm genuinely just curious. And, yes, I meant 100^2, not just 100. (x-y)(x+y)=x(x+y) - y(x+y) = x^2 + xy -y(x+y) = x^2 +xy -(yx + y^2) = x^2 +xy -xy -y^2 = x^2 - y^2 |
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I've just always been a little curious about it. Clearly, 9*11 is not the same as 10*10, yet it sort of seemed that, since you're going the same distance both ways, it should cancel out.
It makes sense now, and I'll never have to think about it on the can again. Thanks again, norske. |
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I guess you're not seeing the relationship that the rest of us are seeing.
I guess I don't understand why you would not understand 3*5 != 4^2. Check out norske's post. norske's post is from Algebra 2 if I recall. Basic rules of factoring polynomials. |
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Because I want to understand how it works.
why is 3*5 != 4^2? why do you need a proof? I guess I don't understand why you would not understand 3*5 != 4^2. I think you mean 3*5 = 4^2 -1^2 which is true. no I mean 3*5 (15) not equal to 4*4 (16). |
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That much is obvious, but 3*5 is equal to 16 - (1*1). And 2*6 is equal to 16 - (2*2). And 1 and 2 are the distances from 4 in the respective equations.
no I mean 3*5 (15) not equal to 4*4 (16). I've got it, now, anyway. I don't think I've used FOIL in 15 years. I must've in calc, I guess, but I don't remember it at all. Come to think of it, I don't remember much of anything from calc. |
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Quoted: I guess I can kind of see the reasoning, but...I've just always been a little curious about it. Clearly, 9*11 is not the same as 10*10, yet it sort of seemed that, since you're going the same distance both ways, it should cancel out. Would you expect the same from 8 * 12? 7 * 13? 6 * 14? 5 * 15? 4 * 16? 3 * 17? 2 * 18? 1 * 19? |
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I've always sort of been curious at why 99*101 is not equal to 100. Obviously, it isn't, but I couldn't get my head around why. Because it's (99 * 100) + (99 * 1) = 9900 + 99.. not... whatever you think it is, which I can't imagine. Why would it be 100 (or 1,000 or 10,000 or whatever you mean)? While thinking about it today, I noticed that the difference will always follow a simple relationship: (x-y) * (x+y) = x - (y^2) This is probably a very simple algebraic relationship, but I don't remember the necessary rules involved. What's the proof for that, or what's a good board to ask that on? [/quote] There is no proof because it's just transposition / simplification. It's like asking "why does 2x+ x = 3x??" |
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No, but I didn't understand why not. Figuring out that the difference was the distance squared only added to the puzzle.
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I guess I can kind of see the reasoning, but...
I've just always been a little curious about it. Clearly, 9*11 is not the same as 10*10, yet it sort of seemed that, since you're going the same distance both ways, it should cancel out. Would you expect the same from 8 * 12? 7 * 13? 6 * 14? 5 * 15? 4 * 16? 3 * 17? 2 * 18? 1 * 19? |
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Because I want to understand how it works.
why is 3*5 != 4^2? why do you need a proof? I guess I don't understand why you would not understand 3*5 != 4^2. I think you mean 3*5 = 4^2 -1^2 which is true. no I mean 3*5 (15) not equal to 4*4 (16). For the record, I was trying to figure out what factorials had to do with all of this.
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I've always sort of been curious at why 99*101 is not equal to 100^2. Obviously, it isn't, but I couldn't get my head around why. While thinking about it today, I noticed that the difference will always follow a simple relationship: (x-y) * (x+y) = (x^2) - (y^2) This is probably a very simple algebraic relationship, but I don't remember the necessary rules involved. What's the proof for that, or what's a good board to ask that on? The reason 99*101 doesn't equal 100^2 has nothing to do with the second equation you posted. (Although as another poster showed the equation is correct using the FOIL method, that equation still doesn't answer the question of the difference between 99*101 and 100^2) Take any number (call it x). (x-1)*(x+1) = x^2 -1 That's always the relationship in that case. If you take any number and multiply the number before it and the number after it, it will always be equal to one less than the square of that number. For example: 24 * 26 = 624 25^2 = 625 The "difference" is always 1. |
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Yes, 1 is 1*1.
The reason 99*101 doesn't equal 100^2 has nothing to do with the second equation you posted. (Although as another poster showed the equation is correct using the FOIL method, that equation still doesn't answer the question of the difference between 99*101 and 100^2) Take any number (call it x). (x-1)*(x+1) = x^2 -1 That's always the relationship in that case. If you take any number and multiply the number before it and the number after it, it will always be equal to one less than the square of that number. For example: 24 * 26 = 624 25^2 = 625 The "difference" is always 1. Take 23 * 27 = 621 625 - 621 = 4 25 - 23 = 2 2*2 = 4 The difference is only 1 for a distance of 1 because 1^2 happens to be 1. If the distance is 2, the difference is 4. If the distance is 3, difference is 9. Etc. It's a funky relationship to me. Odd to see math work. |
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I've always sort of been curious at why 99*101 is not equal to 100^2. Obviously, it isn't, but I couldn't get my head around why. You've added some riddle to this. If you are going to change the two numbers you are mutiplying, it has to be in the same RATIO, not the same AMOUNT. For example, 14x12=28x6. I used a divisor of two on each number. Under your example I'd do 14x12=15x11. That's just ghey. You just happened to pick a combo where the product comes close to give the allusion that something is wrong. |
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The way to think about this is by counting, not by looking at the underlying algebra.
Imagine what 100x100 really means. It means you have 100 rows of 100 squares each. Now count the number of squares. Now think about trying to create 99x101 with the same number of squares. You start with 100 rows of 100 squares, and take away 100 squares, so that you have 100 rows of 99 squares. Now you add a row to the top with the squares that you removed. You are left with an extra square, since each row now only has 99 squares, but you removed 100. Obviously you'd need the algebra to prove this, but I think the counting is more intuitively convincing about why 100x100 is not equal to 99x101. |
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I've always sort of been curious at why 99*101 is not equal to 100. Obviously, it isn't, but I couldn't get my head around why. While thinking about it today, I noticed that the difference will always follow a simple relationship: (x-y) * (x+y) = x - (y^2) This is probably a very simple algebraic relationship, but I don't remember the necessary rules involved. What's the proof for that, or what's a good board to ask that on? Um... dude... (x-y) * (x+y) = x^2 - xy - y^2 And the reason 99*101 is not 100 is because you are MULTIPLYING. i.e., adding up 99 + 99 one hundred one times. Is this a serious question? Please tell me this is not a college student I'm responding to... my 13 yr old can do this in her sleep... no. (x-y)*(x+y) = x^2-y^2
good grief. I hope you're not tutoring her in Algebra 1 and 2 haha! You're right, my bad! 
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Yes, 1 is 1*1.
The reason 99*101 doesn't equal 100^2 has nothing to do with the second equation you posted. (Although as another poster showed the equation is correct using the FOIL method, that equation still doesn't answer the question of the difference between 99*101 and 100^2) Take any number (call it x). (x-1)*(x+1) = x^2 -1 That's always the relationship in that case. If you take any number and multiply the number before it and the number after it, it will always be equal to one less than the square of that number. For example: 24 * 26 = 624 25^2 = 625 The "difference" is always 1. Take 23 * 27 = 621 625 - 621 = 4 25 - 23 = 2 2*2 = 4 The difference is only 1 for a distance of 1 because 1^2 happens to be 1. If the distance is 2, the difference is 4. If the distance is 3, difference is 9. Etc. It's a funky relationship to me. Odd to see math work. Ah. I see what you were getting at. I misunderstood your post. Lot of neat little things like that in math. |