Proof that 1=2

[peleschramm]

Hey guys, check out this proof that shows, using simple algebra, that 1 actually equals 2.

Assume: a = b
1.multiply both sides by a
a^2 = a*b

2. subtract b^2 from both sides
a^2-b^2 = a*b-b^2

3. apply the distributive law to both sides
(a+b)(a-b) = b(a-b)

4. divide both sides by (a-b)
(a+b) = b

5. substitute all a's for b's (remember, if a = b you can do this)/
a+a = a

6. regroup the two a's in the left side, and rename it 2a
2a = a

7. divide both sides by a
2 = 1

My friend showed me this last year. I just remembered it and decided to share. Can anyone find the mistake? Try to find it yourself before seeking help from the responses or any help at all from that matter.

[wickedbob]

The mistake? He used algebra, that was simple. You can't do better than that?

[catinabag1]

Hey guys, check out this proof that shows, using simple algebra, that 1 actually equals 2.

Assume: a = b
1.multiply both sides by a
a^2 = a*b

2. subtract b^2 from both sides
a^2-b^2 = a*b-b^2

3. apply the distributive law to both sides
(a+b)(a-b) = b(a-b)

4. divide both sides by (a-b)
(a+b) = b

5. substitute all a's for b's (remember, if a = b you can do this)/
a+a = a

6. regroup the two a's in the left side, and rename it 2a
2a = a

7. divide both sides by a
2 = 1

My friend showed me this last year. I just remembered it and decided to share. Can anyone find the mistake? Try to find it yourself before seeking help from the responses or any help at all from that matter.

the mistake is you divide by 0.

[peleschramm]

bravo

[catinabag1]

bravo
thank you.:D i love math. my calculus teacher showed this one day in class.

[catinabag1]

sorry i killed your thread so early too.:D

[kington99]

yes without even reading it i can guess it's a div zero problem, these things always are.

[Fooby]

I have an interesting theory that might keep this thread open, but not relating to its name.

Is it possible to find the square route of a negative number, my mathematics teacher keeps telling us it's impossible and the answer doesn't exist, but something in the back of my mind keeps pushing it back into my thoughts. I need an answer!

[Hazmat]

I have an interesting theory that might keep this thread open, but not relating to its name.

Is it possible to find the square route of a negative number, my mathematics teacher keeps telling us it's impossible and the answer doesn't exist, but something in the back of my mind keeps pushing it back into my thoughts. I need an answer!
umm.......errr.........um..... Actually i never thought of that b4. I wonder too if it somewhat possible. :confused:

[kington99]

I have an interesting theory that might keep this thread open, but not relating to its name.

Is it possible to find the square route of a negative number, my mathematics teacher keeps telling us it's impossible and the answer doesn't exist, but something in the back of my mind keeps pushing it back into my thoughts. I need an answer!


Yes you can, it's called a imaginary number.

The root of -1 is called i. This can be multiplied by other number to create other negative roots i.e. the root of -9 is 3i.

[captainkrunk61]

And technically your teacher wasn't lying i stands for imaginary, and imaginary number do no exist, instead they get substituted by the letter i for Algebraic purposes. Now you can go into class and show eerybody up!

[peleschramm]

In algebra, however, you can substitute i for a route of a negative number and then eventually when solving a equation or something it can "turn" back into a real number (I don't know if that made sense). What I am trying to say is that imaginary numbers aren't completely useless, though it has been awhile (about 2 years) since I've covered them in depth.

[unifreak7]

This works fine! Just look at the quatum physics side of things.

[peleschramm]

This works fine! Just look at the quatum physics side of things.
I must confess, I'm afraid I fail to understand what you are trying to say.

[kington99]

And technically your teacher wasn't lying i stands for imaginary, and imaginary number do no exist, instead they get substituted by the letter i for Algebraic purposes. Now you can go into class and show eerybody up!


She also apparently said it was impossible. It is not. Imaginary numbers exist, they just don't fall in to the sub-group 'real numbers'. i is a number in the same way pi is a number, but it cannot be represented on a number line the way pi can, it requires a 2 dimensional Argand diagram.

[peleschramm]

She also apparently said it was impossible. It is not. Imaginary numbers exist, they just don't fall in to the sub-group 'real numbers'. i is a number in the same way pi is a number, but it cannot be represented on a number line the way pi can, it requires a 2 dimensional Argand diagram.
Eh, you can see pi in action in front of your eyes (for those who didn't know, pi is simply the amount of times you multiply the diameter by to get the circumference). To the extent of my knowledge, imaginary numbers cannot be seen in any way really.

Pi is an irrational number, not an imaginary number. Irrational and imaginary numbers are two completely different things.

Edit: but really, the keyword here is imaginary NUMBER, these are all numbers.

[1-wheeled-grape]

wow i am slightly confused now. I got the first thing, but 2 dimensional Argand diagrams? I was told that you can't get the square route of a minus number so dont try, and I havent tried. Teachers aren't that dumb (except from choosing to go back to school:eek: ), when it gets to teaching they usually know what theyre on about so just go with what they say.

[peleschramm]

wow i am slightly confused now. I got the first thing, but 2 dimensional Argand diagrams? I was told that you can't get the square route of a minus number so dont try, and I havent tried. Teachers aren't that dumb (except from choosing to go back to school:eek: ), when it gets to teaching they usually know what theyre on about so just go with what they say.
Are you sure you're teacher didn't say, "don't try this since you don't know how to do this yet"? or something a long the lines of that.

Or even more likely "There is no REAL solution to this". That would be true, there is only an imaginary solution. Perhaps that is what she meant. If she said it was impossible, and there is no way to deal with square routes of negative numbers, I would take away her teaching license, or if I was nice I would just make her teach Kindergarten.

[Seager]

wow i am slightly confused now. I got the first thing, but 2 dimensional Argand diagrams? I was told that you can't get the square route of a minus number so dont try, and I havent tried. Teachers aren't that dumb (except from choosing to go back to school:eek: ), when it gets to teaching they usually know what theyre on about so just go with what they say.

Put yourself in your teacher's shoes. She/He probably very well knows that you can find the square root of a negative using i. She also knows that normal square roots are a hard concept to teach and if she starts throwing out imaginary numbers NOW, a few years before you normally learn them, the students will just get more confused.

Sometimes it's easier to say you can't do it than say "well, you can, but that's super complicated and you'll learn it later." A lot of kids won't accept that answer and it will just screw the whole class up.

I would just not mention them at all, but if someone asked tell them to come up later and that you'll explain it. That way the whole class doesn't get gunked up. Too much information at once is a recipe for failure when teaching math.

[kington99]

Eh, you can see pi in action in front of your eyes (for those who didn't know, pi is simply the amount of times you multiply the diameter by to get the circumference). To the extent of my knowledge, imaginary numbers cannot be seen in any way really.

Pi is an irrational number, not an imaginary number. Irrational and imaginary numbers are two completely different things.

Edit: but really, the keyword here is imaginary NUMBER, these are all numbers.


All true, all irrelevant.

whether you see a number infront of your eyes or not makes no difference to whether it exists or not. Everytime i analyse an electronic circuit i see imaginary numers everywhere.

I used pi as an example of a common algebraic representation of a number. I could have used x where x=2 but it would have required more explanation. The fact that pi is irrational has precisely nothig to do with my argument, it was simply a number that you can spot on a number line, somewhere between 3 and 4.

[phlegm]

Eh, you can see pi in action in front of your eyes...

I've never seen a perfect circle. Have you?

I do have faith in the existence of perfect circles though. :p

[Seager]

I've never seen a perfect circle. Have you?

I do have faith in the existence of perfect circles though. :p

I saw them playing in the Quad Cities a few years ago. They were pretty good.

[peleschramm]

All true, all irrelevant.

whether you see a number infront of your eyes or not makes no difference to whether it exists or not. Everytime i analyse an electronic circuit i see imaginary numers everywhere.

I used pi as an example of a common algebraic representation of a number. I could have used x where x=2 but it would have required more explanation. The fact that pi is irrational has precisely nothig to do with my argument, it was simply a number that you can spot on a number line, somewhere between 3 and 4.
Please explain seeing imaginary numbers.

I was just pointing out that I thought you were confused between irrational numbers (which are a type of REAL number), and imaginary numbers (which aren't)

And Phlegm, I hope you won't be offended if I simply ignore your statement, as I think you were joking and it was sort of irrelevant. I'm sorry if you meant that seriously.

[phlegm]

And Phlegm, I hope you won't be offended if I simply ignore your statement, as I think you were joking and it was sort of irrelevant. I'm sorry if you meant that seriously.

I'm not offended, but I did mean it seriously. Maybe I should've left the :p off. I am a believer in the world of math! :)

[kombi]

http://mathforum.org/dr.math/faq/faq.false.proof.html

[kington99]

Please explain seeing imaginary numbers.



please explain seeing pi. When you look at a circle you think of pi, when i look at a capacitor i think of i.

[peleschramm]

please explain seeing pi. When you look at a circle you think of pi, when i look at a capacitor i think of i.
Yes, But I explained why a circle has anything to do with Pi. You haven't yet explained how a capacitor has anything at all to do with imaginary numbers. I'm just interested.

[BenjaminD]

Btw, check this proof of 2 = 1 (found on the internet):

-2 = -2
4 - 6 = 1 - 3
4 - 6 + 9/4 = 1 - 3 + 9/4
(2 - 3/2)2 = (1 - 3/2)2
2 - 3/2 = 1 - 3/2
2 = 1

greets,Benjamin

[peleschramm]

The problem is that the solution to a square route always has to have a +/- sign.

[surfcolorado]

Btw, check this proof of 2 = 1 (found on the internet):

-2 = -2
4 - 6 = 1 - 3
4 - 6 + 9/4 = 1 - 3 + 9/4

"2(2-3+9/6)=2(1/2-3/2+9/6)
2(2-3+3/2)=2(1/2-3/2+3/2)
2(2-3/2)=2(2-3/2)"

(2 - 3/2)2 = (1 - 3/2)2
2 - 3/2 = 1 - 3/2
2 = 1

greets,Benjamin

1-3+9/4 does not = (1-3/2)2

[peleschramm]

he meant ^2

[surfcolorado]

just saw that you mean (2-3/2)^2 not (2-3/2)*2

peleschramm, you beat me to it

[kington99]

Yes, But I explained why a circle has anything to do with Pi. You haven't yet explained how a capacitor has anything at all to do with imaginary numbers. I'm just interested.

oh ok, fair dos, the impedance of a capacitor is -j/wC (j is used in place of i in electronics to avoid confusion with the use of i for current).
The impedance of an inductor is jwL.

So any calculation involving the relationship between potential difference ad current will involve j.

[peleschramm]

oh ok, fair dos, the impedance of a capacitor is -j/wC (j is used in place of i in electronics to avoid confusion with the use of i for current).
The impedance of an inductor is jwL.

So any calculation involving the relationship between potential difference ad current will involve j.
Ok, I guess I'll take your word for it, as I know very little about electricity.

[kington99]

Ok, I guess I'll take your word for it

no need (http://en.wikipedia.org/wiki/Capacitor#Impedance)

[catinabag1]

wow i am slightly confused now. I got the first thing, but 2 dimensional Argand diagrams? I was told that you can't get the square route of a minus number so dont try, and I havent tried. Teachers aren't that dumb (except from choosing to go back to school:eek: ), when it gets to teaching they usually know what theyre on about so just go with what they say.
actually you can take the square root of a negative number using i.

i=(-1)^(1/2) so (-4)^(1/2)=2i

[Seager]

actually you can take the square root of a negative number using i.

i=(-1)^(1/2) so (-4)^(1/2)=2i

Hey math ninja, you're freakin' killing me. Not only have we established that in this thread already, but we've even discussed the best way to teach or not teach it.

[catinabag1]

Hey math ninja, you're freakin' killing me. Not only have we established that in this thread already, but we've even discussed the best way to teach or not teach it.
well excuse me for not wanting to read through pages and pages of crap before i post.

[Seager]

well excuse me for not wanting to read through pages and pages of crap before i post.

Pages and pages? Dude, you posted halfway down the third page. Also, the answer was posted two posts after the question, ON THE FIRST PAGE. You only had to read three posts.

No hard feelings, I just think it's funny, that's all.

[monkeyman]

Seager, give him a break. Talking at everyone is much easier than talking with them.

I love this thread. You guys are all nerds, and it's great.

http://www.wegotcards.com/cards/friends/insults/nerd.GIF

[Seager]

I love this thread. You guys are all nerds, and it's great.

Dude, you need to read the Math-Fu fight they had in the other math thread. It was ridiculous.

[surfcolorado]

I love this thread. You guys are all nerds, and it's great.


Yeah, I didn't like being a math/science nerd in high school, I wanted to be more in the cool crowd, but now with my professional engineers license and a six figure income in about 5 more years. I don't mind being called a nerd or engineerd.

[Seager]

Yeah, I didn't like being a math/science nerd in high school, I wanted to be more in the cool crowd, but now with my professional engineers license and a six figure income in about 5 more years. I don't mind being called a nerd or engineerd.

Ironically, those of us who are living paycheck to paycheck probably use just as much math everyday trying to budget things out. :)

[surfcolorado]

Oh, I didn't say I was not living paycheck to paycheck right now. My wife is a stay at home mom with two kids. We are definitely living paycheck to paycheck right now. I'm saying my future looks bright. My income is growing, however it would grow better if the house market was better. I actually write a lot more than I thought I would as an engineer. Half of my work is calculating traffic growth and analyzing intersections, half is writing a report of my findings. Five years ago, 90% of time was using math, but back then I designed roads.

[Seager]

Oh, I didn't say I was not living paycheck to paycheck right now. My wife is a stay at home mom with two kids. We are definitely living paycheck to paycheck right now. I'm saying my future looks bright. My income is growing, however it would grow better if the house market was better. I actually write a lot more than I thought I would as an engineer. Half of my work is calculating traffic growth and analyzing intersections, half is writing a report of my findings. Five years ago, 90% of time was using math, but back then I designed roads.

Sooo... If I drive through Ft. Collins and the roads suck I can blame you?

Hey, don't you guys have big city burrito there? I hear it's great!

[surfcolorado]

Sooo... If I drive through Ft. Collins and the roads suck I can blame you?

Hey, don't you guys have big city burrito there? I hear it's great!

I love the big city potato burrito. There is one in Fort Collins, Windsor, and Loveland.

If you drive Airport Road in Longmont from 17th to Clover Basin and it sucks you can blame me. I designed that section of Road (horiztonal, vertical, drainage, turn lanes, etc.)

Yes, I'll take some of the blame if you drive in Fort Collins, Loveland, northern Colorado and you have excessive delays (above the Level of service allowed in the Larimer County Urban Area Street Standards) at intersections. I take no responsibility for the condition of the roads, I only analyze the delays and required improvements (turn lanes, more through lanes, signal phasing, intersection control (i.e signal, stop sign, roundabout)) at intersections.

[peleschramm]

Seager, give him a break. Talking at everyone is much easier than talking with them.

I love this thread. You guys are all nerds, and it's great.

Haha, I swear I really don't enjoy most math. There is very little in it i enjoy.

[monkeyman]

Haha, I swear I really don't enjoy most math. There is very little in it i enjoy.
I was actually saying that I think it's cool how we're all such nerds...but ok then. No nerd points for you.

[peleschramm]

Haha Yeah, I know. I do like such discussions on math. I can tell you I HATE trigonometry, which is what we have been doing most of the year. But I don't like most math, just some of it is somewhat interesting, so I'm only half a nerd ;)

[maestro8]

Haha, I swear I really don't enjoy most math. There is very little in it i enjoy.
God told you to go forth and multiply. You're really going to let him down if you don't do your math homework.

[peleschramm]

God told you to go forth and multiply. You're really going to let him down if you don't do your math homework.
I'm past multiplying... and past believing in god for that matter.

I know you were kidding though.

[unifreak7]

Obvisiouly there is no such real function to prove that any number equals anything else as it stands alone in math. If there was math would fall apart. The only thing you can do is something like this.

(9)^1/2 = 3 and -3

One number = Two different numbers, at the same time. (Closer to quantum here)

1 = 2

Pi is an amazing number, and is linked to a power serious. That's why it won't end.

Instead of dealing with numbers that are concrete, go with why an electron seems to go from one place to another, popping out of exsistence, and coming back, with time and distance faster than C. Or why observation makes an atom be a particle and not a wave (i have a macro concept of that if anyone woud like to hear it).

-SJ

[maestro8]

One number = Two different numbers, at the same time. (Closer to quantum here)
I'm going to have to go ahead and disagree with you there.

What you probably meant to say was:

One equation = two solutions.