3=1?

[Blue The Pikachu 537]

Ok guys, for those of you who are mathematical (Like me) Can someone look at this and tell me what's wrong with this?

Apparently 3=1 in this equation. Sorry if its illegible, i have bad handwriting. I know this is not possible, but this equation makes my head hurt. Can anyone prove this wrong? XD

[PROJECT: Simon 3,954]

OMG. Gotta go tell my math teacher to show that Im smart!

[ThatOneComrade 1,091]

I wish I was smart enough to understand this 

[boiteporte 591]

Easy.

 

Your fifth equal sign is wrong. I will let you figure out why.

Edited November 7, 2014 by boiteporte

[1111 1,284]

mmmm idk mabey the i is just a troll

[Blue The Pikachu 537]

mmmm idk mabey the i is just a troll

Nah, My Pre-calculus teacher showed us this, i just DON'T KNOW where the error is at! XD I don't see error at the fifth equal sign, seems right to me :/

[VitalSpark 1,830]

Really, the fifth? This just applies the rule sqrt(a/b) == sqrt(a)/sqrt(b ), which I believe holds for all a and b where b is non-zero. I think the fifth equality looks sound.

 

I'd say the 6th one is the bad equals sign. The sixth one implies that:

• sqrt(1) == 1
• sqrt(-1) == i

But actually:

• sqrt(1) == 1 or -1
• sqrt(-1) == i or -i

The "or" bits have been conveniently forgotten, and the rest of the chain just assumes the square roots are equivalent to 1 and i respectively.

Edited November 7, 2014 by Vital Spark

[Admiral Regulus 2,769]

I found the error. At the beginning of the bottom row, you have 2 + (1 / i / i). This is still equal to 3, because i / i is 1. However, in the next part, the i / i is changed to i², which is not 1, but -1. i / i is NOT equal to i². Effectively, you're changing this from 2 + 1 to 2 - 1 to get 1 instead of 3.

 

 

Really, the fifth? This just applies the rule sqrt(a/b) == sqrt(a)/sqrt(b ), which I believe holds for all a and b where b is non-zero. I think the fifth equality looks sound.

 

I'd say the 6th one is the bad equals sign. The sixth one implies that:

• sqrt(1) == 1
• sqrt(-1) == i

But actually:

• sqrt(1) == 1 or -1
• sqrt(-1) == i or -i

The "or" bits have been conveniently forgotten, and the rest of the chain just assumes the square roots are equivalent to 1 and i respectively.

 

That doesn't matter, actually. Since it is OR, as long as either component is used, the equality holds true.

Edited November 7, 2014 by Regulus

[CloudFyre 848]

Really, the fifth? This just applies the rule sqrt(a/b) == sqrt(a)/sqrt( , which I believe holds for all a and b where b is non-zero.

 

I'd say the 6th one is the bad equals sign. The sixth one implies that:

• sqrt(1) == 1
• sqrt(-1) == i

But actually:

• sqrt(1) == 1 or -1
• sqrt(-1) == i or -i

The "or" bits have been conveniently forgotten.

*Waves* Vital has it right.

 

There are plenty of mathematical fallacies that can be found online; most of them involve either setting a variable equal to zero somewhere (which people don't think about), or someone misses the fact that square roots have two answers, assuming that the problem definition states that all numbers are being used. The principal square root definition is values greater than 0 (just the positive answer), and the full square root definition includes both the positive and negative answer. If the problem states that you only use positive numbers (X >= 0), then you only need to bother with the positive value.

 

Edit: (Simplified.)

Edited November 7, 2014 by CloudFyre

[boiteporte 591]

Hm, I studied Physics and Chemistry. Not mathematics.

 

Too bad,  right ?

[VitalSpark 1,830]

square roots have two answers.

 

With a bit of interpretation Regulus's post can be used to find a second place in the chain where the same trick gets pulled, which I missed the first time around.

[Snuggie Lord 170]

I have no idea what these people are on about.

[Castle_Cross 331]

Everybody is arguing about the equation, and I'm wondering what's that drawing in the other page xD

[VitalSpark 1,830]

I have no idea what these people are on about.

 

Don't worry. It's all imaginary.

[Snuggie Lord 170]

Don't worry. It's all imaginary.

Sure...Whatever...Bro...

 

I have no idea

Edited November 7, 2014 by Snuggie Lord

[Blue The Pikachu 537]

Everybody is arguing about the equation, and I'm wondering what's that drawing in the other page xD

Its Fluttershy XD I drew her to be on my binder at all times X3

[Admiral Regulus 2,769]

 Edit: This will help as well.

 

To expand further: √[1/1] = 1, and √[(-1)/(-1)] = 1

 

HOWEVER: i = √[-1], so attempting to plug that into √[(-1)/i] is like asking "What is the square root of negative one divided by another square root of negative one?" (Or written in number form, √[(-1)/√[(-1)]])

 

This would be valid IF AND ONLY IF i = -1 (since the group inside of the square root would be -1 divided by -1). Unfortunately, i actually equals √[-1].

 

Actually, I jumped the gun on that one. What I originally had written was actually correct, it was just an unnecessary overcomplication.

[CloudFyre 848]

Actually, I jumped the gun on that one. What I originally had written was actually correct, it was just an unnecessary overcomplication.

I'm in the process of editing mine; I misread a little of what you had written.

 

Those darn parentheses.

[Admiral Regulus 2,769]

This has nothing to do with a square root having two solutions. The error here is, as I said, in the bottom row. You can take the 1, and break it up into 1 / 1. You can take the one on the bottom and break it up into 1 / ( i / i ), which is also 1 * i / i, which is also 1. It is not, however, equal to 1 / i². That's equal to -1, because i² is -1, and 1 / -1 is -1. This sounds much more complicated than it actually is. The mistake is just a simple error of not seeing that 1 / (1 / a ) is equal to a.

 

If you use exponent notation, this becomes easier to see. The first part of the second row can be written as 1 * i^-1 * ( i^-1 )^-1. This is the same as 1 * i^-1 * i¹, which is 1. You just add the positive and negative exponents, and you see the i term is supposed to disappear. In the next part, the error is equating it to 1 * i^-2. It's an error of 1 - 1 being -2 instead of 0.

Edited November 7, 2014 by Regulus

[caramba2654 602]

*Waves* Vital has it right, and I second this. As a CSE major who has taken Calculus 1, 2, 3, Differential Equations, Discrete Math, and Calculus based Statistics, this is the answer you're looking for. (I was just about to post that, but you beat me to the punch.)

 

There are plenty of mathematical fallacies that can be found online; most of them involve either setting a variable equal to zero somewhere (which people don't think about), or someone misses the fact that square roots have two answers.

 

This is the important thing to note:

 

For numbers like √[4], there are two answers: one positive (2) and one negative (-2). In either case, 2*2 or (-2)*(-2) both equal 4. The same goes for the square root of negative 1, but it takes a little more thinking.

 

 

Edit: This will help as well.

 

To expand further: √[1/1] = 1, and √[(-1)/(-1)] = 1

 

HOWEVER: i = √[-1], so attempting to plug that into √[(-1)/i] is like asking "What is the square root of negative one divided by another square root of negative one?" (Or written in number form, √[(-1)/√[(-1)]])

 

This would be valid IF AND ONLY IF i = -1 (since the group inside of the square root would be -1 divided by -1). Unfortunately, i actually equals √[-1].

 

Just to let it be clear:

 

sqrt(4) = 2   (and only 2. -2 isn't in the definition for this)

 

x^2 = 4 -> x = 2 or x = -2   (now -2 is included because it's in the definition)

[CloudFyre 848]

I'll work through it one by one:

3 = 1 + 2                                          ☑ Good   ☐ No Good

1 + 2 = i/i + 2                                   ☑ Good   ☐ No Good
i/i + 2 = 2 + (√[-1])/i                          ☑ Good   ☐ No Good
2 + (√[-1])/i   = 2 + (√[1/-1])/i            ☑ Good   ☐ No Good
2 + (√[1/-1])/i   = 2 + (√[1]/√[-1])/i     ☐ Good   ☑ No Good

This is where we break down. The square root of (1 / -1) IS NOT the same thing as the individual square roots √[1]  divided by  √[-1].

The first case [square root of (1 / -1)] is equal to i OR -i.

The second case [square root of (1) divided by the square root of (-1)] is equal to 1/i OR 1/-1, and those are NOT equal to i or -i.

[VitalSpark 1,830]

OK, here's the super simple explanation of why multiple roots bites you.

 

   2 = √4 = -2

 

Therefore 2 is -2.

[CloudFyre 848]

Just to let it be clear:

 

sqrt(4) = 2   (and only 2. -2 isn't in the definition for this)

 

x^2 = 4 -> x = 2 or x = -2   (now -2 is included because it's in the definition

 

True. I'll edit the post to reflect that.

Edited November 7, 2014 by CloudFyre

[SasQ 1,376]

@, it depends on how much do you know about imaginary numbers, or the infamous square roots of negative numbers, because there are plenty of them in your equation, and that's where the error hides. Actually, it is the same mistake the great Leonhard Euler himself once stumbled upon, when he tried to make a following equality:

 

-1 = i*i = root(-1)*root(-1) = root( (-1)*(-1) ) = root(1) = 1

 

I marked up with red the place where the error is. The rule that you can split a square root of multiplication into multiplication of the square roots of the factors works well with real numbers, but it breaks for imaginary numbers. And it all boils down to the unfortunate notation for square roots. Lots of mathematicians, even those with PhDs, are confused by that. So let me explain where the problem is.

 

Before negative numbers become accepted in mathematics somewhere around 18th century, mathematicians were taking square roots mostly from positive numbers (with some exceptions such as Cardano, Bombelli etc. which already understood negative numbers and imaginary numbers). First they used the Latin word "radix" before a number to symbolize the operation of taking the square root. Then they shortened it to "r.", and to just "r", and started putting the number before the prolonged line of this "r". At that time, "r" symbolized just the operation of taking the square root. E.g. root(4) = 2 in modern notation.

 

But when they realized that there are numbers which are not perfect squares and cannot be reduced further, they were just leaving it in that form, unreduced, considering it a symbolic name for that new type of number, which they called a surd or an irrational number (that is, a number which cannot be grasped by reason). So the sign "r" started to be used also for making numbers.

 

But then negative numbers started to be used as proper citizens of the number realm, people stopped believing that they're "unreal numbers" (less than nothing). And they figured out that one can make 4 not only from 2*2, but also from (-2)*(-2). So the square root of 4 is not only 2, but -2 is too. This already stands a problem with square roots, because if one wants it to be a function, it has to return just one value, not two! What's more important, if the square root operation would generate two answers, there would be no way to write them down in a distinguishable way.

 

This is when things started to become confusing, and they're to this day:

 

Some mathematicians claim that "root(4) = 2 or -2", because both answers, when squared, make 4.

Others noticed that this would make a problem, because how one would then write down the answers for taking the square root of 2? There are two answers, one of which is positive root of 2, and the other one is negative. To be able to write the first one as root(2), and the other one as -root(2), one needs to make a constraint that the "r" symbol is to mean just a single number, the so called "primary root", which is the positive one. Because only then one can use the minus sign before it to write down the other answer, which is then -root(2). These mathematicians say that square root of a positive number is always positive, never negative. So "root(4)" will be just 2. The same mathematicians say that "root(x^2) = +-x = abs(x)", though, which in my opinion is very inconsistent, because one cannot always know whether some symbol is a constant or a variable, and it can depend on the context of which symbol you solve for.

 

The whole confusion seems to come from the fact that there are actually two different thins here: one is the operation of taking the square root, which gives two answers, and the other are the actual numbers, which are called roots. But for both of them, the same notation is used, and this confuses everything. Because in one place the "r" could mean the operation, and in another place it could mean one of its results, a single number, one of two.

 

Things get even more confusing when introducing square roots of negative numbers, so called "imaginary numbers", because then the usual law that you can split the square root of a product into a product of square roots no longer holds, depending on whether you mean the operation or one of the numbers which come out of it. This is what mistaken the great Euler, and many other mathematicians after him. Notice, though, that these answers usually differ just by the sign, and it's because they are opposite numbers. You either get the first answer or the second.

 

Myself I tend to distinguish the operation from its results in this way:

For the operation, I always use fractional powers. E.g. I write 2^(1/2) to mean the operation of taking the square root of 2, but I write "root(2)" and "-root(2)" to designate the results of this operation, which are single numbers (one positive, and the other negative).

 

square roots have two answers.

Are you sure about that?

 

I'd say that the operation of taking the square root has two answers, and I'd even make it more precise: that it has two unique answers, because there are sometimes actually infinitely many answers, modulo 2 pi, each of them producing the same square.

 

But square roots, if you mean the answers, which are single numbers, are... well... single numbers, so they don't have two answers – they're the answers themselves (each one of them).

Edited November 7, 2014 by SasQ

[VitalSpark 1,830]

The second case [square root of (1) divided by the square root of (-1)] is equal to 1/i OR 1/-1, and those are NOT equal to i or -i.

 

The second case is:

 

    √1 / √-1

 

which is:

 

    (1 or -1) / (i or -i)

 

which is:

 

    1/i   or   -1/i   or 1/-i    or    -1/-i

 

which is:

 

    i   or   -i

 

So the same as the first case.