Is Math flawed? Are we?

[Arcesious]

Over the years I've come up with some questions about math, where I seem to think that there are 'flaws' in it. I hope that my questions aren't due to a lack of knowledge on my part about the processes of math. I haven't touched Calculus yet. I know 2 + 2 always equals 4, but that part of math isn't flawed as far as I can comprehend.

 

Here's a few:

 

1/3 = 0.333...

however, 1/3 + 1/3 + 1/3 = 1. Infinity sure is confusing.

 

0.5 x 0.5 = .25

0.3 x 0.3 = 0.09

How can two numbers multipled by each other logically become a smaller number? The math says the answer becomes smaller, but how is that physically possible?

 

10--10 = 20

1--11 = 12

This kind of problem just bothers the heck out of me. An even amount of subtraction signs suddenly makes it addition? How? Physically, subtraction problems like this can't even be possible. You can't get 20 apples by subtracting negative 10 apples. Said negative apples don't even exist if they're in negative amount... Is double subtraction even logically possible as far as human comprehension goes?

 

Shouldn't 1--11 equal -10 instead of 12? Mathematically it equals 12, but it seems to me that no matter how many subtraction signs you have, it still boils down to 1 -11 = -10 and not 1+11. Why is it that 1---11 = -10 but 1----11 = 12?

 

Are there any other 'problems' like these you know of? Do you have solutions that make sense that disprove these examples as problems? What's your opinion? Are we humans simply too limited in comprehension to fully understand the perfect empiricism of math? Or are these problems here because math was created as a flawed process by humans, due to our limited comprehension?

 

My opinion is that math isn't flawed, but simply our understanding of its quantifing processes is flawed. I wonder how our understanding of math would change if we could mentally comprehend the fourth or fifth dimension...

 

So what do you think?

[Jeff]

Think of the multiplication sign as "of" when you say it in your head. 0.3 of 0.3 is 0.09, that makes sense.

[Totenkopf]

10--10 = 20

1--11 = 12

This kind of problem just bothers the heck out of me. An even amount of subtraction signs suddenly makes it addition? How? Physically, subtraction problems like this can't even be possible. You can't get 20 apples by subtracting negative 10 apples. Said negative apples don't even exist if they're in negative amount... Is double subtraction even logically possible as far as human comprehension goes?

 

Shouldn't 1--11 equal -10 instead of 12? Mathematically it equals 12, but it seems to me that no matter how many subtraction signs you have, it still boils down to 1 -11 = -10 and not 1+11. Why is it that 1---11 = -10 but 1----11 = 12?

 

It's the same as a 2x negative. "I ain't got none" doesn't equal "I have none".

[Mono_Giganto]

1/3 = 0.333...

however, 1/3 + 1/3 + 1/3 = 1. Infinity sure is confusing.

 

The whole concept of infinity takes a while to get used to; it's pretty theoretical. Try this one:

 

Start with 1, and add 1/2, then 1/4, then 1/8, then 1/16, etc. Keep doubling that bottom number and adding it to the total. What's the final answer?

 

0.5 x 0.5 = .25

0.3 x 0.3 = 0.09

How can two numbers multipled by each other logically become a smaller number? The math says the answer becomes smaller, but how is that physically possible?

 

Like Jeff said, think about when you take half of something (IE multiplying by one-half) you should end up with a smaller value. So using your apples as an example:

 

0.5 x Apple = Half of an Apple

 

Half of an Apple < Apple

 

 

10--10 = 20

1--11 = 12

This kind of problem just bothers the heck out of me. An even amount of subtraction signs suddenly makes it addition? How? Physically, subtraction problems like this can't even be possible. You can't get 20 apples by subtracting negative 10 apples.

 

Actually, these problems are entirely possible in the real world. Maybe not with something so mundane as the number of apples in your basket, but there are many things that are considered to exist in negative quantities, especially any quantity that also has a direction associated with it.

 

Shouldn't 1--11 equal -10 instead of 12? Mathematically it equals 12, but it seems to me that no matter how many subtraction signs you have, it still boils down to 1 -11 = -10 and not 1+11.

 

The problem here is that -11 =/= 11. They're two different numbers that represent two different things.

 

So, let me ask you this question. Let's say I took a 12-inch ruler. Let's say I subtracted 11 from all of the values marked on the ruler. Now, instead of going from 0 to 12, it goes from -11 to 1. How long is the ruler now?

[jonathan7]

The whole concept of infinity takes a while to get used to; it's pretty theoretical.

 

Infinity is entirely theoretical, even space is not infinite (at least if you accept the Friedmann Lemaitre Big Bang Model). There are no known infinites in the entire universe, indeed I think it is philosophically proovable that the idea of infinity is de fact absurd.

[Litofsky]

I know 2 + 2 always equals 4....

 

So what do you think?

 

I think that Mr. Blair disagrees with you.

[Darth InSidious]

I think that Mr. Blair disagrees with you.

 

Yes, but only because Cherie's rather shady Australian friend tells her it equals giving him all her money...

[igyman]

Shouldn't 1--11 equal -10 instead of 12? Mathematically it equals 12, but it seems to me that no matter how many subtraction signs you have, it still boils down to 1 -11 = -10 and not 1+11.

 

As Giganto said, -11=/=11. Basically, they're something along the lines of polar opposites. Think in terms of protons and electrons if you like chemistry, or perhaps Light Side and Dark Side in Star Wars terms. Point being that when you subtract a negative number from a positive number, you are actually increasing the positive side by removing the negative. Another way to look at it is to observe the opposite situation: -11-1=-12. If that seems logical, there's no reason for 1--11=1-(-11)=12 to seem illogical.

[Arcesious]

Heh, well that solves that... Does anyone know any (nearly) impossible to solve math conundrums?

[ForeverNight]

Alright. In middle school were you ever taught that you can't get a negative number using exponents only?

 

Well, they were wrong.

 

Rather famous equation:

 

e^((pi)i)=-1

 

Interesting, no?

 

I don't know the why behind it (Get's into calculus and I'm only in pre-calc right now) but I do know this equation, and it's more famous cousin, the Famous Five equation:

 

e^((pi)i)+1=0

 

That one has everything that you need in math... two basic 'constants' (i and e), pi, and then two universal numbers 1 and 0. Along with the only three operations that are truly used.

 

Interesting, no?

[Mono_Giganto]

I don't know the why behind it

 

Well it's derived from Euler's formula:

 

e^(i x) = cos x + i sin x

 

Where x = pi, so:

 

e^(i pi) = cos pi + i sin pi = (-1) + i (0) = -1

 

Of course, I suppose this is just explaining it in terms of another equation that now needs explaining.... but personally I don't have the willpower to go into greater detail on this particular equation.

 

I did supply a link to the Wikipedia page about the formula if you're that interested.

[Det. Bart Lasiter]

Heh, well that solves that... Does anyone know any (nearly) impossible to solve math conundrums?

 

http://en.wikipedia.org/wiki/Hilbert%27s_problems

[ForeverNight]

Thanks Mono, when I'm not distracted by school work and an upcoming ACT then I'll give it a good look over!

[acdcfanbill]

Alright. In middle school were you ever taught that you can't get a negative number using exponents only?

 

I don't recall ever hearing anything like that from anyone. Every odd exponent with a negative input will result in a negative output. Any even exponential powers of i give negative results except 4, 8, 12....

[ForeverNight]

Alright, since you have to be so smart about it lemme rephrase:

 

without starting with a negative number can you make a negative using an exponenet?

 

Happy?

[acdcfanbill]

Alright, since you have to be so smart about it lemme rephrase:

 

without starting with a negative number can you make a negative using an exponenet?

 

Happy?

 

Well, in that case, it's pretty much impossible to get a negative number without using one somewhere. If you don't count i as negative, then using that is about your only option. I would think arguments for it's sign could be made both ways though.

[ForeverNight]

No. It's an imaginary number. There's many different types of numbers, but the two you have to pay attention to right now are Real and Imaginary.

 

Real Numbers and the numbers that exist: (sq.rt.)7; 4; -1; 100; 1.39; 1.9999999.....;-24.333333333333; and so on.

 

Imaginary Numbers are numbers that don't technically exist but are used in order to solve important math equations, i is the most famous one that is there... in fact, it's the only one I can think of right now.....

 

Since Negative numbers are real numbers, i is not negative since it is imaginary.

[Darth Avlectus]

@ thread I would say:

 

Flawed, No--incomplete, Yes. Now if you want to view incomplete as flawed, then that is your prerogative but I think that a sweeping generalization discredits the merit it does in fact have.

 

It's a frustrating subject for me, but what I could find about it, fascinated me. I only wish there weren't so many societal distractions.

 

Optics handbooks, though little help to me in my laser hobby, have an intriguing level of math to them. I can hardly understand it, but the descriptions and explanations of the relationships are really something.

 

Maybe one day in my life, at my leisurely pursuit, I'll finally learn the math required to understand the book.

 

Point is, it cannot cover every single subject to come up. Or maybe it isn't supposed to. I'm not sure.

[acdcfanbill]

I understand the difference between real and imaginary numbers and how negative is a subset of reals. Yet squares of imaginary numbers do give negative real results. And since i is a placeholder for sqrt(-1), it can also be -i, that being the nature of square roots.

[Jae Onasi]

 

So what do you think?

I think your understanding of math is incomplete at this point and that's why it's confusing. Jeff has the best explanation for why multiplication of fractions or decimals yield smaller fractions--basically, you're taking a portion of a portion. Example--if you multiply 1/2 by 1/2, you're taking half of that half. Think of a pie. If you take half of that half of pie, you end up with a quarter of that pie, hence 1/2 x 1/2= 1/4. The best way to deal with that is keep doing your math exercises. At some point it'll suddenly click. For instance, when I have to solve for x in an equation, it's sometimes easier to think of x as a box with the answer inside of it, and doing the algebraic formulas are your keys to unlocking that box to discover the answer inside.

 

For negative numbers, sometimes thinking in terms of direction helps. I work with negative numbers all day long so it's second nature for me, now, but if you think of it on an east/west continuum, e.g. the Greenwich mean time is the 0 point, it can help. Negative numbers move you west, positive numbers move you towards the east a certain number of hours. Now, if you have a negative number, say -7 hours, that moves you 7 hours towards the west. Now, take away (subtract) -3 from that. You're taking away 3 western hours from the original (heading back towards the east, now), so now you end up with only -4 hours as a result. I'm not sure if that helps or not.

[VarsityPuppet]

I like your explanation Jae Onasi. Very user-friendly

 

Here's one that's a little hard to wrap your head around.

 

Where n=any number,

 

 

n/0 = undefined

 

 

Hint: You have to change the way you think about division for it to actually make sense..

[ForeverNight]

Alright... so if we divide by zero, that's the same as multiplying by the inverse: so we have n*(1/0) so we still divide by zero... and since zero goes into one an infinite amount of times and yet not once.....

 

Basically just go with undefined.

 

@acdcfanbill: I don't know why we're arguing about what the hell i is, the only number in there that matters if it's positive or negative is e. And since e is positive (~2.71828183) than my assertion and formula still stand. (And -(i^2) would net 1 as opposed to -1 as an answer)

[VarsityPuppet]

Alright... so if we divide by zero, that's the same as multiplying by the inverse: so we have n*(1/0) so we still divide by zero... and since zero goes into one an infinite amount of times and yet not once.....

 

Basically just go with undefined.

 

True, but not what I was getting at.

 

Let's go with the apples again: you have 1 apple to divide amongst 3 people.

aka, you have to divide 1 apple into 3 different parts

 

that of course would be: 1/3= .33333

 

 

But let's say you have 1 apple to divide amongst 0 people.

aka, you now have to divide 1 apple into 0 parts. <<< That, my friends is not possible.

 

 

idk, that's the way I like to think of it. But yeah, if you take into consideration that multiplication is just division reversed, you can come to a similar conclusion anyways.

[Boba Rhett]

TLDR, but just wanted to say that math is just a toolsuite we use for working with numbers and that there are places where it can and will break down.

[Darth Avlectus]

^^^This coming from an electronics engineer.