The Questions Thread.

[Hankka]

2 second googling revealed it to be:

Junior Caldera - The Way

[Ze DreamGirl]

Thanks ^^ Did you actually put all the sentence in the search bar ? I'm always worried it might not work if what i put in there is too long...or too precise...

[Hankka]

Thanks ^^ Did you actually put all the sentence in the search bar ? I'm always worried it might not work if what i put in there is too long...or too precise...

Yep, that's just what I did. It usually works fine with song lyrics.

[Ze DreamGirl]

Ok, it might be stupid but: Can I break a calculator if i try to divide by zero with it ? Not that i want to do it since I'm afraid of what might happen...

[Crimson Penguin]

Ok, it might be stupid but: Can I break a calculator if i try to divide by zero with it ? Not that i want to do it since I'm afraid of what might happen...

Haha, no, you won't break your calculator, even if your math teacher tries to tell you otherwise. ; ) You'll just get an error message, nothing more.

Though that does bring up a question I've been wondering about for a while: Why haven't mathematicians figured out some crazy complex way of dividing by zero yet? I mean, you shouldn't be able to take the square root of a negative number either, but somebody devised a way to do it. Why, exactly, is dividing by zero so off-limits in the world of math?

[Ludwig]

Haha, no, you won't break your calculator, even if your math teacher tries to tell you otherwise. ; ) You'll just get an error message, nothing more.

Though that does bring up a question I've been wondering about for a while: Why haven't mathematicians figured out some crazy complex way of dividing by zero yet? I mean, you shouldn't be able to take the square root of a negative number either, but somebody devised a way to do it. Why, exactly, is dividing by zero so off-limits in the world of math?

This is easiest understood in a wider perspective. You have the more general structure called a field, which is a group of objects and two operations on them that satisfies certain conditions. Let's call these operations + and * (addition and multiplication). Due to the conditions on a field, there exists both an additive neutral element (called 0) and a multiplicative neutral element (called 1), these does by definition satisfy a+0=0+a=a and a*1=1*a=a for all elements a in the field. One of the conditions of it being a field is that a*(b+c)=a*b+a*c, for any a,b,c in the field. Because it's true for all a,b,c it's also true in the special case c=0, which gives a*(b+0)=a*b+a*0, where a*(b+0)=a*b, which gives that a*0=0 for all a in the field.

We now define the additive inverse to the element a as the element b(=-a) satisfying a+b=b+a=0 and the multiplicative inverse to a as c(=a^-1) satisfying a*c=c*a=1. Having this, we can now define subtraction of the element a as adding its additive inverse and similarly for division (multiplying by a^-1). Dividing by 0 is now the same as multiplying with its multiplicative inverse, let's call it d. Because it's the multiplicative inverse we have that 0*d=1, but we also have that 0*d=0 because d is an element in the field. We can now conclude that unless 0=1, 0 doesn't have a multiplicative inverse and that it's therefore impossible to divide by it.
Suppose that 0=1, this means that a*0=a*1=a => a=0, for all a in the field. This means that the field only have one element but part of the requirements of being a field is that it have at least 2 elements (the purpose is to exclude the trivial case of the one element field).

We are still not able to take the square roots of negative numbers btw.

[Crimson Penguin]

This is easiest understood in a wider perspective. You have the more general structure called a field, which is a group of objects and two operations on them that satisfies certain conditions. Let's call these operations + and * (addition and multiplication). Due to the conditions on a field, there exists both an additive neutral element (called 0) and a multiplicative neutral element (called 1), these does by definition satisfy a+0=0+a=a and a*1=1*a=a for all elements a in the field. One of the conditions of it being a field is that a*(b+c)=a*b+a*c, for any a,b,c in the field. Because it's true for all a,b,c it's also true in the special case c=0, which gives a*(b+0)=a*b+a*0, where a*(b+0)=a*b, which gives that a*0=0 for all a in the field.

We now define the additive inverse to the element a as the element b(=-a) satisfying a+b=b+a=0 and the multiplicative inverse to a as c(=a^-1) satisfying a*c=c*a=1. Having this, we can now define subtraction of the element a as adding its additive inverse and similarly for division (multiplying by a^-1). Dividing by 0 is now the same as multiplying with its multiplicative inverse, let's call it d. Because it's the multiplicative inverse we have that 0*d=1, but we also have that 0*d=0 because d is an element in the field. We can now conclude that unless 0=1, 0 doesn't have a multiplicative inverse and that it's therefore impossible to divide by it.
Suppose that 0=1, this means that a*0=a*1=a => a=0, for all a in the field. This means that the field only have one element but part of the requirements of being a field is that it have at least 2 elements (the purpose is to exclude the trivial case of the one element field).

Ah, I think I understand now. Thanks for putting your explanation in terms that even this math-phobic person can understand.

We are still not able to take the square roots of negative numbers btw.

...You mean that all that talk about imaginary numbers in college algebra and trig was a lie? :O (I never did understand what the purpose of learning about those things was...)

[Ludwig]

...You mean that all that talk about imaginary numbers in college algebra and trig was a lie? :O (I never did understand what the purpose of learning about those things was...)

Not all the talk about it. Most of it is true actually. One common mistake taught in college seems to be that i=sqrt(-1), which works in (some/most) applications but is (when strict) not true. i is defined as i^2=-1. The reason this is not the same as i=sqrt(-1) is because of how the square root is defined. We say that a=sqrt(A) when a^2=A and a>0 (or a=0). This is to avoid disambiguity, because (-a)^2=^2=A. We had to choose one of them to make the meaning of sqrt(A) clear and we choose the positive one. The problem with is that, despite i^2=-1, i is not greater or equal to 0.
Allowing the square root of negative numbers would lead to contradiction ex. 1=sqrt(1)=sqrt((-1)*(-1))=sqrt(-1)*sqrt(-1)=i*i=-1. Although it would be possible to define sqrt(-a) as i*sqrt(a) (for a>0), it's decided not to because of the previously mentioned example.

[Gelatino95]

Though that does bring up a question I've been wondering about for a while: Why haven't mathematicians figured out some crazy complex way of dividing by zero yet? I mean, you shouldn't be able to take the square root of a negative number either, but somebody devised a way to do it. Why, exactly, is dividing by zero so off-limits in the world of math?

Analytically, dividing by zero is the same as multiplying by infinity

Proof: When you examine the asymptote of a graph, that limit is approaching infinity, but when you actually go to the function and solve for it, the value is divided by zero. That is what defines an asymptote.

Long story short, you can't multiply by infinity because infinity isn't really a value. Dividing by zero won't give you a real answer. It's like trying to find the value of an asymptote; there is no value because there's nothing real there.

[Ludwig]

Analytically, dividing by zero is the same as multiplying by infinity

Proof: When you examine the asymptote of a graph, that limit is approaching infinity, but when you actually go to the function and solve for it, the value is divided by zero. That is what defines an asymptote.

Long story short, you can't multiply by infinity because infinity isn't really a value. Dividing by zero won't give you a real answer. It's like trying to find the value of an asymptote; there is no value because there's nothing real there.

The limit for 1/x as x->0 does not exists though. For x->0+, 1/x goes towards infinity. For x->0-, 1/x goes towards negative infinity. Also, the graph for 1/x is discontinuous in x=1, so the "limit" is not a valid replacement for the actual value. How the graph acts when it's near x=0 does not tell us anything about what the issue in x=0 is.

[Gelatino95]

That's very true. Since the graph approaches two different points at x=0, there's not really a real limit.

In that case, maybe dividing by zero isn't analogous to multiplying by infinity in all respects.

The point still stands though that neither dividing by zero nor multiplying by infinity will yield a real value.

[Ludwig]

That's very true. Since the graph approaches two different points at x=0, there's not really a real limit.

In that case, maybe dividing by zero isn't analogous to multiplying by infinity in all respects.

The point still stands though that neither dividing by zero nor multiplying by infinity will yield a real value.

You made me curious. What education on mathematical analysis do you have? You have the notion of an asymptote, but it seems like you only have an intuitive understanding of a limit, rather than knowing the exact definition.
Although analysis have its ways to get around division by zero in the cases that matters, it's not the discipline to use in this type of problem. It's a question about the structure of the rational numbers and the discipline about structures is called algebra.

[Gelatino95]

I'm taking my first year of calculus right now.

I know what a limit is and I know that the limit as x approaches 0 for the function n/x (where n is any real number) does not exist because it is unbounded.

I haven't taken any statistics courses or anything involving math theory, so I'm mostly just using my knowledge of the math involved to make these inferences.

For example, I thought that maybe dividing by zero was kind of like multiplying by infinity because as you decrease the denominator of a fraction with a constant numerator, the value of that quantity increases as it gets closer to zero (which pretty much just characterizes the n/x graph)

[Ludwig]

I'm taking my first year of calculus right now.

I know what a limit is and I know that the limit as x approaches 0 for the function n/x (where n is any real number) does not exist because it is unbounded.

I haven't taken any statistics courses or anything involving math theory, so I'm mostly just using my knowledge of the math involved to make these inferences.

For example, I thought that maybe dividing by zero was kind of like multiplying by infinity because as you decrease the denominator of a fraction with a constant numerator, the value of that quantity increases as it gets closer to zero (which pretty much just characterizes the n/x graph)

What is the definition of a limit? The first course usually gives an intuitive description of a limit, i.e. what's happening with the function as x approaches a value. That does not suffice as a strict definition and is not how a limit is actually defined. Maybe it's different on the Dinosaur Planet.
The reason why I thought that you likely don't know the definition of a limit is because you didn't know that 1/x approaches different "values" depending on if x approaches 0 from the right, or from the left. That's more common for first course students than later course students, because later course students have more experience.

You shouldn't bother with statistics courses if you like mathematics. Statistics is application based and you'll get most of the theory in other courses.

[Roseheart95]

When learning a language and watching DVDs or something in that language, is it better to watch them with English subtitles or without? With I find sometimes I learn new words, but usually I get too focussed on the subs and miss what's actually being said, and then without I find myself picking out words that I know. Which is better?

[Gelatino95]

What is the definition of a limit? The first course usually gives an intuitive description of a limit, i.e. what's happening with the function as x approaches a value. That does not suffice as a strict definition and is not how a limit is actually defined. Maybe it's different on the Dinosaur Planet.
The reason why I thought that you likely don't know the definition of a limit is because you didn't know that 1/x approaches different "values" depending on if x approaches 0 from the right, or from the left. That's more common for first course students than later course students, because later course students have more experience.

You shouldn't bother with statistics courses if you like mathematics. Statistics is application based and you'll get most of the theory in other courses.

I definitely knew that it approaches two different directions as x approaches zero, my answer just wasn't detailed enough.

My teacher is very peculiar about how we word our answers: she always wants us to use the word "unbounded" when applicable, probably because she doesn't want us to lose any points on the AP exam due to a formality.

When learning a language and watching DVDs or something in that language, is it better to watch them with English subtitles or without? With I find sometimes I learn new words, but usually I get too focussed on the subs and miss what's actually being said, and then without I find myself picking out words that I know. Which is better?

Subtitles may distract you from the foreign words and may cause them to go over your head when you hear them, so in my opinion I would advise you to stay away from them.

Now, you have to be careful about what kind of video you are using to educate yourself. You must be sure that you understand the material you are watching, foreign or not; whether you've seen it before or you can easily discern what is happening based on the visuals, you should be able to at least have a vague idea of what is going on without knowing the words. With that, it will be much easier to discern many of the words that you hear and will prove to be a more fulfilling experience overall. If you try to learn from something incomprehensible, you will likely be completely lost and everything will just go in one ear and out the other, even if you are trying to pay attention.

[Ludwig]

I definitely knew that it approaches two different directions as x approaches zero, my answer just wasn't detailed enough.

My teacher is very peculiar about how we word our answers: she always wants us to use the word "unbounded" when applicable, probably because she doesn't want us to lose any points on the AP exam due to a formality.

You did write that it approaches infinity and you also wrote that the quantity increases as x approaches 0.
Anyways, back to the question you didn't answer. What definition of a limit did were you taught?

[Gelatino95]

We never got a concise definition

Just a bunch of rules

[Ludwig]

We never got a concise definition

Just a bunch of rules

You wrote that you know what a limit is, but apparently you don't.
The definition of a limit is:
(x to a real number) We say that lim[x->a]f(x)=A when for every e>0, there exists an N>0 such that |f(x)-A|<e whenever |x-a|<N.
(x to infinity) We say that lim[x->infinity]f(x)=A when for every e>0, there exists an N>0 such that |f(x)-A|<e whenever x>N.
(x to negative infinity) We say that lim[x->-infinity]f(x)=A when for every e>0, there exists an N<0 such that |f(x)-A|<e whenever x<N.

[An00bis]

When learning a language and watching DVDs or something in that language, is it better to watch them with English subtitles or without?

It depends on your grasp of the language so far.

If you're a beginner you should probably watch with subs since you can make the connection between certain words easier. It also helps with sentence structure. If you're at least intermediate watch without subs and use a language dictionary to look up words you don't know. At that point I'd suggest pausing the video every couple of lines to check your comprehension.

Oh, and ... you should probably not watch something exciting at first since it can break your concentration.