A contradiction regarding the size of negative numbers

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A contradiction regarding the size of negative numbers
If we divide 1 by b, we know that as long as b is greater than zero, the smaller b becomes, the larger the value of 1/b becomes. For example, 1 divided by 2 is 1/2, 1 divided by 1 is 1, and 1 divided by 1/2 is 2. As the value of b approaches zero, the value of 1/b approaches infinity. So, the smaller we can make b, the larger the value of 1/b should become. Therefore, if we substitute a value for b that is even less than zero, i.e., a negative number, then the value of 1/b should be even larger than positive infinity. Yet, if we replace b with negative 2, then we have negative 1/2 for the value of 1/b, which is less than zero. Therefore, a negative number can simultaneously be larger than positive infinity, while being smaller than zero.
But, how can this be?
But, how can this be?
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Re: A contradiction regarding the size of negative numbers
sum being different than operative integer?
Re: A contradiction regarding the size of negative numbers
1/b tends toward infinity as b tends towards zero not as b gets smaller. It only makes sense to say that 1/b tends towards infinity as b gets smaller if you're dealing exclusively with natural numbers which rules out negative numbers.SecularCauses wrote:If we divide 1 by b, we know that as long as b is greater than zero, the smaller b becomes, the larger the value of 1/b becomes. For example, 1 divided by 2 is 1/2, 1 divided by 1 is 1, and 1 divided by 1/2 is 2. As the value of b approaches zero, the value of 1/b approaches infinity. So, the smaller we can make b, the larger the value of 1/b should become. Therefore, if we substitute a value for b that is even less than zero, i.e., a negative number, then the value of 1/b should be even larger than positive infinity.
Re: A contradiction regarding the size of negative numbers
Hello,John wrote:1/b tends toward infinity as b tends towards zero not as b gets smaller. It only makes sense to say that 1/b tends towards infinity as b gets smaller if you're dealing exclusively with natural numbers which rules out negative numbers.SecularCauses wrote:If we divide 1 by b, we know that as long as b is greater than zero, the smaller b becomes, the larger the value of 1/b becomes. For example, 1 divided by 2 is 1/2, 1 divided by 1 is 1, and 1 divided by 1/2 is 2. As the value of b approaches zero, the value of 1/b approaches infinity. So, the smaller we can make b, the larger the value of 1/b should become. Therefore, if we substitute a value for b that is even less than zero, i.e., a negative number, then the value of 1/b should be even larger than positive infinity.
I have kind of the same question. Obviously, we can't divide by zero. so let's say I divide a very large number (like 1,000,000) but a very negative number (like 9000, let us say).
How come if I decrease the denominator, the result doesn't get larger? I am sure someone here knows the answer.
Last edited by story12 on Mon Nov 19, 2012 8:18 am, edited 1 time in total.

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Re: A contradiction regarding the size of negative numbers
story12, thanks for your interesting post. Here is your starting comment:
Taking this further, 1/b approaches zero from above as b gets larger and larger (approaches infinity), and it also approaches zero from below as b gets negatively larger and larger (approaches negative infinity). These limits are called asymptotes  they're never reached, but can be approached as closely as I want by making b as large (positively or negatively) as I want.
Focusing on what you say here, story12:
If I draw the function 1/b as a graph with a vertical line at b=0 for values of 1/b and a horizontal line for values of b, which divides my paper into four quarters, or "quadrants", the graph looks like two dishes, exact replicas of each other, but trapped in different quadrants. See this web site for the graph: http://www.google.com/search?client=saf ... 8&oe=UTF8
Math isn't always according to my everyday intuition. For example, why does the infinite sum 1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... grow larger and larger, going to infinity, while the infinite sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/2^n + ... approaches 2 but never quite "gets there"? Here, 2^n means 2 multiplied by itself n times.
Cool stuff. For some of us
This points to what is fun (I think) about limit theory in calculus  the odd behavior of numbers when we get close to zero or infinity. If I consider b approaching zero from the right (picture the line of numbers with zero in the middle, positive numbers to the right, and negative numbers to the left), 1/b approaches infinity. At zero I don't have a number, since division by zero is undefined. But close to zero on the left, I have a "large" negative number (infinite in absolute value). That is, if b approaches zero from the left, 1/b approaches "negative infinity", with b=zero again not yielding any number at all.If we divide 1 by b, we know that as long as b is greater than zero, the smaller b becomes, the larger the value of 1/b becomes. For example, 1 divided by 2 is 1/2, 1 divided by 1 is 1, and 1 divided by 1/2 is 2. As the value of b approaches zero, the value of 1/b approaches infinity. So, the smaller we can make b, the larger the value of 1/b should become. Therefore, if we substitute a value for b that is even less than zero, i.e., a negative number, then the value of 1/b should be even larger than positive infinity. Yet, if we replace b with negative 2, then we have negative 1/2 for the value of 1/b, which is less than zero. Therefore, a negative number can simultaneously be larger than positive infinity, while being smaller than zero.
But, how can this be?
Taking this further, 1/b approaches zero from above as b gets larger and larger (approaches infinity), and it also approaches zero from below as b gets negatively larger and larger (approaches negative infinity). These limits are called asymptotes  they're never reached, but can be approached as closely as I want by making b as large (positively or negatively) as I want.
Focusing on what you say here, story12:
This does seem odd! However, this "function", 1/b, where b varies from negative infinity to positive infinity, is discontinuous at b=0. There are many such functions in math  sometimes called "illbehaved" functions. In this case, as b moves through zero, the function "violently flips" from one extreme to the other, and has no definition at zero. In your illustration, b moved from positive to negative through zero, causing 1/b to "flip" suddenly from extremely large positively, to extremely large negatively.Therefore, if we substitute a value for b that is even less than zero, i.e., a negative number, then the value of 1/b should be even larger than positive infinity. Yet, if we replace b with negative 2, then we have negative 1/2 for the value of 1/b, which is less than zero. Therefore, a negative number can simultaneously be larger than positive infinity, while being smaller than zero.
But, how can this be?
If I draw the function 1/b as a graph with a vertical line at b=0 for values of 1/b and a horizontal line for values of b, which divides my paper into four quarters, or "quadrants", the graph looks like two dishes, exact replicas of each other, but trapped in different quadrants. See this web site for the graph: http://www.google.com/search?client=saf ... 8&oe=UTF8
Math isn't always according to my everyday intuition. For example, why does the infinite sum 1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... grow larger and larger, going to infinity, while the infinite sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/2^n + ... approaches 2 but never quite "gets there"? Here, 2^n means 2 multiplied by itself n times.
Cool stuff. For some of us

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Re: A contradiction regarding the size of negative numbers
story 12, in another comment, you wrote:
By the way, in this last comment you have shown the function 1,000,000/b. This function is much like the function 1/b: As b increases without bound in the positive direction, it approaches zero, and as b approaches zero from the right, it goes to infinity. Also, as b increases without bound in the negative direction, it approaches zero, and as b approaches zero from the left, it goes to negative infinity. And like 1/b, it is discontinuous at b=0. The two functions are much alike!
I would be a little clearer about what's happening to the denominator. Starting from 1,000,000/(9,000). "Decrease the denominator" can mean two things  make it a larger negative number, like 10,000, (that is, larger in absolute value), or make it smaller in absolute value (closer to zero, like 8,000). In the first case the ratio is a smaller negative number (smaller in absolute value. In the second case, the ratio is a larger negative number (larger in absolute value). I like to think of absolute value, because for me, "decreasing" and "increasing" are then easier to understand, and I think about the + or  sign afterwards.Hello,
I have kind of the same question. Obviously, we can't divide by zero. so let's say I divide a very large number (like 1,000,000) but a very negative number (like 9000, let us say).
How come if I decrease the denominator, the result doesn't get larger?
By the way, in this last comment you have shown the function 1,000,000/b. This function is much like the function 1/b: As b increases without bound in the positive direction, it approaches zero, and as b approaches zero from the right, it goes to infinity. Also, as b increases without bound in the negative direction, it approaches zero, and as b approaches zero from the left, it goes to negative infinity. And like 1/b, it is discontinuous at b=0. The two functions are much alike!

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Re: A contradiction regarding the size of negative numbers
You do not understand what it means to divide. It is like asking a question how many of these are in this.SecularCauses wrote:If we divide 1 by b, we know that as long as b is greater than zero, the smaller b becomes, the larger the value of 1/b becomes. For example, 1 divided by 2 is 1/2, 1 divided by 1 is 1, and 1 divided by 1/2 is 2. As the value of b approaches zero, the value of 1/b approaches infinity. So, the smaller we can make b, the larger the value of 1/b should become. Therefore, if we substitute a value for b that is even less than zero, i.e., a negative number, then the value of 1/b should be even larger than positive infinity. Yet, if we replace b with negative 2, then we have negative 1/2 for the value of 1/b, which is less than zero. Therefore, a negative number can simultaneously be larger than positive infinity, while being smaller than zero.
But, how can this be?
1/1 =1. Because it asks how many 1s are there in 1, Thus the answer has to be 1.
get over it.
Re: A contradiction regarding the size of negative numbers
so using analogous reasoningchaz wyman wrote:You do not understand what it means to divide. It is like asking a question how many of these are in this. 1/1 =1. Because it asks how many 1s are there in 1, Thus the answer has to be 1.
get over it.
1/1 is 1
so there is 1 1s in 1  that makes no sense
it means nothing to divide
1/1 = 1 because that is what the rules of abstract symbol manipulation require
that is all there is to math  there is no meaning beyond that
i had a major problem with math when i got out of basic arithmetic
and i continued having a problem with it  it just made no sense
until a math teacher explained that math does not have to make sense
all you do is manipulate symbols according to rules there is no sense there to make
once i got that i started doing really well in math
the example he used is this
any number to the power of 0 is 1
why is that
what does that even mean
well it means nothing
but any number x to the power of 3 is x * x * x, to get its value to the power of 2, divide x^3 by x
to gets its value to the power of 1 divide by x again
to get its value to the power of 0 divide by x again and you always get 1 at this point
that just the rule  if you have x^y to get x^(y1) just divide by x
and since then i have done really well in math basically its the easiest subject ever

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Re: A contradiction regarding the size of negative numbers
Kayla wrote:so using analogous reasoningchaz wyman wrote:You do not understand what it means to divide. It is like asking a question how many of these are in this. 1/1 =1. Because it asks how many 1s are there in 1, Thus the answer has to be 1.
get over it.
1/1 is 1
so there is 1 1s in 1  that makes no sense
Duh! It makes perfect sense. THERE IS 1 1 in 1. So fucking obviously
I did not think you were THAT stupid.
Run along to your maths teacher. Let's hope he is smart enough to put you right.
it means nothing to divide
1/1 = 1 because that is what the rules of abstract symbol manipulation require
that is all there is to math  there is no meaning beyond that
i had a major problem with math when i got out of basic arithmetic
NO shit
and i continued having a problem with it  it just made no sense
until a math teacher explained that math does not have to make sense
all you do is manipulate symbols according to rules there is no sense there to make
Sounds like a poor maths teacher  don't run along to that one. This method of teaching maths is shite.
once i got that i started doing really well in math
the example he used is this
any number to the power of 0 is 1
why is that
what does that even mean
well it means nothing
but any number x to the power of 3 is x * x * x, to get its value to the power of 2, divide x^3 by x
to gets its value to the power of 1 divide by x again
to get its value to the power of 0 divide by x again and you always get 1 at this point
that just the rule  if you have x^y to get x^(y1) just divide by x
and since then i have done really well in math basically its the easiest subject ever
Re: A contradiction regarding the size of negative numbers
yes, you could say that there is one 1 in 1chaz wyman wrote: Duh! It makes perfect sense. THERE IS 1 1 in 1. So fucking obviously
I did not think you were THAT stupid.
but to say that there is 1 one in 1 makes no sense  which is what i said if you are going to attack what i say attack what i actually say
there is a reason why there was so much resistance among mathematicians towards negative numbers at first  they make no sense  they are a convenient way of doing some kinds of math but they mean nothing
[qoute]Sounds like a poor maths teacher  don't run along to that one. This method of teaching maths is shite.[/quote]
as a substitute gym teacher you are not in a good position to criticize how others teach
what he teaches is pretty much accepted among mathematicians
math is a formal system that is all  that is not exactly a controversial claim
you can disagree of course but try to actually offer some arguments beyond well you are stupid
and i have seen this math teacher's results  i am not the only one whose ability to do math improved from his explanations

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Re: A contradiction regarding the size of negative numbers
as a substitute gym teacher you are not in a good position to criticize how others teachKayla wrote:yes, you could say that there is one 1 in 1chaz wyman wrote: Duh! It makes perfect sense. THERE IS 1 1 in 1. So fucking obviously
I did not think you were THAT stupid.
but to say that there is 1 one in 1 makes no sense  which is what i said if you are going to attack what i say attack what i actually say
there is a reason why there was so much resistance among mathematicians towards negative numbers at first  they make no sense  they are a convenient way of doing some kinds of math but they mean nothing
[qoute]Sounds like a poor maths teacher  don't run along to that one. This method of teaching maths is shite.
what he teaches is pretty much accepted among mathematicians
math is a formal system that is all  that is not exactly a controversial claim
you can disagree of course but try to actually offer some arguments beyond well you are stupid
and i have seen this math teacher's results  i am not the only one whose ability to do math improved from his explanations[/quote]
Like I said, take it up with your maths teacher.
Oh  whilst you are there, then ask him what is the Square root of minus 1.
In other words which number when multiplied by itself gives the answer 1.
You should have hours of fun with that one.
PS . I've never been any kind of a gym teacher. So go and fuck yourself with the rough end of a pineapple. Its about all you are good for.
Re: A contradiction regarding the size of negative numbers
that would be ichaz wyman wrote:Like I said, take it up with your maths teacher.
Oh  whilst you are there, then ask him what is the Square root of minus 1.
no need to ask a math teacher this one is common knowledge
what kind of teacher were youPS . I've never been any kind of a gym teacher. So go and fuck yourself with the rough end of a pineapple. Its about all you are good for.
i know from watching british tv and movies and reading harry potter that british teachers are all rather daft in some way or another but usually they have some redeeming quality
what is yours?

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Re: A contradiction regarding the size of negative numbers
Chaz reveals the real level of his intellect ...chaz wyman wrote:So go and fuck yourself with the rough end of a pineapple.
The forum software can help people to ignore hatemongers and other troublemakers automatically. Just visit their profile page—there's a button which takes you directly to a poster's profile at the bottom of each post—and click on the [Add foe] option under their login name. Suddenly it is as if they were no longer members of your forum !

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Re: A contradiction regarding the size of negative numbers
Whilst she insults me, then I do so back with knobs on.mickthinks wrote:Chaz reveals the real level of his intellect ...chaz wyman wrote:So go and fuck yourself with the rough end of a pineapple.
The forum software can help people to ignore hatemongers and other troublemakers automatically. Just visit their profile page—there's a button which takes you directly to a poster's profile at the bottom of each post—and click on the [Add foe] option under their login name. Suddenly it is as if they were no longer members of your forum !
For those that hold their tongue, so do I.
That is my intellectual position. Live with it!

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Re: A contradiction regarding the size of negative numbers
.Kayla wrote:
i know from watching british tv and movies and reading harry potter that british teachers are all rather daft in some way or another but usually they have some redeeming quality
I Know you are not too bright, but gleaning an opinion about British teachers from Harry Potter is an all time low.
Bullshit in, bullshit out.
You would be so lucky as to achieve qualifications sufficient for the ability to teach gymnastics.
Last edited by chaz wyman on Mon Dec 10, 2012 10:13 pm, edited 2 times in total.
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