A contradiction regarding the size of negative numbers

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SecularCauses
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A contradiction regarding the size of negative numbers

Post by SecularCauses » Sun Oct 07, 2012 11:48 pm

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?

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The Voice of Time
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Re: A contradiction regarding the size of negative numbers

Post by The Voice of Time » Mon Oct 08, 2012 1:36 am

sum being different than operative integer?

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John
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Re: A contradiction regarding the size of negative numbers

Post by John » Thu Oct 11, 2012 1:18 pm

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.
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.

story12
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Re: A contradiction regarding the size of negative numbers

Post by story12 » Tue Oct 30, 2012 1:17 am

John wrote:
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.
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.
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? 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

Post by Mike Strand » Wed Oct 31, 2012 2:28 pm

story12, thanks for your interesting post. Here is your starting comment:
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?
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.

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:
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?
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 "ill-behaved" 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.

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=UTF-8

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 8)

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Re: A contradiction regarding the size of negative numbers

Post by Mike Strand » Wed Oct 31, 2012 3:41 pm

story 12, in another comment, you wrote:
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?
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.

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

Post by chaz wyman » Thu Nov 01, 2012 12:41 am

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?
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.

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Kayla
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Re: A contradiction regarding the size of negative numbers

Post by Kayla » Sat Dec 08, 2012 5:42 pm

chaz 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.
so using analogous reasoning

-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^(y-1) 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

Post by chaz wyman » Sun Dec 09, 2012 6:22 pm

Kayla wrote:
chaz 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.
so using analogous reasoning

-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^(y-1) just divide by x

and since then i have done really well in math basically its the easiest subject ever

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Kayla
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Re: A contradiction regarding the size of negative numbers

Post by Kayla » Mon Dec 10, 2012 1:56 am

chaz wyman wrote: Duh! It makes perfect sense. THERE IS 1 -1 in -1. So fucking obviously
I did not think you were THAT stupid.
yes, you could say that there is one -1 in -1

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

Post by chaz wyman » Mon Dec 10, 2012 3:39 pm

Kayla wrote:
chaz wyman wrote: Duh! It makes perfect sense. THERE IS 1 -1 in -1. So fucking obviously
I did not think you were THAT stupid.
yes, you could say that there is one -1 in -1

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.
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[/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.

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Kayla
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Re: A contradiction regarding the size of negative numbers

Post by Kayla » Mon Dec 10, 2012 7:39 pm

chaz 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.
that would be i

no need to ask a math teacher this one is common knowledge
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.
what kind of teacher were you

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

Post by mickthinks » Mon Dec 10, 2012 8:38 pm

chaz wyman wrote:So go and fuck yourself with the rough end of a pineapple.
Chaz reveals the real level of his intellect ...

The forum software can help people to ignore hate-mongers 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 :D !

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Re: A contradiction regarding the size of negative numbers

Post by chaz wyman » Mon Dec 10, 2012 9:20 pm

mickthinks wrote:
chaz wyman wrote:So go and fuck yourself with the rough end of a pineapple.
Chaz reveals the real level of his intellect ...

The forum software can help people to ignore hate-mongers 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 :D !
Whilst she insults me, then I do so back with knobs on.
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

Post by chaz wyman » Mon Dec 10, 2012 9:22 pm

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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