Alpha's Science Quiz #8 (1 Viewer)

[alphamale]

(math - sue me)

The positive integers are the set of numbers 1, 2, 3 .....
The next greatest integer after 7 is 8.

The real numbers are those that can be expressed as a decimal number,
e.g. 3.1415

What is the next greatest real number after 3.1415?

 

[FallingPianos]

(math - sue me)

The positive integers are the set of numbers 1, 2, 3 .....
The next greatest integer after 7 is 8.

The real numbers are those that can be expressed as a decimal number,
e.g. 3.1415

What is the next greatest real number after 3.1415?

if we use this function: f(x)= x

the next greatest real number after 3.1415 is the limit of f(x) as x approaches 3.1415 from the right.

also, your definition of real numbers is incorrect. you just defined rational numbers.

 

[Kandahar]

The real numbers are those that can be expressed as a decimal number, e.g. 3.1415

Real numbers are any numbers that aren't imaginary. Pi and e are both real numbers although you can't express them as decimals.

What is the next greatest real number after 3.1415?

3.1415 + (infinitesimal positive number)

 

[alphamale]

if we use this function: f(x)= x

the next greatest real number after 3.1415 is the limit of f(x) as x approaches 3.1415 from the right.

I'll accept that!

also, your definition of real numbers is incorrect. you just defined rational numbers.

Wrong. Rational numbers are the set of numbers that can be represented as a ratio of integers, and are a subset of real numbers. I didn't exclude decimals with infinite decimal places.

 

[alphamale]

Real numbers are any numbers that aren't imaginary. Pi and e are both real numbers although you can't express them as decimals.

Yes you can, with an infinitely long decimal number. And defining a number set by what they are not is kind of weak.

3.1415 + (infinitesimal positive number)

Is there only one infinitesimal positive number?

 

[FallingPianos]

The real numbers are those that can be expressed as a decimal number, e.g. 3.1415

your definition of real numbers is incorrect. you just defined rational numbers.

Rational numbers are the set of numbers that can be represented as a ratio of integers, and are a subset of real numbers.

thats exactly what I said. rational numbers are the set of real numbers that can be represented as a ratio of integers. this includes all decimal numbers with an infinite patturn such as .333333 (there needs to be a line over the last three, but I dont know how to type that) - all of which can be expressed as fractions, and all decimal numbers with a finite number of digits - which can also all be expressed as a fraction.

by contrast real numbers are the set of all numbers that arent imaginary. they include both rational and irrational (infinite number of digits with no patturn) numbers.

 

[FallingPianos]

Yes you can, with an infinitely long decimal number. And defining a number set by what they are not is kind of weak.

when mathematicians speak of being able to express a number as a decimal, they arent counting infinately long numbers unless there is a patturn in which case you can simply express the patturn. for example .3 with a horizontal line over the three to express that the threes go on infinitly is equal to 1/3.

 

[alphamale]

thats exactly what I said. rational numbers are the set of real numbers that can be represented as a ratio of integers. this includes all decimal numbers with an infinite patturn such as .333333 (there needs to be a line over the last three, but I dont know how to type that) - all of which can be expressed as fractions, and all decimal numbers with a finite number of digits - which can also all be expressed as a fraction.

by contrast real numbers are the set of all numbers that arent imaginary. they include both rational and irrational (infinite number of digits with no patturn) numbers.

No, decimals with infinite decimal places that do not have infinitely repeating decimals can't be expressed as a fraction.

 

[alphamale]

when mathematicians speak of being able to express a number as a decimal, they arent counting infinately long numbers unless there is a patturn in which case you can simply express the patturn. for example .3 with a horizontal line over the three to express that the threes go on infinitly is equal to 1/3.

Never heard of that - nobody would claim that pi is not a real number, even though it's decimal places are only known up to a finite number.

 

[FallingPianos]

Real numbers are any numbers that aren't imaginary. Pi and e are both real numbers although you can't express them as decimals.

Yes you can, with an infinitely long decimal number.

when mathematicians speak of being able to express a number as a decimal, they arent counting infinately long numbers unless there is a patturn in which case you can simply express the patturn. for example .3 with a horizontal line over the three to express that the threes go on infinitly is equal to 1/3.

Never heard of that - nobody would claim that pi is not a real number, even though it's decimal places are only known up to a finite number.

no one was claiming that pi is not a real number. it is a real number. its just not a rational number. its been mathematically proven that its infinatly long and with no patturn.

 

[FallingPianos]

rational numbers are the set of real numbers that can be represented as a ratio of integers. this includes all decimal numbers with an infinite patturn such as .333333 (there needs to be a line over the last three, but I dont know how to type that) - all of which can be expressed as fractions, and all decimal numbers with a finite number of digits - which can also all be expressed as a fraction.

by contrast real numbers are the set of all numbers that arent imaginary. they include both rational and irrational (infinite number of digits with no patturn) numbers.

No, decimals with infinite decimal places that do not have infinitely repeating decimals can't be expressed as a fraction.

i dont know what your disagreeing with, I never said they could.

 

[Scarecrow Akhbar]

The real numbers are those that can be expressed as a decimal number, e.g. 3.1415

And then he was told that the definition he gave was for "rational numbers", a proper correction since e is a real number not expressable as a ratio of integers, which it what a decimal number is.

Of course, he's probably thinking he's pretty tricky by using the left truncation of the first six digits of the decimal expression for pi, but I don't think anyone was fooled. Naturally, 3.1415 is most accurately described as a rational number, anyway, since the rationals are a subset of the reals.

However, to prove his ignorance of the topic, Alphamale proceed to fire the carbine into his toes:

Wrong. Rational numbers are the set of numbers that can be represented as a ratio of integers, and are a subset of real numbers. I didn't exclude decimals with infinite decimal places.

Clearly, since e, pi, the square root of 8, and log 22 cannot be expressed by any finite decimal expansion, nor by any ratio of integers, but are indubitably real, he's merely proving his barest comprehension of terms by his waffling at this point.

He should apologize and retreat from the field in shame.


=====

So, are there more integers than real numbers, or less?

 

[alphamale]

And then he was told that the definition he gave was for "rational numbers", a proper correction since e is a real number not expressable as a ratio of integers, which it what a decimal number is.

That was fairly garbled - I gave a 3.1415 as an example of a real number, and it is. Another person said no, it's a rational number, which is pretty silly because all rational numbers are real numbers. The person made a REAL BIG DEAL over my happening to choose a real number which was also a rational number, completely inconsequential for the problem stated.

Of course, he's probably thinking he's pretty tricky by using the left truncation of the first six digits of the decimal expression for pi, but I don't think anyone was fooled. Naturally, 3.1415 is most accurately described as a rational number, anyway, since the rationals are a subset of the reals.

More inapropos stream-of-consciousness babble. :2razz:

Clearly, since e, pi, the square root of 8, and log 22 cannot be expressed by any finite decimal expansion, nor by any ratio of integers, but are indubitably real, he's merely proving his barest comprehension of terms by his waffling at this point.

The difference between real and rational numbers is inconsequential to the problem stated - the only thing being proven here is that you have a reading comprehension problem, plus possibly an intermittent fugue state in your brain.

He should apologize and retreat from the field in shame.

You should apologize to your parents for shaming them for your pussified use of the third person in a discussion forum - the sure sign of a gutless (and in this case witless) wuss.

 

[Kandahar]

That was fairly garbled - I gave a 3.1415 as an example of a real number, and it is. Another person said no, it's a rational number, which is pretty silly because all rational numbers are real numbers. The person made a REAL BIG DEAL over my happening to choose a real number which was also a rational number, completely inconsequential for the problem stated.

No one has disputed that 3.1415 is a real or rational number; you are correct that it is both. However, the definition you offered for real numbers is actually the definition of rational numbers. Pi is real, but not rational. It cannot be expressed as a decimal.

REAL NUMBERS: Numbers greater than, less than, or equal to zero. Examples include -5, 0, pi, 1/3, sqrt(2), e. (Subsets: Rational/irrational)

RATIONAL NUMBERS: Real numbers that can be expressed as a fraction and/or terminating or repeating decimal. Examples include -90, 0, 4.2, 1/9. (Subsets: Integers/decimals)

IRRATIONAL NUMBERS: Real numbers that continue infinitely with no pattern. Examples include pi, e, chi.

 

[nkgupta80]

yes, by saying something can be expressed in decimal forms, you are automatically refering to rational numbers. Irrational numbers can only be truly expressed in symbolic terms (i.e) pi, e, sqrt(2), h, etc.

 

[Scarecrow Akhbar]

That was fairly garbled - I gave a 3.1415 as an example of a real number, and it is. Another person said no, it's a rational number, which is pretty silly because all rational numbers are real numbers. The person made a REAL BIG DEAL over my happening to choose a real number which was also a rational number, completely inconsequential for the problem stated.



More inapropos stream-of-consciousness babble. :2razz:



The difference between real and rational numbers is inconsequential to the problem stated - the only thing being proven here is that you have a reading comprehension problem, plus possibly an intermittent fugue state in your brain.



You should apologize to your parents for shaming them for your pussified use of the third person in a discussion forum - the sure sign of a gutless (and in this case witless) wuss.

So it's true what they say, isn't? That the men with the smallest units are least able to admit their errors and apologize for being stupid?

 

[alphamale]

No one has disputed that 3.1415 is a real or rational number; you are correct that it is both. However, the definition you offered for real numbers is actually the definition of rational numbers. Pi is real, but not rational. It cannot be expressed as a decimal.

Read more carefully - I didn't define real numbers.

REAL NUMBERS: Numbers greater than, less than, or equal to zero. Examples include -5, 0, pi, 1/3, sqrt(2), e. (Subsets: Rational/irrational)

This is weak - the definition of real numbers is actually very complex.

IRRATIONAL NUMBERS: Real numbers that continue infinitely with no pattern. Examples include pi, e, chi.

Useless - you don't know if pi, e, chi would repeat if calcualted to enough decimal places.

 

[alphamale]

yes, by saying something can be expressed in decimal forms, you are automatically refering to rational numbers. Irrational numbers can only be truly expressed in symbolic terms (i.e) pi, e, sqrt(2), h, etc.

"1", "0", and "5" are all symbolic terms.

 

[Kandahar]

Read more carefully - I didn't define real numbers.

Yes you did. Here:

The real numbers are those that can be expressed as a decimal number, e.g. 3.1415

That definition is wrong; it's the definition of rational numbers.

This is weak - the definition of real numbers is actually very complex.

I know, but I provided the simple one-sentence definition.

Useless - you don't know if pi, e, chi would repeat if calcualted to enough decimal places.

Yes I do. It's been mathematically proven that they're irrational.

 

[FallingPianos]

Yes you did. Here:



That definition is wrong; it's the definition of rational numbers.



I know, but I provided the simple one-sentence definition.



Yes I do. It's been mathematically proven that they're irrational.

I was going to reply to him saying the exact same thing, but you beat me to it. :2wave:

 

[alphamale]

Yes you did. Here:



That definition is wrong; it's the definition of rational numbers.

Wrong - I gave a characterization of one of the properties of real numbers - I didn't offer a definition - and you continue to post as if I said "a decimal number with only a finite number of decimal places", which I did not.

Yes I do. It's been mathematically proven that they're irrational.

And that proof is?

 

[FallingPianos]

And that proof is?

it was first poven in 1761 by Johann Heinrich Lambert. there are several proofs out there, here is one of them: Proof That PI is Irrational