How about you on the same subject, but mathematically, not philosophically:

Can two numbers

*x*and

*y*written in decimal expansion differ in every decimal place, yet be equal?

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Guest Message by DevFuse

Started by bonanova, Jul 05 2012 06:50 AM

16 replies to this topic

Posted 05 July 2012 - 06:50 AM

Title is fav quote by Kipling, on equality.

How about you on the same subject, but mathematically, not philosophically:

Can two numbers*x* and *y* written in decimal expansion differ in every decimal place, yet be equal?

How about you on the same subject, but mathematically, not philosophically:

Can two numbers

*Vidi vici veni.*

Posted 05 July 2012 - 07:44 AM

Spoiler for how about it ?

Posted 05 July 2012 - 09:24 AM

0.999......

1.000...... are not "almost equal" ... they are EQUAL

proof 1)) 1/3 = 0.333......

3*1/3 = 0.333*3 but also 3*1/3 = 1

0.999.... = 1

proof 2)) let x = 0.999.....

10x = 9.999....

subtract the two equations.... 9x = 9 ... or x=1, but assumption 1, x=0.9999 ..... thus 0.999.... = 1 ....or if you like, 1.0000.....

1.000...... are not "almost equal" ... they are EQUAL

proof 1)) 1/3 = 0.333......

3*1/3 = 0.333*3 but also 3*1/3 = 1

0.999.... = 1

proof 2)) let x = 0.999.....

10x = 9.999....

subtract the two equations.... 9x = 9 ... or x=1, but assumption 1, x=0.9999 ..... thus 0.999.... = 1 ....or if you like, 1.0000.....

Posted 05 July 2012 - 10:13 AM

Spoiler for Please use spoilers

Posted 05 July 2012 - 12:47 PM

the difference may be infinitesimally small but it exists...

1. The difference of two real numbers numbers always exists and is also a real number.

2. A real number cannot be "infinitesimally small".

3. Please don't answer "thanks for the explanation, but I'll stick to... "

Posted 05 July 2012 - 03:17 PM

Spoiler for Got one

Posted 05 July 2012 - 03:24 PM

1. The difference of two real numbers numbers always exists and is also a real number.

2. A real number cannot be "infinitesimally small".

3. Please don't answer "thanks for the explanation, but I'll stick to... "

To what class of numbers do those that are infinitesimally small belong?

*Vidi vici veni.*

Posted 05 July 2012 - 03:34 PM

In standard mathematics 1.000... and 0.999... are considered to be two different ways to express exactly the same number.

Posted 05 July 2012 - 03:38 PM

Spoiler for how about it ?

Spoiler for a discussion of almost

*Vidi vici veni.*

Posted 05 July 2012 - 03:39 PM

There are infiniesinally small real numbers in the non-standard real number system of Abraham Robinson which he used in his non-standard analysis. That system allows a form of calculus where the derivative is the ratio of two infinitesimals.

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