16 replies to this topic

[bonanova]

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?

[ujjagrawal]

Spoiler for how about it ?

0.9999999999999999999999...
1.0000000000000000000000...
almost equal

[envy]

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.....

[ujjagrawal]

Spoiler for Please use spoilers

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.....

Thanks for the proofs...

Even though I will stick too... they are not "EQUAL" but almost equal ...

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

[witzar]

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... "

[dpalmer]

Spoiler for Got one

I can think of one example: 1.000... and 0.9999... are equal but differ in every decimal place! This is the only one that comes to mind for me.

[bonanova]

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?

[jim]

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

[bonanova]

Spoiler for how about it ?

0.9999999999999999999999...
1.0000000000000000000000...
almost equal

Spoiler for a discussion of almost

Let x = 0.999... y = 1.000... and z = y - x.

[1] describe your understanding of the notation "..."
[2] describe z.

[jim]

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.