I'm no math guru

[teenix]

I just came across this question

Solve for X
X = X / 5

Transposing, I get
5X = X
Gives same result for X

Transposing again, I get
5 = X / X
Now it looks broken

cheers

Tony

[KeithB]

The only number that works for that is X=0.

[Albert Chan]

(12-02-2024 08:12 PM)KeithB Wrote:  The only number that works for that is X=0.

(±∞) = (±∞) / 5, does that count?

[teenix]

Yep, it was a silly question, but the Forum wouldn't let me delete the post.

My old mind pondered you cannot divide by zero, but then zero divided by any number is zero.

The old calculators don't like it though. Do the newer one's figure this out?

cheers

Tony

[EdS2]

If
5x=x
then (for finite x!)
4x=0
so
x=0

Generally speaking, dividing both sides by something can only be done if you add the proviso that the something isn't zero. You can then treat that as an alternate case, if the something is zero.

[KeithB]

Is the HP-71 new enough? 8^)
Depending on flag settings, you can get 9.999999E499 or Inf.

[KeithB]

(12-02-2024 08:20 PM)Albert Chan Wrote:  

(12-02-2024 08:12 PM)KeithB Wrote:  The only number that works for that is X=0.

(±∞) = (±∞) / 5, does that count?

Sure, why not.

[born2laser]

(12-02-2024 10:00 PM)KeithB Wrote:  

(12-02-2024 08:20 PM)Albert Chan Wrote:  (±∞) = (±∞) / 5, does that count?

Sure, why not.

No, because ∞ is not a number

[toml_12953]

(Yesterday 02:34 AM)born2laser Wrote:  

(12-02-2024 10:00 PM)KeithB Wrote:  Sure, why not.

No, because ∞ is not a number

There is a branch of mathematics in which infinity is considered a number, specifically a type of number called a "transfinite number," It's called set theory; particularly within set theory, the work of Georg Cantor developed a system for dealing with different sizes of infinities, where "aleph-null" (ℵ₀) represents the cardinality of the set of natural numbers, considered the "smallest" infinity.
It can still be worked with in set theory. ∞ is known as a transfinite number. Thus

∞ < ∞ + 1

[AnnoyedOne]

Of course zero (0) is a real number. ±∞ is a mathematical concept and not real.

So the answer depends on the context. If you live in the real world x=0. If you live in the world of theory ±∞ works as well.

A1

[KeithB]

"the work of Georg Cantor developed a system for dealing with different sizes of infinities, where "aleph-null" (ℵ₀) represents the cardinality of the set of natural numbers, considered the "smallest" infinity. "
Loves me some diagonal proofs!

[Nigel (UK)]

I always thought that going beyond sheep-counting numbers was a big mistake. Yan, tan, tethera...

Nigel (UK)

[Josef]

(12-02-2024 08:07 PM)teenix Wrote:  I just came across this question

Solve for X
X = X / 5

Transposing, I get
5X = X
Gives same result for X

Transposing again, I get
5 = X / X
Now it looks broken

cheers

Tony

Nowadays, just ask e.g. chatGPT and they will solve similar tasks instantly with justification.

[Ren]

Neither am I,
which makes me a bit of a misfit around here...
B^)

Ren
dona nobis pacem

[Maximilian Hohmann]

Hello!

(Yesterday 08:47 PM)Josef Wrote:  Nowadays, just ask e.g. chatGPT and they will solve similar tasks instantly with justification.

Today it was in the news here in Germany, that students who use AI tools learn better and have a better learning experience than those who don't!

So let's go for it! For this, I asked "Claude" because lately the free version of ChatGPT is quite slow due to large demand.

Please solve X = X / 5 for x

Claude:
Let me solve this algebraic equation step by step:
X = X / 5
To solve for X, we'll multiply both sides by 5:
5X = X
Subtract X from both sides:
4X = 0
Divide both sides by 4:
X = 0
Therefore, the only solution to the equation X = X / 5 is X = 0.
Let's verify:

When X = 0:
0 = 0 / 5
0 = 0 ✓

This means that 0 is the only value that satisfies the equation X = X / 5.

Would infinity also solve this equation?

Let me carefully examine this claim.
If X = X / 5, we want to check if infinity (∞) works:
∞ = ∞ / 5
This reveals an interesting mathematical nuance. In standard real number arithmetic, this is undefined because:

Division of infinity by 5 would still be infinity
But the equation ∞ = ∞ / 5 suggests 5 × 1 = 1, which is mathematically inconsistent

In the extended real number system or in some interpretations of limits, this might behave differently. However, in standard algebraic manipulation, infinity is not a number that solves X = X / 5.
The rigorous solution remains X = 0, which is the only real number that satisfies the equation:

0 = 0 / 5
0 = 0 ✓

The key point is that for the equation to be true, the value of X must be zero, where dividing by 5 reduces the value to itself. Infinity does not meet this criterion.

An answer that even I (not in the least a math guru...) can understand!

Regards
Max

[born2laser]

(Yesterday 09:34 PM)Maximilian Hohmann Wrote:  Hello!

(Yesterday 08:47 PM)Josef Wrote:  Nowadays, just ask e.g. chatGPT and they will solve similar tasks instantly with justification.

Today it was in the news here in Germany, that students who use AI tools learn better and have a better learning experience than those who don't!

So let's go for it! For this, I asked "Claude" because lately the free version of ChatGPT is quite slow due to large demand.

Please solve X = X / 5 for x

Claude:
Let me solve this algebraic equation step by step:
X = X / 5
To solve for X, we'll multiply both sides by 5:
5X = X
Subtract X from both sides:
4X = 0
Divide both sides by 4:
X = 0
Therefore, the only solution to the equation X = X / 5 is X = 0.
Let's verify:

When X = 0:
0 = 0 / 5
0 = 0 ✓

This means that 0 is the only value that satisfies the equation X = X / 5.

Would infinity also solve this equation?

Let me carefully examine this claim.
If X = X / 5, we want to check if infinity (∞) works:
∞ = ∞ / 5
This reveals an interesting mathematical nuance. In standard real number arithmetic, this is undefined because:

Division of infinity by 5 would still be infinity
But the equation ∞ = ∞ / 5 suggests 5 × 1 = 1, which is mathematically inconsistent

In the extended real number system or in some interpretations of limits, this might behave differently. However, in standard algebraic manipulation, infinity is not a number that solves X = X / 5.
The rigorous solution remains X = 0, which is the only real number that satisfies the equation:

0 = 0 / 5
0 = 0 ✓

The key point is that for the equation to be true, the value of X must be zero, where dividing by 5 reduces the value to itself. Infinity does not meet this criterion.

An answer that even I (not in the least a math guru...) can understand!

Regards
Max

ugh!
The answer to the first question is correct, maybe because it had enough textbooks in the training set, but sort of misses the second one because it argues the question both ways.
The implicit assumption is that we are looking for solutions in the set of real numbers (or complex for that matter). If the chosen set is different the number of solutions is different, for example the equation has no solutions in the set of natural numbers, because that does not include the zero.
If you use the set that treats infinity as a number with all the weird properties that entails, you have to play by those rules and accept infinity as a solution.
I will take the position that -assuming the equation represents some real world problem- we are looking for solutions that may have real application, and thus I'll stick with the set of real numbers and zero is the only solution

Juan