Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

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Welcome back!

The linear Diophantine equation *ax* + *by* = *c* (where *a*, *b*, *c* are integers) has a solution if and only if *c* is a multiple of the greatest common divisor (aka the highest common factor) of *a* and *b*.

For example, 15*x* + 12*y* = 500 has no solutions because the GCD (or HCF) of 15 and 12 is 3, which doesn't divide 500.

And 17*x* + 29*y* = 1000 must have a solution in the integers because the GCD (or HCF) of 17 and 29 is 1, which divides 1000.

Also, once you've found a solution (*x*, *y*), then all the other solutions have the form

for an arbitrary integer *k*.

KerimF wrote:

You may like to solve for example:

107*A + 109*B = 10,000

You can extend this trick to find integer solutions to much more complicated equations (e.g. quadratics, cubics) but you'd need to do a bit more work. For example, you can take modulo 4 of both sides of the equation

to show that this doesn't have any integer solutions.

KerimF wrote:

Based on this definition, you mentioned, you are totally right.

I just added what I know because in my designs, if I ended up with the equation:

x^2 - 9 = 0

X^2 = 9

x = sqrt(9) = ±3

I had to consider both results.But in case you believe that I do wrong, and I have to ignore always the negative result, please accept my apology.

Actually, you have the right idea, it's just that the ± symbol needs to come slightly earlier. Instead of:

you want to say:

And what amnkb is saying is that .Welcome!

Hi Bob,

I vote yes.

Bob wrote:

Thanks Phro. Dealt with. Normally I'd fully remove the poster straight away but his answer is correct .....

Bob

Looks like ChatGPT was used to produce the content of Alexander_Thomas's post -- probably with navyboy484's post entered as a prompt. That writing style is unmistakable. Ever since the software was released, bots are using it to try to create more convincing/human-like responses.

For example, if I type the following into ChatGPT:

```
navyboy484 wrote:
Just an easy problem for you guys to solve (I don’t need help with this just wanted to post something easy here to pass the time).
Solve for “x”
2.1x = 23.1
```

...then ChatGPT 3.5 gives me this:

```
Sure, I'd be happy to solve the equation for "x"!
The equation is:
2.1x = 23.1
To solve for "x," we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 2.1:
(2.1x) / 2.1 = 23.1 / 2.1
The 2.1 on the left side cancels out, leaving us with:
x = 23.1 / 2.1
Now, let's calculate the value of "x":
x = 11
So, the solution to the equation is x = 11.
```

Not exactly the same as post #3, but we can see that the format is very similar.

Welcome!

Welcome!

Welcome!

Welcome!

MaxT wrote:

Yes - that's what I thought was right - a scalar of 16.5

So, is the operation I wrote out - which I saw someone else do - not a valid operation at all?

I didn't think it was but it left me confused.

It's a perfectly valid operation, just that the answer should have been a scalar (rather than a vector), i.e.

Jack Omar wrote:

In the link about derivatives it is shown that it is 0/0 but I should take a really small difference and then shrink it to 0 so it is the same as 0/0 ???

I don't get that part with shrinking to 0, which is pretty much the same as 0/0.

Hi Jack Omar,

Careful -- taking the limit as something approaches 0 is not quite the same as 0/0! We're only interested in what happens as getsThe question we want to answer is: how do we describe the 'slope' of a function at any given point on its graph? Well, as the pictures you've added show, we can

(a) Take the first point on the curve, find another point on the curve, then draw a straight line through them...

(b) ...then calculate the slope of the line you've just drawn.

Post #1 wrote:

I think I made a mistake in last equation but I don't know why.

Looks OK to me -- isn't the last line the same as the first one?

Post #3 wrote:

No -- the division by is for the previous step. They are saying that:Second question isn't there an error in the result ? There is 2x + Δx, then it is devided by Δx, 2x÷Δx + Δx÷Δx then 2x÷Δx + 1 ? Shrinking it into Δx to 0 I get 2x÷0 + 1 ?

I saw that you asked some questions about integration as well -- but let me know if this makes sense first and then we can move on.

Hi MaxT,

Welcome to the forum.

Almost -- the dot product of two vectors is a scalar, not a vector. In this case we need to add together the results, i.e.

9 + 7.5 = 16.5

Welcome!

Welcome!

Zazu, it's perfectly OK if you'd like to share a link to another maths resource and there is a dedicated subforum -- Maths Teaching Resources -- for doing just that (and we will be happy to support you in doing so).

What isn't OK is for you to pretend to ask for help, return to your post a day later and edit it to insert a hidden link to your website. It gives off the impression that you are using our forum as nothing more than a bulletin board to redirect traffic to your site. Please stop overriding my edits -- if you keep doing so I will have no choice but to restrict your access to the forum.

Just a precautionary note -- while Bob's approach in post #6 is correct, we can only do this because the series converges. In other words, the logic would be:

(1) First prove that the series converges, e.g. by looking at the sequence of partial sums (that's the sum of the first N terms) and showing that that sequence converges as N gets larger.

(2) Now that we know the sum converges, we can then do the rearranging of terms that Bob outlined in post #6 to identify what it converges to.

If the series did not converge then this kind of manipulation is not allowed -- take for example the divergent series 1 - 1 + 1 - 1 + 1 - ... which can be rearranged to 'converge' to lots of different things.

Similar reasoning is also required for differentiating/integrating the infinite series in post #2. We can do this here because p is a probability (so is between 0 and 1, within the series' radius of convergence) and so the series is integrable within that radius of convergence (and its integral also converges).

Hi jadewest,

The key bit is here:

jadewest wrote:

each number increase on the scale indicates an intensity 10 times stronger than the previous number on the scale.

In other words, if you've got an earthquake A measuring 1 on the Richter scale and earthquake B measuring 2 on the Richter scale, earthquake B has an intensity 10 times stronger than earthquake A. Does that make sense?

What would happen if earthquake A measured 2 on the Richter scale and earthquake B measured 4 on the Richter scale -- what would the difference in intensity be then?

More generally any cubic can be reduced to a depressed cubic by substitution -- and then the method above can be applied.

What have you tried? In what context were you given this problem?

The answers to these questions will help us understand what type of solution you are looking for.

Jeremy Desmond wrote:

Thank you Ganesh for that extra interesting information about pi. But you still haven’t explained how we know pi has an infinite number of decimal places containing non-repeating digits. You stated that computers have calculated pi correct to 1.33 x 10^13 decimal places but how do we know the end is not just round the corner? Is it possible that one day a computer will calculate the exact value of pi to a finite number of decimal places?

Hi Jeremy,

Welcome to the forum.

The reason that pi doesn't have a final digit is because it is an irrational number, and irrational numbers have decimal expansions which continue forever. In other words, to answer your question it suffices to:

(1) Prove that pi is irrational, and then

(2) Prove that all irrational numbers have an infinite, non-recurring decimal expansion.

There are quite a few proofs that pi is irrational, some of which are more complex than others. One of the most common proofs is Lambert's, where he essentially (a) shows that all infinite continued fractions are irrational and then (b) finds an infinite continued fraction for pi, which automatically implies that it must be irrational from part (a). If you're interested in continued fractions I've got some videos about it on my YouTube channel (although it doesn't discuss the continued fraction of pi). This takes care of (1).

Now for (2). Suppose instead that you could find an irrational number which didn't have an infinite, non-recurring decimal expansion. Let's say for example, the number 0.12345123451234512345... This decimal expansion has an infinitely recurrent pattern (the '12345' bit). But we see that if we let x = 0.12345123451234512345... then:

Multiplying both sides by 100000:

Subtracting x from both sides:

And finally, dividing both sides by 99999 gives us:

But hang on -- if we can express it as a fraction of two integers, it must be a rational number! (This is pretty much the definition of what it means for a number to be rational.) Similarly, if pi had a 'final digit' we could also express it as a fraction -- for example, if pi terminated after 5 decimal places, i.e. 3.14159, then we could write that as 314159/100000, which is a rational number.

Let me know if that makes sense -- happy to clarify anything if needed.