Basic: Misunderstanding of some division concepts

[AbedeuS]

Hey, I see a lot of things going on with some mathematical equations that even I would do, but I'm always confused by them for some reason, such as:

$$\frac{\sigma-\delta}{\sigma}$$

Wouldnt this just be equal to:

$$\frac{\sigma}{\sigma} - \frac{\delta}{\sigma} = 1 - \frac{\delta}{\sigma}$$

I know it doesn't actually work like this, but I'm always helplessly confused at the logic (well recently anyway).

Thanks for your (basic :D) help.

-Abe

 

[CRGreathouse]

You can do that, it's not wrong. You're just distributing here -- if it's easier, think of it as

$$\frac1\sigma(\sigma-\delta)=\frac1\sigma\cdot\sigma-\frac1\sigma\cdot\delta=1-\frac\delta\sigma$$

 

[AbedeuS]

Well it just seems random to write anything other than 1-fraction then, in chemistry we use equations like:

$$\frac{\sigma_{ref}-\sigma_{observed}}{\sigma_{ref}}$$ for chemical shift.

Seems a bit confusing >,<

 

[matt grime]

What do you mean, 'it doesn't work like that'? It does, but it is just a convention some people adopt to use the X/Y format, since it is, for whatever reason, deemed more useful or attractive. In the case you cite it is useful to leave it like that as it is very suggestive that the quantity is a normalized differene.

 

[AbedeuS]

What do you mean by normalised difference? I've always wondered why nmr uses a ppm axis method rather than just llabeling the frequency of absorption:

EDIT: Because TMS is the reference chemical usually this has extremely shielded hydrogen nuclei, which means the resonant frequency of it is going to be lower then almost all of the other samples loaded into an NMR spec, so the overall outcome of that equation should, for a start, be negative?

 

[CompuChip]

Matt meant, that from the equation
$$\frac{ \sigma_\mathrm{ref} - \sigma_\mathrm{obs} }{ \sigma_\mathrm{ref} }$$
it is immediately clear that you take the difference between the observed and reference quantity, and that this is what you really want to know. It's more like a convenience that we normalise it by $\sigma_\mathrm{ref}$, so that we can compare the quantities for different measurements (with different $\sigma_\mathrm{ref}$).

Of course it's always allowed to write it as
$$1 - \frac{ \sigma_\mathrm{obs} }{ \sigma_\mathrm{ref} },$$
but it's (supposedly) much less clear where this expression came from and what it is you're actually doing.

 

[AbedeuS]

Wouldnt it make more sense to do:

$$\frac{\sigma_{obs}-\sigma_{ref}}{\sigma_{ref}}$$

if I was trying to find the observed difference?

 

[matt grime]

That is 'the difference of the reference from the observed', when you _probably_ want the difference fo the observed from the reference. One is just -1 times the other and it is completely immaterial - you are focusing on the wrong things entirely.

 

[AbedeuS]

Good point ¬_¬

But the Observed frequency is usually higher, so it would be best to have shift positive rather than neg

 

[CompuChip]

If you're comparing it to the reference quantity, then it seems logical that the result is negative if there is a decrease with respect to that quantity.
Otherwise it would better if it had been defined as
$$\frac{ \sigma_\mathrm{obs} - \sigma_\mathrm{ref} }{ \sigma_\mathrm{obs} }.$$

 

[AbedeuS]

Ah, I see, so because we define reference as the most shielded (lowest number) it just makes logical sense that any other value (higher number than reference) would make the overall equation negative, so the "Shift" can be perceived as shielding (which decreases the value) so a more negative shift == less shielding.

I probably would have preferred to have it going up with less shielding though, indicating that the frequency observed was higher, and therefore the shielding was lower.

I find it pretty confusing however that on nmr spectrographs the ppm appears to get larger numbers, but there not negative, the axis is written correctly, with 0 on the right at any higher numbers on the left indicating a decrease, but its never llabeled to make it clear its negative, or is this just a freaky convention?

Also, is normalisation performed on an equation when there would be some other variables involved that could disproportionately change the two values, so you normalise it, so that if a variable is changed (in the case of NMR, the external magnetic field is changed) the shift values should still retain linearity with respect to eachover?

Thanks for the help ^_^ I'm dopey.