question about notation

Waestrel

Things to the minus power I get, no probs. But when I see units like cm, seconds to a negative power I get confused. Can some one explain to me how cm to the minus 2 or day ^-1 makes sense? I cant grasp it.

locum-motion

Take [latex]5g[/latex] of salt (Sodium Chloride or NaCl). Dissolve it in about [latex]75ml[/latex] of water, and then top up the solution with more water until you have precisely [latex]100ml[/latex] of solution.

What is the concentration of your solution?

Well, you can express it in various ways. One of the easiest is to call it "NaCl 5% w/v (aq)". This translates to "Sodium Chloride 5 percent (by weight in volume) in an aqueous solution"

Another way to express it would be [latex]NaCl 5g/100ml (aq)[/latex], which is "Sodium Chloride five grams per one hundred millilitres of aqueous solution"

[latex]5g/100ml[/latex] is equal to [latex]50g/1,000ml[/latex] which is equal to [latex]50g/l[/latex], so we can also call our solution [latex]NaCl 50g/l (aq)[/latex] also known as "Sodium Chloride fifty grams per one litre of aqueous solution"

In expressing the concentration in [latex]g/l[/latex], there is (obviously) a part of the unit over the line, and a part under the line. For some reason (perhaps it's just a convention, I'm not sure) it is considered preferable to remove the 'below the line' element. If we multiply the top and bottom of the expression by the same factor we don't change the value of it. If the factor by which we multiply both parts is the reciprocal of the bottom part, then the bottom part becomes 1, and it vanishes. [latex]l^{-1}[/latex] ([latex]l[/latex] to the power of minus one) is the reciprocal of [latex]l[/latex].

So:

[latex] g / l[/latex]

= [latex]g * l^{-1} / l * l^{-1}[/latex] (multiplying top and bottom by [latex]l[/latex] to the power of minus 1)

= [latex]g * l^{-1} / 1[/latex] (solving for [latex]l[/latex] times its own reciprocal)

= [latex]g * l^{-1}[/latex] (removing the 'divided by 1', since anything divided by 1 equals itself)

= [latex]gl^{-1} [/latex] (taking out the unneccessary * symbol)


So, our solution can be described as [latex]NaCl 50gl^{-1} (aq)[/latex]. If I were reading that out, I would automatically translate the negative power into the word 'per' in my head, and I would still call it "Sodium Chloride fifty grams per litre (aqueous)"

Next: What is a "litre"? The "litre" isn't actually one of the SI units at all, it's just an expression of an amount of space that is actually defined in terms of other units. A "litre" is the volume of a cube that has sides of [latex]10cm[/latex]. So that's

[latex]10 cm * 10 cm * 10 cm[/latex]

which of course can also be expressed as

[latex]10*10*10 cm*cm*cm[/latex]

or

[latex]10^3 cm^3[/latex]

or

[latex]1,000 cm^3[/latex]

So, one litre is one thousand cubic centimetres, which we all know.

But, don't forget that [latex]10cm[/latex] is also equal to [latex]0.1m[/latex] or to [latex]1dm[/latex] (one decimetre).

So, therefore a litre is also equal to

[latex]1dm*1dm*1dm[/latex]

or

[latex]1dm^3[/latex]

Thus, one litre is also equal to one cubic decimetre.


We can then substitute this into our expression of the concentration of the salt solution, because [latex]1l[/latex] = [latex]1dm^3[/latex].

So our solution is now called [latex]NaCl 50gdm^{-3} (aq)[/latex]. Once again I translate the negative power into the word 'per' in my head when I read it, so it's "Sodium Chloride fifty grams per cubic decilitre(aqueous)"

If you just remember to do that little translation in your head (negative power equals 'per'), then you should have no problem.

Apart from concentrations, another unit that it might crop up in is acceleration. Remember, 'velocity' or 'speed' is the rate at which a thing changes position per unit of time, so you might say that an object has moved 10 metres per second or [latex]10ms^{-1}[/latex]. 'Acceleration', however, is the rate of change of velocity per unit of time. So, if an object is moving at [latex]5ms^{-1}[/latex] at a particular moment in time, and 3 seconds later it is moving at [latex]41ms^{-1}[/latex], then its velocity has changed by [latex]36ms^{-1}[/latex] in 3s, or we could say that its 'acceleration' was "12 metres per second per second". This can also be described as "12 metres per second squared" or [latex]12ms^{-2}[/latex].

Once again, just translate the negative power into the word 'per' in your head, and you'll be fine.

Pherekydes

Waestrel wrote: »

Things to the minus power I get, no probs. But when I see units like cm, seconds to a negative power I get confused. Can some one explain to me how cm to the minus 2 or day ^-1 makes sense? I cant grasp it.

This is usually shorthand for "per cm squared" or "per day".

Yakuza

When you see a unit to a negative power, it's because the quantity is being divided by that unit. As said above, when you say a unit with a "per" in it, the quantity being expressed is something divided by something else :

metres per second - [latex] "m/s" [/latex] or [latex]ms^{-1}[/latex]

A4 paper is 80gsm = 80 grams per square metre - [latex]80 g s^{-2}[/latex]

Waestrel

THanks guys, so simple really. Dont know why i had such difficulty. I haev an honours science degree, but i never learned this.