Mathematic notation (doubtless a stupid question)

Hello,
When writing, for example, "metres per second per second" for acceleration, it is noted as ms^-2 (where the caret symbolises the -2 is in superscript).
I understand that, and why, "per second per second" is "seconds squared", but in the notation, why is it superscript minus 2? Why not just superscript 2?
If I was ever taught this in maths lessons, the info is no longer in my head.
Thanks in advance for any help,
David Paste.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload

Because its "per", and so you are dividing. which gives you a negative coefficient.
E.g.:
Miles/hour is speed, and if I say I go at a certain miles/hour for a certain time, the answer must be miles. So, 30 miles/hour for two hours is 60 miles. That is:
30 m/h x 2 h = 60 m
So the /h and the h cancel, meaning that as the h is really h^1, the /h must be h^-1 in order for the coefficients to add up to zero (meaning there's no h left in the final expression). When you multiply things you add their coefficients, as in 10^2 x 10^3 -> 10^5.
--
"That which can be asserted without evidence, can be dismissed without
evidence."
  Click to see the full signature.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload
wrote:

The notation m s^-2 and kg m^-3 always strike me as perverse: what was wrong with m/s^2 and kg/m^3? I remember being taught m/s^2, and then when I changed school to one that did a Nuffield physics syllabus, the notation changed to kg m^-2 which it claimed to be "better" in some unspecified way.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload

I'm inclined to agree.
--
"If you're not able to ask questions and deal with the answers without feeling
that someone has called your intelligence or competence into question, don't
  Click to see the full signature.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload
On 12/09/2019 09:00, Tim Streater wrote:

I disagree. Mainly as the superscripts reduce the risk of mistakes where there are multiple terms. Eg
    kg.m^−1.s^−2     kg/m.s^2
--
Robin
reply-to address is (intended to be) valid
  Click to see the full signature.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload

Certainly if you are analysing a problem and checking the dimensions ( a useful technique, not always taught these days although I made a point of teaching it), indice notation is probably far easier to use.
I suspect it is a question of what you are used to. A physics teacher when I was a pupil first introduced indice notation for units, so it is the one I’ve used most. I have used others but much prefer indices.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload
On 12/09/2019 10:14, Brian Reay wrote:

Yes.
I never liked subtraction (or division) because it was non-commutative as opposed to addition (or multiplication) which are commutative. Commutative operations are far easier to handle algebraically.
When my son was little I taught him addition of negative numbers rather than subtraction, he liked the idea. You were a teacher, weren't you? I always wondered why this approach wasn't more common in schools?
Indicies emphasise multiplication by an inverse instead of division and hence are commutative. Personally, I don't seem to care about the notation, I use both.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload

There are several techniques.
Opinions on which are easy to grasp vary.
The same is true for other areas of Mathematics. This is why, for example, long division wasn’t taught for many years in primary schools which has had a ‘knock on’ impact even now. Some primary teachers can’t handle long division with confidence so avoid it (they never learned it themselves).
You may think, so what but it is key to algebraic division which you need later. Yes, I was a teacher, in a Specialist Mathematics School. I qualified late in life as a second career, I’ve degrees in Engineering and another in Mathematics. I always tried to bring real world examples into my lessons.

Once you are confident, coping with different notation etc isn’t generally an issue. Ditto terminology. However, when people are learning something or less confident, consistency is important.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload
wrote:

I agree about the usefulness of dimensional analysis.
--
I was brought up to believe that you should never give offence if you can avoid
it; the new culture tells us you should always take offence if you can. There
  Click to see the full signature.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload


If I can't quite remember a formula, I use dimensional analysis. Is the dimension that the formula produces different to the dimension of the answer that I would expect? Am I adding together terms which are not the same dimension? If so, check that I've remembered the formula correctly.
v^2 = u^2 + 2 as
m^2.s^-2 = m^2.s^-2 + m.s^-2 . m [ie m^s.s^-2]
So if I had brainfade and mis-remembered it as v^2 = u + 2as, I'd soon realise that something was wrong.
Likewise with
s = ut + 1/2 at^2
m = m.s^-1 . s [ie m] + m.s^-2 . s^2 [ie m]
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload

Those are good examples.
Dimensional analysis used to be included in Physics, I’m not sure if it was at O or A level, it is sometimes difficult to recall when you learned something you seem to have ‘always’ known. Speaking to a colleague when I was still teaching, it was included superficially in Physics (I think A level but possibly GCSE) but he was delighted when he learned I taught it in my classes.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload


It goes hand-in-hand with checking that the answer you get is *roughly* what you'd expect. If you calculate the volume of a hot-water cylinder and you get an answer of a couple of millilitres (a thimbleful) or a million litres (a swimming pool) then you've probably made a mistake ;-)
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload
On 12/09/2019 12:27, Brian Reay wrote:

I find just being able to think in terms of dimensions - even if not doing an accurate analysis on paper, it a very useful tool to catch errors, and would also stop lots of silly mistakes.
Like how many times you see people talk about kW when they mean kWh or vice versa. Just a basic understanding that the unit of energy is the Joule, and that power in Watts is a measure of the rate of flow of energy. Hence 1 Watt equates to one J/sec. Or energy over time. If you then multiply by time, the two times dimensions cancel out and you are left with just energy.
--
Cheers,

John.
  Click to see the full signature.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload
On Thursday, 12 September 2019 17:26:28 UTC+1, John Rumm wrote:

but for what they are talking about it rarely matters to average users. What does the Kwh mean on fride/freezer consumption. such as 124Kwh
note that oa.com uses an uppercase K argos use kWh/year lower case K updercase W
When buying a kettle I found the figure of 3kw more useful than the fridge version and more useful than 10.8 mega joules of energy.
calories is an even more weird one.
a drink I have here per 100ml is 6kJ or 1 Kcal an average adult 8400 kJ or 2000 kcal
Is that 2000 calories or 2000 kilo calories which is 2 mega calories :-)

I'm left knackered ;-)
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload
On 12/09/2019 12:27, Brian Reay wrote:

I'm pretty sure we covered it in O-Level Physics.
SteveW
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload
On 12/09/2019 09:53, Robin wrote:

My understanding is that this is why the "simpler" notation is deprecated. Also it is better when you are typing stuff from scratch. In the old old days where manuscripts were hand-written and subsequently transcribed by typists, or you were exchanging hand written calculation sheets with colleagues the slash notation is slightly easier to read. Especially if you express the formula as a fraction, with a full horizontal line.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload

As long as you imply brackets according to the standard BODMAS rules - ie kg/(m.s^2) rather than (kg/m).s^2 - then you'll be OK in either notation. So in slash notation you put all the units that are multiplied before the slash and all the units which are divided (ie have a negative index) after the slash. Accelerations in m/s/s looks absurd but if you parse it as (m/s)/s then it's obvious that it's a rate of change of speed wrt time; mind you, even I wouldn't go so far as to quote accelerations that way.

I think the slash notation tends to match more closely how you would say the units - "metres per second" rather than "metres seconds-to-the-minus-one" ;-) Index notation is for purists (like IUPAC names for organic compounds) whereas slash notation is for everyday use (like pre-IUPAC names): most people know that vinegar contains acetic acid, whereas only people who have studied chemistry will know that its "Sunday School name" is ethanoic acid. The latter is logical and makes it obvious that it contains an ethyl radical like ethane and ethanol do, but it's not the common "lay public" name.
At least no-one's perpetrated the ultimate howler: whereas it makes sense to refer to a volume (kg/m^3 or kg m^-3) in "kilogrammes per cubic metre", it is ludicrous to refer to an acceleration in "metres per square second". The former uses a unit of physical linear dimension, and everyone can visualise an area in square metres or a volume in cubic metres, but "square seconds" is ludicrous.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload
On 12/09/2019 08:20, NY wrote:

It normalises notation to exponential form, that is all. The - in the exponent is equivalent to the / of division
--
Any fool can believe in principles - and most of them do!



Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload

Yes I understand what it means: m.s^-2 or kg.m^-3 are identical to m/s^2 or kg/m^3. All I was saying is that the former is a slightly cack-handed way of expressing it (IMHO) and one which has to be translated back again if you are going to refer to it in spoken form. But it does make it easier to check that powers of units match throughout an equation.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload

its to do with the fact that mathematically speaking:
1 / x is equivalent to x^-1
So therefore
1/x^2 is equivalent to x^-2.
so once the 1 is replaced by another quantity we have:
for density say:
kg / m^3 is equivalent to kgm^-3
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Add image file
Upload

HomeOwnersHub.com is a website for homeowners and building and maintenance pros. It is not affiliated with any of the manufacturers or service providers discussed here. All logos and trade names are the property of their respective owners.