Science Math Notation

[BDanielMayfield]

For what practical purpose, if any, are mathematic expressions in scientific literature often made less comprehensible to the average person? For example, many times I've noticed that per unit of time rates that could be simply expressed as /s are instead written as s^-1 or s^-1. Accelerations that could be expressed as /s² are written as s^-2. Why?

Bruce

[Chris Peterson]

For what practical purpose, if any, are mathematic expressions in scientific literature often made less comprehensible to the average person? For example, many times I've noticed that per unit of time rates that could be simply expressed as /s are instead written as s^-1 or s^-1. Accelerations that could be expressed as /s² are written as s^-2. Why?

It's often less ambiguous to avoid having division operations in the units. That is, it's clearer if every component of the unit is multiplicative. You could write specific heat capacity, for instance, as

J/(K kg)
J/K/kg (works, but breaks SI convention)
J K^-1 kg^-1

I think the latter is clearest, and also reflects what you'd see if you were carrying your units through a calculation using typical unit analysis. But the SI conventions allow the use of a division sign, as long as only one is used (which means you frequently need parentheses as well).

[BDanielMayfield]

I guess I just need to stop thinking negatively about negative exponents.

[geckzilla]

The average person doesn't even like fractions. They can deal with whole numbers, decimals and percents all day but if you throw a fraction out there it's game over. So if you are at that point you may as well use the negative exponents too.

[Ann]

I have to agree with Bruce, unfortunately. But I don't question the fact that to people who work with math in their jobs, or otherwise use it regularly in a non-simplistic way (see Geck's definition of simplistic math) Chris' favorite way of expressing heat capacity (and similar functions) is the clearest.

Ann

[BDanielMayfield]

I have to agree with Bruce, unfortunately.

, but I understand why you would put it that way, I think.

[Ann]

Sorry, Bruce! No offense intended!

Ann

[Nitpicker]

For complicated combinations of units, it is most certainly simpler to express them with negative exponents and no denominators. For simple units like "metres per second", one can either be consistent in the use of negative exponents, or make an exception and use a denominator. Engineers are perhaps more likely than scientists to make an exception, as engineers tend to write for an audience that isn't so technically minded. A new name for a unit will be invented, if the components become too cumbersome for its popularity.

1 kg.m.s^-2 = 1 N = 1 newton ~ the weight of an apple on Earth (by coincidence).

100,000 N.m^-2 = 100 kPa = 100 kilopascals ~ the pressure of 100,000 apples on Earth per square metre ~ the pressure of Earth's atmosphere at sea level.

Edit reason: denominator, not divisor.

[BDanielMayfield]

Sorry, Bruce! No offense intended!

Ann

Absolutely none taken.

Bruce

[BDanielMayfield]

For complicated combinations of units, it is most certainly simpler to express them with negative exponents and no divisors. For simple units like "metres per second", one can either be consistent in the use of negative exponents, or make an exception and use a divisor. Engineers are perhaps more likely than scientists to make an exception, as engineers tend to write for an audience that isn't so technically minded. A new name for a unit will be invented, if the components become too cumbersome for its popularity.

1 kg.m.s^-2 = 1 N = 1 newton ~ the weight of an apple on Earth (by coincidence).

100,000 N.m^-2 = 100 kPa = 100 kilopascals ~ the pressure of 100,000 apples on Earth per square metre ~ the pressure of Earth's atmosphere at sea level.

You say "it is most certainly simpler to express them with negative exponents", but how do we typically verbally express them if we are reading aloud? Don't we say "meters per second" which I would write as m/s, (which is simpler to write or type out than m.s^-1) rather than saying "meters times seconds raised to negative one"? If I had to verbally explain what a newton is I would say "one kilogram meter per second squared" rather than "one kilogram meter times seconds to the negative second power". So would you, I suspect.

This issue ain't that important though. I'll get over my distaste for negative exponents as soon as the rest of the world gets over the fear of division.

[Nitpicker]

You say "it is most certainly simpler to express them with negative exponents", but how do we typically verbally express them if we are reading aloud? Don't we say "meters per second" which I would write as m/s, (which is simpler to write or type out than m.s^-1) rather than saying "meters times seconds raised to negative one"? If I had to verbally explain what a newton is I would say "one kilogram meter per second squared" rather than "one kilogram meter times seconds to the negative second power". So would you, I suspect.

This issue ain't that important though. I'll get over my distaste for negative exponents as soon as the rest of the world gets over the fear of division.

You quoted me out of context. "Metres per second" is not complicated. But there are very complicated combinations of units. They are so complicated that no one really tries to read them out loud in polite company. It is merely a notation that is easier to read. And some people prefer to extend it to even the simplest of units, in order to be consistent.

Consistency is the practical purpose, to answer your original question.

[Nitpicker]

FWIW, my mind reads kg.m.s^-2 as "kay gee em ess to the minus two", but I've been doing it for a while. I prefer newtons.

[BDanielMayfield]

Consistency is the practical purpose, to answer your original question.

That's a sound, logical reason which I can understand and accept. Consistency is a good thing where math is involved. It would be why actual apples wouldn't be used down at the Office of Standard Weights and Measures to define a newton.

Though, by irrelevant coincidence, I happen to prefer apple newtons over fig newtons.

Bruce

[Nitpicker]

Consistency is the practical purpose, to answer your original question.

That's a sound, logical reason which I can understand and accept. Consistency is a good thing where math is involved. It would be why actual apples wouldn't be used down at the Office of Standard Weights and Measures to define a newton.

Though, by irrelevant coincidence, I happen to prefer apple newtons over fig newtons.

Bruce

I had to look up what a fig newton is. I had heard of them, but couldn't place them. Here in Oz they are typically known as fruit pillows, but I used to call them "squashed fly biscuits" (where biscuit = cookie) as a kid; actually I still do. Others insist that Garibaldi biscuits are the true squashed fly biscuits (but it has to be said that the British generally have no idea about good biscuits).

[Beyond]

Unless they are uncommonly fresh, I find that fruit pillows AKA squashed fly biscuits, are too dry for my taste, no matter what the fly filling is.

[Chris Peterson]

FWIW, my mind reads kg.m.s^-2 as "kay gee em ess to the minus two", but I've been doing it for a while. I prefer newtons.

I read negative exponents as "per". So this naturally reads to me as "kilogram meters per second squared".

Newtons are fine as long as we are restricting the discussion to force (as when the above unit is the result of a calculation). But if you are using the above in a calculation, it's best to stick with the base units. In general, all derived units should be converted to base units before using them in calculations. Otherwise, you end up with somewhat nonsensical final units that can disguise what's actually going on.

(BTW, SI convention doesn't support using a period in the units the way you have done. You have to separate multiplicative units with a space or a middle dot, and divisive units with a solidus if you don't use a negative exponent. There is a Unicode value for the middle dot, obtained with ALT-0183 on Windows computers. Thus, the newton is kg·m·s^-2.)

[BDanielMayfield]

Unless they are uncommonly fresh, I find that fruit pillows AKA squashed fly biscuits, are too dry for my taste, no matter what the fly filling is.

No matter what the fruit is, leave the flies out please.

[Nitpicker]

FWIW, my mind reads kg.m.s^-2 as "kay gee em ess to the minus two", but I've been doing it for a while. I prefer newtons.

I read negative exponents as "per". So this naturally reads to me as "kilogram meters per second squared".

Newtons are fine as long as we are restricting the discussion to force (as when the above unit is the result of a calculation). But if you are using the above in a calculation, it's best to stick with the base units. In general, all derived units should be converted to base units before using them in calculations. Otherwise, you end up with somewhat nonsensical final units that can disguise what's actually going on.

(BTW, SI convention doesn't support using a period in the units the way you have done. You have to separate multiplicative units with a space or a middle dot, and divisive units with a solidus if you don't use a negative exponent. There is a Unicode value for the middle dot, obtained with ALT-0183 on Windows computers. Thus, the newton is kg·m·s^-2.)

Well, you've got me on a technicality, so I shall respond with a technicality on the technicality:
http://en.wikipedia.org/wiki/Internatio ... eral_rules

General rules for writing SI units and quantities apply to text that is either handwritten or produced using an automated process

Whilst I am in the habit of using a dot-operator (·) when hand writing, I am inclined to break the general rule with a period (.) in typed text. Until the dot operator and/or the interpunct becomes more accessible on the US keyboard layout favoured here, or until I am replaced with an automated machine, I will only bring out the dot operator for special occasions, like vector notation. In short, life is too short and compound units are too common. The period, the asterisk and even the letter "x" are all acceptable, de facto symbols for communicating multiplication in common contexts.

(My favourite unit is the totally incongruous one for the roughness coefficient, n, in Manning's formula: s·m^-1/3. It is amazing that it works so well. Manning made a pretty good engineer for an accountant.)

[Chris Peterson]

Well, you've got me on a technicality, so I shall respond with a technicality on the technicality:
http://en.wikipedia.org/wiki/Internatio ... eral_rules

General rules for writing SI units and quantities apply to text that is either handwritten or produced using an automated process

That disclaimer does not appear in either the NIST or SI brochures. My interpretation is that you should follow the convention for both handwritten or typeset methods.

Whilst I am in the habit of using a dot-operator (·) when hand writing, I am inclined to break the general rule with a period (.) in typed text. Until the dot operator and/or the interpunct becomes more accessible on the US keyboard layout favoured here, or until I am replaced with an automated machine, I will only bring out the dot operator for special occasions, like vector notation.

Well, a simple ALT-0183 isn't very challenging. No different than what you'd do for other common symbols like degree or copyright. But if you don't want to take the trouble, that's fine. You can simply use a space, which is easily accessible on the keyboard and is fully within the SI conventions. No need to actually use something that is in the "wrong" list.

In short, life is too short and compound units are too common. The period, the asterisk and even the letter "x" are all acceptable, de facto symbols for communicating multiplication in common contexts.

But not in scientific contexts, which I think are worth attempting to observe in a forum like this.

Anyway, use what you choose. I was commenting mainly on the fact that there's an easy way to get the correct symbol if you like dots better than spaces.

[Nitpicker]

I am surprised that you are a stickler for the dot operator, whilst maintaining opposition to the convention of genitive form in Bayer designations.

In my world, the need to communicate "multiplied by" is an order of magnitude more common than the need to express "degrees" or "copyright". I also think that the use of a period in place of a dot operator is never ambiguous in a compound unit.

My general rule for all conventions, is to bend them when it is significantly more convenient to do so. The good people who come up with the rules for SI units should perhaps consider that. Or maybe the good people who devise standard keyboard layouts, and provide for "plus" (+), "minus" (-) and an acceptable variant of "divided by" (/), but not "multiplied by" are the culprits. Either way, I refuse to suffer because of these bad decisions. <high horse emoticon>

[Chris Peterson]

I am surprised that you are a stickler for the dot operator, whilst maintaining opposition to the convention of genitive form in Bayer designations.

I'm not. I usually use a space. If I have any complaint, it's against using the period as an alternative- not only because it's wrong and its use serves no good purpose, but because I find the result to be harder to read than the alternatives. A period actually reduces the clarity of the units.

[Nitpicker]

(A) kg.m.s^-2
versus
(B) kg·m·s^-2
versus
(C) kg m s^-2

Well I guess opinions differ. I see very little difference between (A) and (B), but I find (C) more difficult to read.

[Chris Peterson]

(A) kg.m.s^-2
versus
(B) kg·m·s^-2
versus
(C) kg m s^-2

Well I guess opinions differ. I see very little difference between (A) and (B), but I find (C) more difficult to read.

I find (B) easiest, then (C), and (A) I actually find jarring, as the period simultaneously makes me think of both decimals and symbolic objects in programming (that is, the s member of the m member of the kg object).

[BDanielMayfield]

(A) kg.m.s^-2
versus
(B) kg·m·s^-2
versus
(C) kg m s^-2

Well I guess opinions differ. I see very little difference between (A) and (B), but I find (C) more difficult to read.

Of those choices, B is definitely best. Chris, I'm glad to know about ALT-0183. Is there a good reference to non-keyboard symbol codes?

[Nitpicker]

If you must muddy the water with programming syntax, let's revert to: kg*m*s^-2

In my convention of using the "low dot operator", it acts as a separator of factors within a term. Dots have many uses. I'm a dot.