MATHCAD AND MAPLE – WORKING WITH UNITS

(Russian expanded text – http://twt.mpei.ac.ru/ochkov/Unit_MC_MP/Unit_MC_MP.htm)

Valery Ochkov

Others articles of V.Ochkovhttp://twt.mpei.ac.ru/ochkov/work2.htm (in Russian)

http://twt.mpei.ac.ru/ochkov/work2_eng.htm (in English)

Today solving engineering and technical problems on computer the users tend to work with universal mathematical packages (Mathcad, Maple, MatLab, Mathematica and others) rather than use programming languages. The mathematical packages enable computation projects to draw up much faster. The same speeding-up occurred when machine code was changed to the programming languages.

Mathcad lets us work with physical quantities and it stands out against the listed packages. The seventh Maple version appeared in summer 2001 with the only new feature allowing us to work with dimensions. The article compares working with dimensions in these two packages.

Speaking language of mathematics Mathcad and Maple built-in tools allowing us to work with dimensional quantities are sets overlapping but not coincident. To compare the “dimensional” pluses and minuses in these physical and mathematical packages is not quite well-posed problem. First of all, dimensional quantities appeared in Maple later than in Mathcad. Of course, Maple designers had to consider ten-year experience with using dimensions in Mathcad. But shall try to solve such improperly posed problem to compare these packages in relation to work with dimensional quantities (and some more).

After loading the library of work with dimensional quantities Maple displays built-in functions and operators that are applied to it. Mathcad does not give such information even in reference manual. Just in “Tips of the day” was a passing mention that we can not work with dimensional quantities for example in Mathcad programs. This restriction was removed in last versions but the tip not to use dimensional units remains.

The main disadvantage of Maple and Mathcad is that a number of functions and operators entered before the tools for working with dimensional units work improperly with dimensional arguments and operands. Essentially, a user often forgets about these restrictions and obtains unintelligible result. Mathcad and Maple designers should have applied dimensional units to all functions and operators or lock (for example indicating error message) using dimensional arguments (operands) in these functions (operators). The tools for working with dimensional units can be considered as undocumented (poorly documented) tools which we have to apply very guardedly checking the result in every debugging and testing. The condition (restriction) concern not only “exotic” functions and operators (solving differential equations and system of differential equations, ordinary and «unordinary» – with partial derivatives) but also rather routine tools Mathcad and Maple: tabular dependence approximation (representative example of the mistake in Mathcad see in Fig. 1a) or creating plots (representative example of the mistake in Maple see in Fig. 2a).

One comment from Waterloo Maple, Inc.:

The problem of Units in ODE-solutions – see Fig. 4d.

2. The Mathcad characteristic which boosts popularity is handy user interface. Particularly, Mathcad document looks almost like that created in text editor (for example Word) or scientific editor (LaTeX) and has not some conventional programmer symbols which irrelevant to problem. It entirely concern dimensional units: In Mathcad Physical quantities are displayed in their common abbreviations 25 N, 40 cm2, 6250 Pa etc. (Fig. 3). In Maple it is burden with additional symbols (25*unit(N), 40 [cm^2], 6250 [Pa]Fig. 4) that make difficult to see purport and solution of a problem. In Mathcad the change of a unit passes imperceptibly for the user, more correctly, "reader" of the Mathcad-document (Fig. 3a). In Maple this operation needs «the special invitation» functions convert – see Fig. 4a. All this "charge" physics of a task, its physical essence.

One comment from Waterloo Maple, Inc.:

Units package in Maple has two variations:
with(Units[Standard]) requires more syntax (example: 10*Unit(cm^2)) but separates unit names from variable names.
with(Units[Natural]) requires less syntax (example: 10*cm^2) but does not separate unit names from variable names – see. Fig. 4b.
Thus, Maple offers both tradeoffs. Users can use
Units[Natural] if they don't like the extra symbols.

3. In Maple a numerical constant and an individual physical quantity can be connected only with visible multiplication mark (for example, 25*Unit(N)) or with that invisible (for example, 6250 [Pa]). At that, the numerical constant is first co-factor and dimensional unit or dimension – the second (for example, 6250 [Pressure]). Mathcad has not this restriction: the gap between the number and dimensional unit can be multiplication mark or more complicated functional dependence, prefix or postfix operators, etc. This allow us to work easily not only with absolute measuring scale but with relational also. For instance, Figure 5 shows how to define temperature and it difference in Celsius in Mathcad. In the first case (t1 := 90 ºC) the gap between the numerical constant (90) and two symbols (ºC) means call of function (ºC) in the form of postfix operator x f and in the latter case – multiplication. Temperature, one of seven SI units, has a feature that it often measured not on the absolute scale (Rankine or Kelvin) but on relational (Celsius or Fahrenheit). This can cause confusion in calculations: let’s imagine a drawing of constructed building in which some of dimensions are measured in meters from the foundation and others – in foots from sea level. In Mathcad prefix and postfix operators save metrological situation as well as invisible symbol on screen (Fig. 6). The user functions and constants for working with relational temperature scales can be hidden at which equations become usual. A user can only change the source data (green constants) to get new result (Fig. 6a). Although Maple has six built-in temperature scales, two absolute and four relational, the later are slightly tangled: Figure 7 shows temperature has a certain double definition – temperature (scale) and temperature difference (degree). That is because only the multiplication mark can be placed between the numerical constant and the temperature unit; here for example how we can type the normal man temperature in Maple t:=36.6*unit(degC). We set the variable t not the temperature value (scale) but temperature difference between 0o Celsius and 36.6o Celsius which nevertheless can be displayed in the form of temperature also. (Author emphasizes the temperature because he had a difficulty with it taking part in Internet project WaterSteamPro in which functions returns heat and physical properties of water and steam in Mathcad http://twt.mpei.ac.ru/orlov/watersteampro). The logarithmic scales in Maple are in a mess too: we can convert between built-in bels, decibels, nepers but it is not clear how to use it in practice. In Mathcad the infix operator allows to make calculations without difficulties – the first operator of that is numerator of the logarithm (determined physical value) and the second is denominator (the base, reference of logarithmic scale, see Fig. 8).

Comments to “temperature scales” of Maple Unit Developer

"Although Maple has six built-in temperature scales, two absolute and four relational, ..."

This is incorrect. Maple has 6 units of temperature. I refer to the document "The International System of Units (SI)" from the Beuro International des Poids et Mesures", 7th edition, 1998. On page 100 in the English version, it shows that 1 degree Celsius equals 1 Kelvin. You are incorrect to state that Celsius is a relational temperature scale. It is completely correct to state that
1 calorie per Kelvin = 4.184 joules per degree Celsius.
Therefore, it is necessary and correct that Maple gives
> convert( 1, units, kelvin, Celsius );
1
just as:
> convert( 1.0, units, calories/kelvin, joules/Celsius );
4.184000000
is correct. A separate feature is given to convert relational temperatures:
> convert( 32.0, temperature, Fahrenheit, kelvin );
273.1500000
It would be a complete disaster if Maple were to treat "joules per degree Celsius" the same as "joules per 275.15 kelvin".

I believe that Maple's comment regarding "2 absolute and 4 relational" temperature scales is incorrect. While their example of using the Celsius temperature is correct, in of itself, Celsius temperatures can rarely be used directly in physical equations, since most physical equations require absolute Kelvin temperature.
Additionally, Celsius is indeed a "relational" temperature scale, since it's defined as 1/100th of the temperature difference between the triple point of water and its boiling point at STP.
I'm not convinced there's a clear-cut solution, since an automatic conversion to absolute temperature internally would muck up the example they gave. Somehow, it would need to be clever enough to know when you are requiring an absolute calculation and when you're merely interested in a relative calculation. Because of the offset nature of the two scales, there is no readily usable solution, unlike the situation between feet and meters.

Comments to “logarithmic scales” of Maple Unit Developer

"The logarithmic scales in Maple are in a mess too: we can convert between built-in bels, decibels, nepers but it is not clear how to use it in practice."

This is probably an ambiguity in translating the word "base" from English to Russian, which has several meanings in English. It appears that you are translating "base" to mean the "base of the logarithm". In this particular case, we meant "base" to be something else. Bels or nepers always represent a relative power or voltage ratio, respectively. Thus, in the case of bels, "base" is defined as some power relative to a given base power. Sorry for the confusion.
For example:
> with(Units[Natural]):
> ln(3.91*mW / (4.73*uW(base)));
3.358683726 [Np]
> convert(%, units, dB);
29.17315617 [dB]

Thus a power source of 3.91 milliwatts relative to a base power of 4.73 microwatts is approximately 29 decibels.
The conversion:
> convert( 1, units, bel, nepers );
1/2 ln(10)
is given on page 105 of "The International System of Units (SI)".  This book states that "the bel is used to express values of such logarithmic quantities as
field level, power level, sound pressure level, and attenuation."  Nepers reflect, for example, voltage ratios, and power is proportional to the square of the voltage and is therefore reflected in the above expression by the factor of 1/2.

4. The user variables and that attached with built-in units are put “in one basket” in Mathcad and similar to each other if a user does not divide it by using styles of variables and constants. This Mathcad feature can be considered as a sort of charge for “external naturalness” discussed in item 2. Figure 9 shows the typical “metrological” mistake connected with redefining voluntary or not variables m (unitary meter), and L (liter), and that, resulted in a mistake in the answer, 1m2L in place of 1000 L. In Maple such “outrage” is impossible (Fig. 10) — the user variables m and L and units of the same name are not mixed. But as mentioned above, we have to pay for it with complication of a document: compare m:= 100 cm or m:=100*unit(cm).

5. Mathcad has four unit systems (SI, MKS, CGS and U.S.) marked on Fig. 1. In Maple we can work with eight built-in systems already (Fig. 11). Besides, in Maple we can create user unit systems. But we do not consider that Maple feature as the author prime interest is the problem of returning dimensional units (of the physical quantities) in engineering computations. Nevertheless, we should note that Maple is very useful in the theoretical analysis of the metrological problems (e.g. development of formula by means of variable units).

Comments to the Item 7 of Maple Unit Developer

"The Maple designers did not rack their brains over the metrological problem and solved it simply (Fig. 16): the angles in trigonometrical functions lose it dimensions "insensibly".

This is true. It would better to have arcsin return units of radians.  To fix this temporarily before the next version of Maple, you can add them manually like this:
> with(Units[Natural]):
> a := sin(53.2*deg);
a := 0.8007313711
> arcsin(a);
0.9285151623
> sin(%);
0.8007313711

CONCLUSION

Computers played a spiteful joke on physical and engineering calculations: dimensional units were squeezed out from it. Today we observe a certain local computer “Renascence” — dimensional units return into calculations. Mathcad was a pioneer in the process, now Maple have joined in the work. Both packages do not finish it. Many functions and operators demand (expect) when the designers will teach it to work with physical quantities. For the present users of the packages are in the same expectation.

AFTERWORD

This article is some “extract” of the book “Physical and economic quantities in Mathcad and Maple” (http://twt.mpei.ac.ru/ochkov/Units/Forword_book.htm) preparing for publication by “Finance and Statistics”. This is how the book has been appeared. The author often looked into student problems solved in Mathcad. Physical or technical and economic problems were frequently solved without dimensional units that resulted in mistakes almost every time. Besides, rejecting, to put it mildly, does not make clear the problem. The causes to reject dimensional units are: students do not know how to work with it or face the difficulties forced to withdraw it from almost finished calculation. The author collected the “difficulties”, supplemented it with solutions and description of “dimensional calculations” technology in Mathcad/Maple.

Appendix Units in Mathcad & Maple

Dimension

Maple (569 units)

Активность (activity)

Bq

Время (time)

day, hr, min, s, sec & yr

Wink, shake, svedberg, blink, second (s), minute (m), hour (h), day (d), pentad, week (wk), fortnight, month (mo), lune, lunour, year (yr), biennium, triennium, quadrennium, quinquennium, decade, century, millenium & eon

Вязкость динамическая (dynamic viscosity)

poise

Poise, reyn & rhe

Вязкость кинематическая (cinematic viscosity)

stokes

Stokes (St)

Давление (pressure)

atm, in_Hg, Pa, psi & torr

bar, atmosphere (atm, atmos, at), barye, pieze, vac, pound_per_square_inch (psi), meter_mercury (mHg), foot_mercury (ftHg), inch_mercury (inHg), meter_water (mH2O), foot_water (ftH2O) & inch_water (inH2O)

Деньги (currency)

USD

Длина (length)

cm, ft, in, km, m, mi, mm & yd

meter (m), bohr (a0), mil, line, barleycorn, digit, inch (in), nail, palm, hand, finger, span, foot (ft), cubit, step, yard (yd), ell, pace, fathom, rod, pole, perch, furlong, mile, league, link, chain, cable, league, point, pica, astronomical_unit (AU, UA), parsec, light_year (ly), hubble, spat, fermi, micron, bicron, x_unit, microinch, calibre, marathon & quadrant

Доза (dose)

Gy & Sv

Емкость (capacitance)

Заряд (charge)

С, coul & statcoul

Coulomb (С), abcoulomb (abC), statcoulomb (statC), electron (e) & franklin (f)

Индуктивность (inductance)

H, henry, mH, mH & stathenry

henry (H), abhenry (abH) & stathenry (statH)

Индукция магнитного поля (magnetic flux density)

gauss, stattesla, T & tesla

Квант действия (action)

planck & h_bar

Количество вещества (Substance)

mole (mol)

mole (mol)

Количество информации, информация (Amount of information)

bit, crumb, nibble, byte, nickel, word, double_word, gawble & block,

Линейная массовая плотность (linear mass density)

Denier, drex & tex

Линейная частота (linear frequency)

abbe

Магнитный поток (magnetic flux)

statweber, Wb & weber

weber (Wb), abweber (abWb), statweber (statWb), maxwell (Mx) & unit_pole

Масса (mass)

gm, kg, lb, mg, oz, slug, ton & tonne

gram (g), tonne (t), electron_mass (em) atomic_mass_unit (amu, u), grain (gr), drachm (dr), ounce (oz), pound (lb, lbs), clove, stone, quarter, cental, quintal, hundredweight (cwt), wey, kip, ton, pennyweight (dwt), scruple, ton (AT @), carat (c, ct), millier, mite, mancus, mast, slug, hyl, crith & jupiter

Молярный электрический заряд (molar electric charge)

Момент дипольный электрический (electric dipole moment)

debye (D)

Мощность (power)

hp, kW, W & watt

watt (W), poncelet, donkeypower, horsepower (hp), ton, abwatt (abW) & statwatt (statW)

Напряженность магнитного поля (magnetic field strange)

Oe & oersted

oersted (Oe)

Облучение (exposure)

roentgen (R)

Обратная длина (Inverse length)

diopter & kayser

Объем (volume)

fl_oz, gal, galUK, L, liter & mL

liter (L, l), stere, gallon (gal), drop, minim, drachm, ounce, tablespoon (tbsp, Tsp, tblsp), dessertspoon (dsp), teaspoon (tsp), pinch, dash, gill, cup, pint, fifth, quart, pottle, peck, bucket, bushel, strike, bag, sack, coombe, seam, butt, last, ton, register, barrel, bulk & barge

Освещенность (illuminosity)

lx

Плотность магнитного потока (magnetic flux density)

tesla (T), abtesla (abT), stattesla (statT) gamma & gauss (G)

Площадь (area)

acre & hectare

are (a), hectare (ha), centare (ca), barn (b), shed, mil, rood, acre, lot, section & township

Поверхностная плотность энергии (surface energy density)

Langley & jansky

Поглощенная доза (absorbed dose)

Потенциал (potential)

kV, KV, mV, statvolt, V & volt

Volt (V), abvolt (abV) & statvolt (statV)

Поток молярный (Molar Flow)

katal (kat)

Прирост логарифмический (Logarithmic Gain)

B, neper (Np) & bel (B)

Проводимость (conductance)

mho, S, siemens & statsiemens

siemens (S), absiemens (abS) & statsiemens (statS)

Проницаемость (permeability)

perm

becquerel (Bq), curie (Ci) & rutherford (Rd)

Расход (volume flow)

gallon_per_minute (gpm) & cubic_foot_per_minute (cfm)

Расход топлива (fuel consumption)

mile_per_gallon (mpg)

Световой поток

Im

Сила (force)

dyne, kgf, lbf, N & Newton

newton (N), gramforce (gf), pond, sthene, dyne (dyn), ounceforce (ozf), poundforce (lbf), tonforce (tonf), kipf & poundal

Сила света (luminosity), свет (light)

cd

candela (cd), carcel, Hefner_unit (HK), lumen (lm), nit, lambert (Lb), footlambert (ftLb), skot, stilb, apostilb (asb), lux (lx), footcandle, nox & phot

Скорость (speed, velocity)

kph & mph

Kyne, mach (M) & mile_per_hour (mph)

Скорость передачи информации (Speed of transfer of information)

bit_per_second (bps)

Сопротивление (resistance)

KW, MW, ohm, statohm & W

ohm (W), abohm (abW) & statohm (statW)

Сопротивление удельное (electric receptivity)

preece

Температура (temperature)

K & R

kelvin (K), Celsius (degC), Rankine (degR), Fahrenheit (degF), Reaumur (degRe) & centigrade (degc)

Теплоизоляция (heat insulation)

Clo, Rvalue & tog

Коэффициент теплопередачи (heat transfer)

Ufactor

Ток (current)

A, amp, KA, mA, mA & statamp

ampere (A), abampere (abA), statampere (statA) & biot gilbert (f)

Угловая скорость (Angular Speed)

revolution_per_minute (rpm)

Угол (angular)

Revolution (rev), angle (°), gon, grade, mil, circle, semicircle, quadrant, sign, point & hour

Угол телесный (Solid Angle)

str

Ускорение (acceleration)

g (acceleration of free fall)

acceleration_of_free_fall (gn) & Galileo (Gal)

Частота (frequency)

GHz, Hz, Hza, kHz, KHz & MHz

hertz (Hz) & fresnel (f)

Эквивалентная доза излучения (dose equivalent)

sievert (Sv) & rem

Энергия (energy)

BTU, cal, erg, J, joule & kcal

Joule (J), hartree (E0), erg, watt_hour (Wh), ton, calorie (cal), Calorie (Cal), British_thermal_unit (Btu), therm (te), quad, Q_unit & Celsius_heat_unit (Chu)

Comments to the Appendix of Maple Unit Developer
You have some entries under Maple missing:
Luminous Intensity         candela (cd), carcel, Hefner unit
Luminous Flux              lumen (lm)
Luminous Luminance         nit, stilb, lambert, footlambert,
apostilb, skot
Illuminance                lux (lx), phot, nox, footcandle
You grouped all of these under light, and it includes nothing under illuminosity.
Also note that in some cases, MathCad does not allow those standard unit symbols used by the reference I give, as well as  "Standard for Use of the International System of Units (SI): The Modern Metric System", IEEE/ASTM SI 10-1997.  For example, MathCad uses 'gm' and 'str' and instead uses 'g' for acceleration of
free fall.
In some cases, you capitalize Maple's unit names when they should not be.  IEEE states that all unit names should be lower case, and unit symbols should only be
upper case if the unit name is a person's name.  One exception is the "liter" which accepts the symbol L.

One remarks from James

I would like to suggest the following as a way to handle units and constants in a way that seems to me to be more consistent and useful than the present method. (There is a long discussion thread on the pros and cons of this in this forum, too long to wade through.)

Use of Styles

The built-in styles should be extended to include 'Units'. The 'Variables' and 'Constants' styles would remain, but take on some more specialized meaning, as follows:

The 'Variables ' style would be used for variables, and no Mathcad built-in constants or units would have this style.

'Constants' would be used for built-in constants and user defined constants. Once a constant is defined in the work sheet it is global and cannot be redefined. Built-in constants would also be global, but could be redefined at only one place in the worksheet. Built-in constants would be defined ONLY in this style (currently they exist in all styles).

'Units' would have much the same restrictions as 'Constants'. Again, built-in units would be defined ONLY in this style.

There should be some visual distinction among these styles (italics is often used for variables in printed work, but italics does not look as good on a monitor for plain text (suggestions?).

Maintaining this degree of rigor would allow more conformance to normal use with out significant extra burden on the user. There are some inconsistencies at the moment. For example 'g' is used for the standard gravitational acceleration constant, but this means that 'gm' has to be used for gram, rather than the standard 'g'. Also, if you define m:=2.5, say, then later evaluate an equation with length units then 'm' for meters appears in the evaluated result with no distinction between the two m's. (In evaluations, units appear with the 'Constants' style, but even if the m in the assignment has the 'Constants' style applied this inconsistency still persists.

UNITS

Mathcad should have a greatly increased set of built-in units. In particular it should cover all the multipliers for SI units (there are 20). It is quite common to use such things as TW, fF, nH, etc. Whether this done by explicitly including all the units with multipliers or by combining base units with multipliers is up to the programmers. I do not use non-SI units much, so I do not know what would be most appropriate there. The kg would have to be carefully handled since this is a 'base' unit with a multiplier. Perhaps the list of multipliers should be the powers of ten and these would be appropriately converted to the correct combination of prefix and unit. Eg, 10-6 kg would become mg and not ukg.

Since styles would have to be applied each time units are used, this needs to be made as simple as possible for the user. Some suggestions for this are:

(A) Automatic typing

(1) If a character (for simplicity I use 'character' but it could be a series of characters) is typed on a blank piece of the worksheet or on the left side of the assignment operator the default style would be 'Variables'

(2) If a character is typed on the right side of an assignment it is assumed to be a variable if that variable is defined already in the work sheet. If it is not defined as a variable in the worksheet, and it matches a built-in or user-defined unit then it is assumed to be a unit. If it does not match a defined unit, but it matches a built-in or user-defined constant then it is assumed to be a constant. If it matches no unit or constant it is assumed to be a variable.

(3) If a character is typed on the right side of an evaluation after the numerical result then it is assumed to be a unit, a constant, or a variable, in that order depending on which of those are already defined.

(B) User defined typing

Since the above will not always give the desired type, the user has to be able to change the applied style as required. This could be done by:

(1) In an empty place holder, press Ctrl-U. This would bring up the current dialog box. (Since I would like to have all multipliers available, it may be necessary to have the multilpliers and base units in two different colums to avoid excessively long lists).

(2) On an existing variable press Ctrl-U. In this case the dialog box would only be shown if the unit does not correspond to a built-in or user-defined unit. Otherwise it would simply apply the 'Units' style.

(3) Right-click on the variable and select "make unit" from a context sensitive pop-up menu.

(4) Click on the measuring jug which would have the same effect as (1) or (2).

Units and Functions

Units should be rigorously carried through functions. In general I believe that this is true for most Mathcad functions. Many functions mathematically require dimensionless quantities, but some can accept dimensions (eg, fitting functions). I believe that I have had problems using units in solve blocks, but I have not tried using them recently.

Units and Matrices

It should be possible to have matrix elements with different units. This is commonly encountered in physics.

CONSTANTS

I do not see a need to differentiate between mathematical constants and physical ones, except that it would allow some standard nomenclature to be used where there is a clash between the two, eg, `e` for elementary charge and for the base of the natural logarithm.

The way of using constants would be essentially the same as for units. The Ctrl-C key could have the same functionality for constants as the Ctrl-U key would have for units.

Other Issues

What about dimensionless units? Eg radian/degree/arcsec...? I think that these would work fine in the above scheme.

Could degC -> degF -> kelvin be implemented?

James

 Developed by MathSoft Engineering & Education. Inc, (www.mathsoft.com)

 Developed by Waterloo Maple. Inc, (www.maplesoft.com)

 The idea “plus-minus” is not adequate “well-badly”. All is comparative: pluses become minuses under certain conditions and vice versa.

 Introducing the tools for work with dimensional quantities into the mathematical packages turn it into physical and mathematical not only in contents (physical problems are solved on computer for a long time) but in form also. Derive (http://education.ti.com/product/software/derive/features/features.html) can work with physical quantities as well but the author does not know it well.

 We will nolens volens touch on general aspects of working with Mathcad and Maple. The author compared these packages already: see chapter “From Mathcad to Maple” of the book “Mathcad for students and engineers”. (http://twt.mpei.ac.ru/ochkov/mc8Pro.book/index.htm)

 In Maple it is called FPS – Foot, Pound, Second.

 Symbol mathematics is built in Mathcad from Maple.

 Tips which appeared when we start Mathcad. As a rule users disable the option.

 To tell the truth, it is very time-taking task. Nobody will make over the functions created ten years ago – it should be created and tested anew.

 Mathcad does so with the new functions. Here is an example. Financial functions introduced into Mathcad 2000 display an error message if the argument was dimensional.

 The gap in physical quantity can also be composition: 10 h. 20 min. By the way, in this case the division sigh can be placed (10:20) instead of the gap burdening with another function, composition. Here is one more example – kilowatt - hour. The minus (hyphen) is the same multiplication here. In Mathcad we can easily bring about these “human” conventionalities , in Maple – we can not.

 This is the “favorite” physical unit of the author. He is the Moscow Power Engineering Institute graduate and works there. Hence, such interest in temperature, the “basic” physical quantity in thermodynamics and heat transfer.

 Two other scales, Reaumur and centigrade scale, were added to these two relational scales.

 Let us remind that bel is a common logarithm of the ratio of two physical quantities with the same dimensions, neper is a natural logarithm of this ratio, decibel is a one tenth of a bel.

 We can introduce, for example, the variable L1 instead of the user variable L. But it is important there to the variables the name traditional for one or another branch of science. The solution is to use variable style mechanism.

 The process of energy production (steam pressure in the boiler, rotor velocity in steam turbine) is of little interest but everyone deal with such political-technical-economical quantity as electricity charge.

 To tell the truth the candela is dragged into SI by the years.

 Becquerel (1852-1908) with husband and wife Curie got Nobel Prize for radioactivity discovery.

 There is a “metrological” joke. The frequency of the current from heat power station is measured in hertz, that from nuclear station – in becquerel.

 To compute using dimensional units prevent us from many mistakes of incorrect work with dimensions – we can not sum up meters and kilograms.

 Seven because… it is a fine number. The article has seven items.

 In Mathcad and Maple except for standard horse force (hp) there are also other "horse" units of capacities having circulation in Great Britain and in Northern America - bhp ("boiler" horse force), ehp ("electrical" horse force), mph ("metric" horse force), hpUK (British horse force), whp ("water" horse force) etc. The developers Mathcad and Maple need to agree upon the certain standard: we have in Mathcad hp, and in Maple HP. Why?

 In Mathcad except for unit BTU there are also other thermal units having circulation in Great Britain and Northern America - BTU15 (unit, adhered to water at 15 degrees Celsius), CBTU (Canadian thermal unit), IBTU (thermal unit of the international organization of the standards - ISO), mBTU ("basic" thermal unit) and tBTU ("thermodynamic" thermal unit). Except for a standard calorie cal, in Mathcad other thermal units - cal15 (unit adhered to water at 15 degrees Celsius), cal20 (unit adhered to water at 20 degrees Celsius), dcal ("dietary" calorie), mcal ("basic" calorie) and tcal ("thermodynamic" calorie) are built-in also. The people in these calories finally have got confused and have decided(solved) to begin with «of a pure(clean) sheet » - with joule.