One good example of Error by using units in
calculation: Gimli
Glider

I tell you a funny story as an epigraph:

Dialogue during an exam:

**Teacher:**
What does a horsepower mean?

**Student:** It
is the power that a horse of one-meter growth and one kilogram weight has*.*

**Teacher:** But
where have you seen such horse*?!*

**Student:** But
you cannot look at it so simple*. It keeps
in *

Now days nobody is wonder at the fact that not
only candidates, engineers and students but also pupils solve their problems on
computer. But choice of program for these purposes astonishes. The question:
Why do a lot of people more often use Mathcad for calculation their problems.
Reply may be next: Mathcad has unique possibility[1]. Mathcad can operate as with simple
values as with ** physical values**. Mathcad can be named

Work in Mathcad is the
third step (and now days it is top one) of using computer engineering for
solving physic-mathematical, technical and educational problems. Two previous
steps are work with computer codes (for example with assembler language) and
with programming languages (BASIC, Pascal, C, FORTRAN etc). These two
technologies of scientific and technical calculations (computer codes and
programming languages) played spiteful joke with these calculations.
Dimensionalities of physical values and its units (meters, kilograms, seconds
and other ones) were excluded. Handmade solution of a
physical problem (school or student ones) as rule demanded and demands using of
dimension values[2].
Automation of such calculations (that is composition of programs for computer)
excludes «physics» of a problem. These programs keep only the numerical values
of variables, but corresponding them units programmer
has to «keep in mind». So when we interpret a calculation into computer
language it is necessary to follow the strict rule – all physical values must ^{ 3}, 10^{6} etc. This strict rule
caused next inconveniences:

1.
International
system SI will never be unique one though it is widely used. For instance,

2.
We
can image the process of creation of a program without its debugging. The main
tool of debugging is output intermediate results. So analyzed these results we’^{6}, 10^{3}.etc
– see the table??? in the third part of the book). Taking into account all
advantages of the system SI but it was took root as a «set of presents». The
part of units (kilograms, meters, seconds) used and use without any problems,
the other partèñïîëüçîâàëèñü è èñïîëüçóþòñÿ áåç êàêèõ-ëèáî çàòðóäíåíèé, äðóãèå æå («loading») did not settle down as dominating (main) units. For
instance, in heat-and-power engineering[5], pressure of a steam in a boiler is
measured and expressed in atmosphere[6], but pressure in a capacitor is measured
in millimeters of mercury. «Main» unit of pressure (pascal
is

3.
In
programs we have to put in some formulas that are derived basing both on
theoretical analysis of a problem and getting after static treatment of
experimental data. As rule coefficients of such formulas (for instance, see
picture. 1.28) are rigidly connected with fixed system of measure. These
coefficients demand conversion for using in the programs. It can bring in
additional error hence it will be cause of mistakes.

Some words about the design of the book.

When this book will be published at the same time
its Internet-version will appear (see http://twt.mpei.ac.ru/ochkov/unit/index.htm). Because of Internet version the design of
the book is special. For instance, hyper-links* are displayed*: see picture. 1.3, see __below__, see __footnote 3__
etc. Working with the book if a reader meets hyper-link he will have to go to
necessary page himself. In Internet-version the reader can go to new place and
come back easier (with help of help-link) – he has to click his mouse by
hyper-link. In Internet-version author will make alterations and addition. Author hope that readers will send them to address ochkov@twt.mpei.ac.ru.

Exclusion of units of physical values from the
calculations has told on the work in Mathcad. Many inexperienced users begin to
form the document in Mathcad-this way

They write dimension of some physical value as
a commentary but not as a factor of numerical constant. So they exclude the
dimension of this physical value from the subsequent calculations. For
instance, if we remember the language BASIC, above mentioned example will have
next form in it: P = 20: Rem Pressure in MPa. Here units are written in
commentary (in note – Rem). Using this algorithm a programmer can forget that pressure is in MPa, but not in bars. Finally he will get a mistake.

Mechanism of
the work in Mathcad with units of physical values allows:

·
*to put in* basic data in necessary system of
measure, in necessary units with necessary factors (kilo, milli
etc);

·
*to put in* a control of dimensions in
formulas, that we use in our calculation (for instance, we can
not sum kilograms and meters)[9];

·
*to** put out *calculated* *values in necessary system and units,
to use convenient scale of the axes of plots etc.

One of the ways to solve this
problem is a failure of dimensional values. We can go to *non-dimensional numbers* that keep quantitative assessment of
physical phenomenon: Re[10], Nu, etc (see the table ??? in the third part of the book). When we say that
the pressure in a boil equals 50 atmospheres then we operate with
non-dimensional value. We mean that the pressure 50 times as much of
atmospheric one. And more radical way is from numerical characters we go to
qualitative assessments. We can say that pressure equals some number of
atmospheres, but we can also say that pressure is *normal (high, low etc) *and form *theory of careless ranges* will guide
this computer-based system**[11]***.*

Mathcad completely support mathematics of work with dimension variables.
We’ll speak about it below and look through some simple examples.

[1] This uniqueness is not absolute
one. With dimensional values Ñ Another popular mathematics program Derive can
work with units.

[2] When the author
tell about it he has remembered one seminar of transmission. They had to
do some calculations using difficult «dimensional» formulas (see, for instance, picture. 2.18). At that time only one tool of
automation of such calculation was a slide-rule. The process of calculation
consists of three stages: the slide-rule gave out mantissa of a reply, but
degree and dimensional of this reply they had to do yourselves.
Students had to do it without some digital computer or analog-computing device.
Because of it seminars of transmission turned into lessons … of technology of
calculation and metrology. So there was almost no time to discuss «physics» of
the problem. Soon electronic calculators and computers appeared. These
computers gave out a reply that containing degree of a number. But the units as
usual were «overboard» of the process of automation of calculations.

[3] This abbreviation we’ll use for
designation of British system of units.

[4] For instance, from Mathcad we can
go to Internet-forum that is called Collaboratory.
Here you can find both questions and replies and Mathcad-documents with
solution of some problems touching on different spaces of science and
engineering. The address of the forum is http://collab.mathsoft.com/~mathcad2000.

[5] Author is
lecture of Moscow Power Engineering Institute.

[6] For instance, this fact was fixed
in marking of water steam turbine: Ò-100-130 (extraction, electric power is 100 megawatt, pressure of live
water steam is 130 atmospheres), Ê-300-240 (condensation, 300 megawatt, 240 atmospheres). Apart many heat electropower stations so far have got old manometers
graduated in atmospheres.

[7] In 70-th years some attempts during
weather forecast to tell atmospheric pressure in «correct» hecto-pascal
took place. It was not common unit of pressure. Its innovation caused numerous
protests. Of course!, and how! Hypertensive-men could
miss to take their medicine if they had not understood weather forecast. Then
pressure became to name both in hecto-pascal and in
millimeters of mercury, but then hecto-pascal was
forgotten.

[8] Certainly control of
power-generating unit is automated, but a man always has the right to interfere
in it work.

[9] Account of a dimension often allows
deriving formula. For example, vibration frequency of a pendulum depends on
only the length of the pendulum (certainly we speak about simple pendulum) and
free fall acceleration. From these conditions it is easy to derive the formula.

[10] Character of the flow of water in a
duct (turbulence, laminarity) can be estimated by
three different parameters: speed, diameter, and ductility. But we can do it
also by one non-dimensionals parameter – by number Re.

[11] Notion «normal» pressure is some
range of values. Concrete value of pressure belongs to this range with certain
power of plausibility. It is described corresponding *function of accessory *(see the article of the author «Mathcad and
the theory of careless ranges» – see http://twt.mpei.ac.ru/ochkov/F_sets.htm).