Previous Chapter: 10 Describing & Changing
Calculations
1 Words are Not Enough
Imagine describing a picture by saying there is a patch of blue, a bit of red
or orange there and yellow here. Words alone do not usually describe a picture
fully. The picture must be seen. A picture seen conveys or presents information
which words alone cannot capture. Similarly, formulas and arithmetic
calculations are better seen than described with words alone. Pictures and
formulas are often worth more and may be easier to understand than a thousand
words. Words cannot replace pictures and formulas.
2 Decimal Notation in Arithmetic
Decimal notation is used in the representation of whole numbers, fractions
and irrational numbers. Decimal notation allows us to write down these numbers
easily. Further, before decimal notation appeared, addition, multiplication,
division and subtraction were complicated operations. People had to use the
abacus or Roman numerals or other means for calculating. Imagine, if you can,
how to do arithmetic without decimal notation. Decimal notation is the key to
our doing arithmetic easily. Decimal notation is itself a shorthand notation for
the representation of numbers.
Since the 16-th century A.D. popularization of decimal notation, rules and
methods have been invented or found to say how to write, add, multiply, divide
and subtract numbers and fractions with this notation. Today these methods
or variants of them are taught in elementary school.13
Good or understandable shorthand helps people to describe calculations and
even to do them. The ease with which we write or record numbers and calculations
affect the way in which calculations are done and if they are done. Before the
use of decimal notation, basic arithmetic operations were difficult to do and
record, few people mastered arithmetic.
In talking and writing, our language and vocabulary and experience limit and
guide what we can say or even think of. Similarly, in mathematics the language
(shorthand conventions) and previous experience limit and guide what can be said
or attempted.
3 Shorthand Notation in Algebra
The use of shorthand notation (letters or symbols) in the description of
calculations is only several centuries old in Europe. Prior to the 12-th century
A.D., calculations were described with words alone. The use of shorthand
notation for describing calculations is not old. In geometry letters have been
used by some Greeks, two thousand years ago, to label the sides or vertices of
triangles, but not to describe calculations.
As in arithmetic, algebraic shorthand notation records and describes
calculations in a written, visual and non-verbal manner. Here words are not
enough to describe simple calculations. For instance think how you would
describe the calculation of the area of a rectangle or a triangle to a young
child with words alone. But in the description of more complicated calculations,
words become inconvenient and awkward to use. The description is best given with
the so-called algebraic shorthand notation of mathematics. For examples of more
complicated calculations, consider the compound interest formula and the
quadratic formula.
- The compound interest formula:
- The quadratic formula:
You may try to explain with words alone, the calculations described by one or
both of these formulas. Also explain with words alone, why each calculation is
done. These tasks are hard. Just to make it harder, the use in your explanation
of the shorthand notation including the letters that appear in these formulas is
forbidden. (All of this is too difficult and too awkward for me to think about
further. The same may hold for you.)
Complicated formulas are written to be seen. They are not easily read aloud.
Their meanings are not easily described without the use of shorthand notation.
The shorthand notation in mathematics gives a written code in which calculations
are efficiently and briefly described and/or changed. It is a code which is
better seen than read aloud.14
The algebraic shorthand provides a language and environment in which
calculations are changed or manipulated into new ones.
4 Seeing is Better Than Hearing
The saying seeing is better than hearing usually means that seeing and
witnessing an event is better or more reliable than hearing about it. In this
section, we will give a new meaning to this saying.
Our ability to see is more powerful than our ability to hear. Words are
typically understood one at a time. After that they need to be remembered. In
contrast, a picture, a calculation or a formula can be seen all at once. They
will remain in front of you for further thought and observation while you are
looking. No memory is required. Once a picture, formula or calculation has been
put on paper, the details can be seen all at once. The paper used this way,
serves as an immediate or quick extension of our minds or memory.
Eyesight together with algebraic shorthand notation and geometric diagrams
provide more powerful ways for the communication and recording of mathematical
thoughts than words alone. All this explains why mathematics is better seen and
written than spoken aloud. Words alone are not enough for the communication of
mathematics.
Footnotes:
13The
development of arithmetic during the period 1300 to 1637 A.D. is described in
Chapter XI of the book A Short Account of the History of Mathematics written
by W. W. Rouse Ball (Publisher Dover, New York 1960). A copy should be
in your town or school library. The book was written in the last part of the
19-th Century.
The book A Source Book in Mathematics by David Eugene Smith, first
printing 1929, Dover reprint 1959, has a chapter on the 1585 A.D contribution of
Simon Stevin (1548-1620) to the popularization of decimal notation. More recent
accounts of the history of arithmetic and decimal notation may modify or correct
the historical impression in the above references.
14We need a
code which could be read aloud more easily. |