12. Shorthand Usage Guide

Volume 2, Three Skills for Algebra

Shorthand notation is useful for describing and doing calculations. The first shorthand you meet in mathematics is decimal notation.¹⁵

  ¹⁵Computer scientists may prefer base 2, 4 or 16 instead of base 10

 Imagine having to write the longhand form three hundred and twenty-five instead of the decimal notation 325 in an addition, multiplication, division or subtraction involving this number. We use decimal notation, like 325, to represent and write numbers briefly. Without decimal notation, methods for doing arithmetic by hand would be awkward or much longer. A second shorthand notation, algebraic and symbolic, is used to describe calculations. Flexible guidelines for the selection and invention of shorthand notation in describing calculations, are explained in this chapter.

 Pronouns, Labels and Place-Holders

1.1  Pronouns in Ordinary Language

In writing and talking, we often use pronouns. For instance, we may be writing about a person Samuel Henry, junior. In describing Samuel Henry, junior, we may speak of his intellect, his height, his personality, his girth or his activities. He might for instance like walking and swimming, or playing chess. He might also be interested in theater and politics.

Imagine how much longer the last paragraph would be if we were not able to use the pronoun he. Then for each occurrence of the pronoun he, we would have to write Samuel Henry, junior. Here pronouns are convenient. The use of pronouns lessens the amount of writing which we have to do and simplifies our communication.

Now when we speak of two or more people, we need to say clearly to whom the pronouns he, she or they refer. We cannot repeatedly use the word it when we have more than one object to speak about. Pronouns can make speaking and writing easier. But each pronoun must be clearly identified. We have to know to what each refers. When misused, pronouns lead to confusion.

1. Pronouns, Labels and Place-Holders

1.2   Pronouns in Mathematics

For example, recall the formula A = W·L used to calculate the area A of a rectangle with width W and length L. We can make a link or analogy between the use of letters (like A, L and W) and the use of pronouns. In place of the single pronoun it, we use three letters A, L and W. Instead of over-using the pronoun it (we would need several) to refer to numbers and quantities, we use these and other letters. When using the word it we must be careful. What the word it represents in each sentence should be obvious from the context. We must take similar care with substitutes (above A, L, and W ) used for the word it.

More Examples: You may have seen shorthand notation in the formula A_circle = pr² for the area A_circle of a circle of radius r. You have seen shorthand in the formula for the area A_triangle = ½bh for the area of a triangle with base length b and height h. The letters appearing in these formulas with and without subscripts are examples of shorthand. They serve as mathematical pronouns.

Shorthand in mathematics is often given by symbols or letters. The letters are often taken from the Roman, German, Greek or Hebrew alphabets. The letter p comes from the Greek alphabet. The selection (and corruption) of letters from many alphabets increases the number of pronouns, labels and place-holders for the numbers and quantities we mentioned.

The shorthand notation for a number or a quantity may become a temporary or permanent pronoun for it. A name for an object provides a pronoun with a short or long term attachment. We try to use one name or pronoun per item when writing and speaking.

In summary, the role of shorthand notation in mathematics is like that of pronouns in everyday speech. Instead of describing fully a number or quantity when we refer to it in conversation or a calculation, we introduce a shorthand symbol for it. The main guide in the use of pronouns or shorthand is clarity. Avoid overuse and clearly say what each mathematical pronoun means. In any situation, each pronoun and symbol should represent one and only one number or quantity. To avoid confusion, care has to be taken so that different symbols, our new pronouns, refer to different numbers and quantities. Plays in which an actor plays more than one role without the audience being told can be confusing as well - Can actors be viewed as living pronouns?

2  Choice, Selection and Invention
of shorthand symbols for numbers & quantities

In the following table we identify some numbers and quantities in the first column, and suggest a shorthand symbol for it in the second. Other symbols could be used instead of those given.

Here we have used letters and other symbols as shorthand to represent an object (quantity or number). Our shorthand for the weight of a truck could also have been T or q or Q or anything you please. To avoid ambiguity and confusion, the only rule is: don't use the same symbol twice for two different objects whenever both appear in a single problem or circumstance.

Is there any confusion about the use of any letter or symbol above? Look at the letter A. Its meaning has changed. So far in this book, it represented the area of a rectangle, a circle and then a triangle. Does the letter A have any meaning now? Should the meaning of A come from our last use of it? Can we change it?

3  Overuse of Shorthand

Look at a list of first names of students in a class. If the name Tom occurs once on that list, then only Tom responds when a teacher calls his name. On the other hand, if the name Catherine occurs more than once, confusion could result. With such a class list, when a teacher says Catherine, please stop that, a few Catherines may hear. Here the innocent Catherines may feel wronged or hurt. A case of mistaken identity can lead to confusion.

To avoid this confusion, to be clear, we may ask them to call themselves by different names or nicknames. Each Catherine could be asked to use a middle name instead. Alternatively, we could have a Catherine I and a Catherine II. Which one gets to be I and which one gets to be II is a problem beyond the scope of this text.

Names and shorthand notation can be chosen in any way we please. In the modern classroom, many students have identification numbers besides their own names. Would student numbered or labeled 7281170 identify him- or herself?

It is a challenge for mathematicians to invent a shorthand notation which can be read aloud without ambiguity.

4  Big and Small Letters

For example in speaking about two rectangles, the lower-case letter, small a, could stand for the area of the first, while the upper-case letter, big A, could represent the area of the second. The letters big A and little a sound alike when spoken aloud, but they look different. We can and do use them as shorthand for different numbers and quantities.

This convention unfortunately reinforces the situation in which mathematics notation is better written and read silently far more clearly than it is spoken or read aloud. Shorthand notation needs to be seen to be understood.¹⁶¹⁶It is a challenge for mathematicians to invent a shorthand notation which can be read aloud without ambiguity.

5  Subscripts Etc 
why subscripts appear

When we start to run out of symbols or imagination for our shorthand, we can always use subscripts (usually whole numbers, letters or words) on our symbols. Subscripts mean more writing, but they give more (compound) symbols to employ as shorthand for quantities. For instance, suppose we have a list of say five numbers. The first number on the list could be denoted by x₁, the second number on the list could be represented by x₂, the third number on this list could be indicated by x₃, the fourth number on this could be designated by x₄, the fifth number on this list could be symbolized by x₅. More subscripts could be used as needed. Here are some more examples.

The choice of shorthand is often a matter of taste. Tastes do differ.

6  An Exercise

The following table gives more examples of numbers and quantities we can speak about without doing any arithmetic. Here is an exercise to be done now or later as you like.

1. Identify what entries in the table are given by numbers.
2. Identify those entries which are numbers and which are quantities. Also identify units in which the quantities can be measured.
3. Suggest shorthand notation for the numbers and quantities in question. 

Remember that each quantity is given by a number times a unit. Some answers are given at the end of this chapter.

7  The Letters M, x, y and z

On calculators, a button marked M often refers to a memory location or the last number stored in that location. A number can be stored in and recalled from the memory location M. We may use the current value of M in calculations. For instance, the number described and given by the expression 5*M+4 can be computed on a calculator. The result obtained depends on the value saved for M. The expression 5*M+4 describes the calculation: multiply the value given or saved for M by the number 5 and then add 4.

We often meet the letters x, y and z in algebra. When you meet them, think of three boxes or memory spots, one for each. We can now speak and talk about the numbers (or quantity) in the first, second and third box, whatever they might be. To write about these numbers briefly, without too many words, we use the shorthand x, y and z. The letter x is our name or shorthand for the number in the first box. Similarly, the letters y and z become shorthand for the contents of the second and third boxes, respectively. The shorthand expression x ·y is now shorthand for the action: multiple the number in the first box by or with the number in the second box. The shorthand expression x·y describes a calculation which could be done if we knew what numbers or quantities were stored in the x and y boxes.

More generally, suppose the physical meanings or connotations of one or more letters are not given. Here we can imagine there are several containers labeled with these letters. Then we refer to the content of each box, a number or quantity, by the box label. The number q now refers to the number or quantity in the box or container labeled q - the latter also refers to the box itself.

The initial convention as begun in the 12-th century A.D. was to use vowels to represent unknown numbers and consonants to represent given ones. This stems from the convention in Arabic or Hebrew of writing only the consonants and then finding the vowels from the context of the words in which they appear.

8  Offspring Naming Conventions

Parents and mathematicians both may have difficulty naming their children. The names of grandparents and uncles, etc, are often used along with numerals, for instance, Richard III, Frederick IV etc, to name children. Similarly mathematicians and you may lack imagination when naming numbers and quantities. Typically we, a teacher or s textbook may use, reuse and recycle endlessly the symbols x, y, z, x₁, x₂, x₉₉, a, b, c as shorthand (pronouns) for numbers and quantities.

In any situation, the shorthand we use for a quantity or number is unimportant. We just have to employ different symbols for different numbers and quantities. Letters by themselves from the language of your choice, or letters with subscripts can serve as shorthand symbols.

Traditionally, letters at the start of an alphabet stand for numbers or quantities that will be given or will remain constant¹⁷. In the same tradition, letters at the end of the alphabet have been used for numbers or quantities which are unknown or variable. This tradition or guideline is often broken. You are allowed to break and introduce other conventions as convenient, for each problem you meet. Here there is only one rule or requirement. Statements like

1. let L be the length of my foot,
2. let A be the area of the circle or
3. let M be a whole number

9  A Review

A symbol or letter can serve as a shorthand label, a name, a place-holder or a pronoun for some number or quantity. A symbol, letter, label etc., can be written whenever the number or quantity is to be used or called upon. For clarity, within the same context or problem, different letters and symbols are to be used for different quantities, people and objects.

We have much choice in selecting a shorthand notation. But the same letters can be used in different problems. Once we realize or allow this, all we have to do is to find a statement, picture or some other indication of the shorthand roles of each symbol and letter in each formula, problem or context.

Answers to Exercise

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science