The computation of the area of the rectangle can be rewritten with shorthand notation as follows. To introduce shorthand notation, we say the area A of a rectangle is given by its width W times its length L. Here, we use A as shorthand for the area of a rectangle, L as shorthand for its length and W as shorthand for its width. The formula (recipe) for calculating the area A of a rectangle can be written more briefly as

A = L×W

or

A = L ·W

The shorthand notation takes less space to write than the word-only description. Read the symbols × and · as times or multiply. The symbols × and · are both shorthand codes for times or multiplication. The dot symbol · is preferably to the times symbol × when the latter could be confused with the letter x. Confusion can occur because the letter x which many people write is too similar to the multiplication symbol ×.

Shorthand notation provides a code for the description of calculations. Formula decoding is required. The shorthand formula A = L ·W is more compact (takes less room) than the word-only description. This formula is meaningless for us when the role of the letters in this shorthand description is not explained. To understand and to use the shorthand description or formula, you need information. You need to know or find what numbers or quantities the symbols mean or represent. In the above formula, L stood for the length of a rectangle. This has to be said to you or you have to ask. To anyone without this information, the formula remains mysterious.

Talking about and describing computations almost gives us the power to do them. In the area calculation, the area A is obtained from the recipe A = L×W provided the length L and W are given or can be found. Without this information, we can describe or understand a calculation but not use it. The above rectangle example reminds us of the following:

1. We can talk about quantities or numbers without doing any arithmetic. We can speak about numbers and quantities even if we have not measured them or do not know their values exactly.
2. We can describe calculations without performing them. This description can be done with words alone (throw out the letters) or with mathematical shorthand notation, as convenient.

1.2  Triangles

In words, the area of a triangle is given by one half the length of a base of the triangle multiplied by the height of the triangle. This formula can be justified but at this moment we will not worry about why it holds. We may also write more briefly

(Area of triangle) = 1 2 [ (base length) ·(height of the triangle)]

We may write still more briefly that the area of a triangle is given by

A = 1 2 [B ·H]

This involves some shorthand notation: the letters A, B and H. When you read or decode this shorthand notation, remember B stands for the length of a base of the triangle. Also remember H stands for the height of the triangle above this base. Lastly, remember A stands for the area of the triangle.

We have used single letters in this shorthand description of the calculation. Any mark or squiggle or symbol you can draw and name can serve as shorthand for some number or quantity.

Perhaps, we should use A_triangle or another symbol, since we have already used the letter A in the previous rectangle example. Alternatively, we adopt the following rule: while you are reading this triangle example, we use the letter A here as shorthand for the area of the triangle only. More will be said on using and reusing (recycling) shorthand symbols (for example, letters) and the roles they take. Think of them as actors which can perform many parts. They may take only one role in any scene, except for stories and scenes involving identical twins or cases of mistaken identities.

1.3  Circles

To rewrite or encode this formula in shorthand form, we will first describe the code. Let A be shorthand for the area of a circle.

¹Here we must forget the previous meanings and roles of the letter A as the area of a rectangle or the area of a triangle.