Language of Algebra

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Pattern Language is the language of Algebra. In using that language, we have used certain conventions which need explanation.

Why Letters of the Alphabet

We used letters of the English alphabet to represent a &ldquo;changing number&rdquo;.

A language is for communicating certain kinds of ideas. In English language all words are made up of letters, which is collectively known as the alphabet of English language. Each language has its own alphabet.

If we consider Arithmetic also as a language how do we express ideas in it? We use numerals (0 to 9) and operating symbols (+, -, X, etc). Hence the alphabet of Arithmetic is the collection of numerals &amp; operating symbols.

As a language, algebra also needs its own alphabet. But the alphabet of algebra has to express both &ldquo;changing&rsquo; and &ldquo;not changing&rsquo; numbers. Numerals can be used to represent a &ldquo;not-changing&rdquo; number. But it needs a representation, different from numerals, for &ldquo;changing&rdquo; numbers.

We could use the numeral shapes from a different language like Greek or even Chinese! But the alphabet of Algebra also had to be easy to learn, so that the focus remains on the structure and logic of the algebra language and not on remembering the symbols of the alphabet itself.

For these reasons, mathematicians settled on the English alphabet itself as they were familiar as well as different from numerals. Hence algebra started by using &ldquo;English alphabets&rdquo; in place of numerals to represent &ldquo;changing&rdquo; numbers.

In the examples shown in the previous chapter, we have mostly used the letter &ldquo;a&rdquo;. Actually any letter of the alphabet can be used.

Representing Constants 

A constant or a &ldquo;not-changing&rdquo; number can be of 2 kinds.

&ldquo;Changing&rdquo; numbers &amp; &ldquo;unknown&rdquo; &ldquo;not-changing&rdquo; numbers
 * 1) It can be already known as 3 or 7. In such a case it can be represented by that number.
 * 2) It can be not-changing&rdquo; but not known. An example is the age difference between Ram and his brother. We know that the age difference will not change. But the age difference may not be known. Here again we need to use a letter from the English alphabet to represent &ldquo;unknown&rdquo; but &ldquo;not-changing&rdquo; numbers.

Both of these are to be represented with letters of the English alphabet. But mathematicians agreed on the following convention.

Operating Symbols in Algebra
 * 1) Letters from the end of the alphabet, like x, y &amp; z are used to represent &ldquo;changing&rdquo; numbers or Variables.
 * 2) Letters from the beginning of the alphabet, like a, b, c are used to represent &ldquo;not-changing&rdquo; but &ldquo;unknown&rdquo; numbers.

All operators used in Arithmetic could be used in Algebra also.

If a variable does not have any operating symbol preceding it, then it is assumed to be &ldquo;+&rdquo;. If the pattern language is &ldquo;b – 5&rdquo; then b is assumed to be &ldquo;+b&rdquo;.

The multiplication sign &ldquo;X&rdquo; could be mistaken for variable x. Hence mathematicians agreed that the &ldquo;X&rdquo; sign could be dropped without any loss of clarity. Another convention was that while writing the product of a variable and a number (or a constant), the number or the constant would be written first, followed by the variable.

Hence 2Xy would be written as &ldquo;2y&rdquo;. z X b would be written as &ldquo;bz&rdquo;.

All other signs were retained as they were, since there was no confusion between them and the letters of the alphabet.

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