Posted by: John Colby | Sunday May 11 2008

Lazy Maths and Formulae

I’m lazy.

At least, that’s what I tell my students. I’m lazy in that I don’t like doing a job twice. I like to reuse part of the job I’ve already done which of course saves time and reduces errors. I like using efficient ways of describing what I’m doing and not being tautological. I like a single source of information (database) so that the records are not out of step.

One of the ways I’m lazy is that I use formulae, mathematical shorthand. To me it expresses in the most efficient way the essence of the problem at hand. It’s also general, not limited to a single case of specific numbers.

This, of course, leads to algebra.

I’m not alone in being lazy, nor modern. Robert Recorde, of Tenby, is credited with in invention in 1557 of the equals sign.

“…to auoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicause noe .2. thynges, can be moare equalle.”

Loosely translated and abbreviated this can be taken to mean:

“I’m fed up with writing the same thing over and over so am using a symbol.”

His publication which introduced the sign was:

The Whetstone of Witte, which is the second part of Arithmetike, containing the Extraction of Rootes, the Cossike Practice, with the Rules of Equation, and the Woorkes of Surde Numbers (London, 1557).

Recorde is also credited, via an earlier publication, with the introduction algebra into England.

Algebra is Arabic in origin.

“… True Algebra was born in the Arab world in the 9th century (the word is originally Arabic: al-jabr, meaning the reduction), and came to flower in Renaissance Europe with the development of a specialised language. Instead of a calculation involving specific number, say 3 + 5 = 8, a symbolic language representing pure ideas and processes mathematicians to focus on the ideas themselves: x + y = z.

“Algebra took the fundamental operations of arithmetic (addition, subtraction, multiplication and division) and the basic concepts (set theory, properties of real and complex numbers and some others) and added rules and definitions – the ‘grammar’ of the language – and symbols – its ‘vocabulary’ – so that calculations could be made without ambiguity. Algebra discusses and investigates numbers in general terms, and thus reveals their nature.

“The basic language of algebra is quite simple, Symbols refer to any quantity, set or type of number, as the operations defines it; parentheses (brackets) keep the order of operations clear or group certain elements together. A power is identified by a smaller number attached to the main number at its upper right. The times sign has gone: multiplication is indicated by two terms or quantities placed side by side (xy means x times y). Division is represented not by a division sign but by the line seen in fractions, separating numerator and denominator. Over time other terms and symbols have been added. Instead of concrete quantities we have the idea of the quantity itself expressed as constants (for known quantities) and variables (for undetermined quantities) as well as the ideas of magnitude and direction.”

(from Guedj, 1996)

So why, in terms of me being lazy, is algebra an advantage?

I’ll quote myself from above:

I’m lazy in that I don’t like doing a job twice. I like to reuse part of the job I’ve already done which of course saves time and reduces errors. I like using efficient ways of describing what I’m doing and not being tautological.

In order to understand the language of algebra, I consider that you have to appreciate a very limited number of things:

  • The concept of the variable
  • The concept of the constant
  • Knowledge of the precedence of arithmetic operations (BODMAS)
  • Knowledge of the shorthand forms that comprise the normal arithmetic operations
  • Knowing that, given the above, you can rearrange algebraic expressions
  • Ability to work things out by going from left to right and top to bottom

As a set of examples, let’s take the best fit straight line in a linear regression problem, usually given as:

y = a + bx

where b is the slope and a the intercept on the y axis.

a and b are given by the formulae:

b = \displaystyle\frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x}^2 - (\sum{x})^2}

a = \displaystyle\frac{\sum{y} - b\sum{x}}{n}

Applying normal precedence of operators, and knowing that the Σ (Sigma) symbol means a sum, it is possible to calculate the equation representing this straight line.

In English, in longhand, the expression:

b = \displaystyle\frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x}^2 - (\sum{x})^2}

means:

  • take the number of data point pairs (n)
  • multiply this by the sum of the product of each of the data point pairs x and y (product means multiply them together)
  • subtract from that all the x values added together multiplied by all the y values added together
  • then when you have this all worked out and totalled, divide it by n multiplied by the sum of all the squares of the x value subtracting the square of the sum of all the x values added together.
  • That then equals b

Somehow I think the formula is shorter, considerably more explicit and more direct.

The problem, of course, comes when students who have not been taught these techniques at school and therefore have no confidence of their own abilities to do this at university level.

So to explain this I get them to recognise that “where there’s a sigma then use a table” in order to perform the calculation. So draw out the table, put in the numbers and do the individual calculations.

  • Then add them up.
  • Then put them in the formula.
  • Then remember where the brackets go when you’re using the calculator
  • and so produce the answer.

Getting to the stage where all these are going right can be a challenge.

But they’re getting there.

Which is good.

P.S. I think Robert Recorde is one of my heroes.

Reference

Guedj, D.(1996) Numbers, The Universal Language. London: Thames and Hudson.

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