Mathematic Principles

Introduction

[page last updated on August 27, 2010]

In primary and secondary school, and for many reasons both good and poor, we’re taught arithmetic not mathematics. Arithmetic defines processes for using numbers to complete specific tasks, for example,

Q: If Alice has ten apples and eats four, how many apples does Alice have left for her friend Bob?

A: Alice has six apples for Bob since, 10 – 4 = 6

We find arithmetic useful in mundane tasks, however, when it comes to more complex situations, the arithmetic we’re used to starts to fail. Consider the following example,

Q: Alice farms apples. Bob manages an import/export company specializing in moving apples to high-demand markets, and Charlie runs a grocery store down the street. If Alice can produce only N apples per week, and if X apples can be exported by Bob, then how many apples Y can Alice sell to Charlie?

A: Alice can sell N apples to Charlie minus X apples exported by Bob, or, N – X = Y

To solve that word problem, you need to think in terms of variables; simple number manipulation cannot complete the task. In real life, the problem becomes more complex when you place restrictions on variables (eg. “N < X, where N and X are natural numbers”) which in turn forces you to refine your approach to the problem.

This habit of definition, precision, and refinement are examples of what I mean by “mathematical principles.”

Definition

In arithmetic, your told which objects and operations you can use to solve a problem. We call these objects “numbers” and these operations “addition,” “subtraction,” “multiplication,” and “division.” The teacher defines these objects and operations for us, thus making it easier to do things with numbers, yet we can greatly benefit by thinking about why objects and operations have been defined and why.

  • Numbers symbolize quantities. Natural numbers ( 1, 2, 3… ) symbolize whole positive quantities.
  • Operations manipulate symbols. In other words, when we perform an operation we take a symbol and do something to it, producing a different symbol. Operations always produce the same results from the same inputs ( 1 + 1 = 2 always).
    • Addition increases the initial quantity by a given quantity, producing a new, higher quantity.
    • Subtraction decreases the initial quantity by a given quantity, producing a new, lesser quantity. Inverse to addition.
    • Multiplication increases the initial quantity by a given quantity, producing a new, higher quantity.
    • Division decrease the initial quantity by a given quantity, producing a new, lesser quantity. Inverse to multiplication.

(Note that “inverse” does not necessarily imply “opposite.”)

In theory, we only need addition and subtraction to construct a reliable arithmetic process – but doing so would require far too many steps to be useful. Defining math from the start makes doing everyday math much more useful.

But then why learn math? Why not just build a calculator, which does the pre-set operations for us? The reason we need to think about how objects and operations get defined is that we can then take that understanding to build new sets of objects and operations.

Precision

We have to be very careful when we use our mathematical tools. If we stray outside our defined use or if our definitions are too broad, the tasks we attempt may fail. Thus a vital aspect to mathematical thinking involves precision.

When we say a definition or process is precise, we mean that any application of the definition or process will yield predictable results. Any application which yields unexpected results must be questioned; and all applications which yield contradictory results must be rejected — the imprecision indicates that something has gone awry, and we cannot get the answer we seek.

Thus, the mathematician must make an effort to use definitions and operations only in the way she first describes them. Otherwise, she effectively invents new mathematical tools — not a bad thing, mind you, but a dangerous genesis.

Refinement

Let’s say Bob needs to count all the soup cans on his pantry shelf. Bob opens his pantry, unloads the cans from the shelf by armfuls, and then replaces them back onto the shelf, counting each. Soon Alice comes along and stops him (she can’t bear the sight of Bob’s inefficiency). She helps Bob rearrange the cans on the shelf into neat rows, then multiplies the number of cans in each row by the number of cans to the back of the pantry, yielding a full count. Alice has not refuted the correctness of Bob’s counting process, but she has refined his process in order to make it faster. She replaced an additive process with a multiplicative process.

By refining mathematical procedures, we can not only improve the speed with which we execute a task, but also give ourselves new tools for completing more complicated tasks.

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