Understanding Integers 2

We saw that integers are numbers which can be negative or positive. They are also numbers which have both magnitude as well as “direction”. Mathematicians like brevity. So they decided to incorporate the idea of “direction” also with the familiar symbols “+” & “-“. Hence in integer arithmetic we have the confusing use of double symbols like “++” or “+-“. Let us unravel the knot systematically.

Integers – an introduction to vectors

Integers are also an introduction to the idea of “vectors” whose direction as well as magnitude are important. If you revisit the real-life problem discussed at the beginning of the previous chapter, you would realise that the answer to the problem can be any number, including fractions, between 2 and 10, depending on the direction in which the steps are taken! Just for the convenience of school mathematics, mathematicians restricted the movement to 2 directions mutually opposite of each other.

Hence, they are vectors in a limited sense, in that the directions are limited to two which are mutually opposite.

The first problem in the beginning of this chapter (involving chocolates) did not involve vectors. For them only the magnitude mattered. These are called “scalar” quantities.

The second problem involved “movements”: which are vectors as they have both magnitude & direction.

The use of + & - in representing “direction” of quantities is easier to understand. We will deal with the issue of operations in a subsequent chapter.

Multiple Interpretations of + & -

Normally “+” and “- “represent addition and subtraction operations on numbers. We could call these the “verb” interpretations of these operators.

In integers, mathematicians use the same + & - signs, to indicate the directions as well. This is never clarified to school students, thus becoming the root cause of the confusion students have in understanding integers.

Representing Directions

The word “direction” can be interpreted in integer math in a variety of ways depending on the context. It could be a movement in the East-West direction, a movement up or down or a change in value in opposite directions.

If one of the directions was represented by a “+”, then the other was automatically a “- “. Conversely, if one of the directions was represented by a “- “, then the other was automatically an “+”. The restriction of the movement to two opposite directions facilitated this notation.

If +3 was a movement in the upward direction, then -3 would be a movement in the downward direction. If +3000 is considered as a wealth of Rs 3000, then -3000 could be considered as a debt of Rs 3000. If +5 is an increase in the temperature, -5 could be a decrease in the temperature. If +3 was climbing 3 steps down a well, -3 could be climbing 3 steps up. Which is “+” and which is “-“ is relative. The only requirement is that + & - represent diametrically opposite entities.

Hence one interpretation of the + & - symbols was as verbs and the other was as adverbs, which gives more information to the verb!

Integers - A Brief History

Integers were accepted as numbers by European mathematicians, only in the 19thcentury. Before that many mathematicians took the view that they did not exist.

But they had been proposed by Brahmagupta in his Brahmasphutasiddhanta in the 7thcentury itself. He even gave rules of operations with integers. He termed a positive quantity as a “fortune” and a negative quantity as a “debt”. It is a pity that this historical fact is not known even in India.

It was not until the 19th century when mathematicians began to investigate the 'laws of arithmetic' in terms of logical definitions that the problem of negative numbers was finally sorted out.

Integers - A Break With Physical Reality

The invention of integers marks the critical stage where math breaks away from its ties with physical reality. Unlike whole numbers & fractions, negative numbers cannot be related to operations in the physical world. We can draw some parallels; like debt & assets.

It marks the point where math starts really becoming abstract. The logic of math could no more be tested with real events. It started depending on an internal logic of its own. If there was an equation 2 + 3 = 5, math dared to ask what about 2-5?

This idea has been caught beautifully in this quote by Derek Muller; "Only by giving up math’s connection to reality, could it guide us to a deeper truth about the way the universe works"