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Appreciating the four arithmetical operations


We all know that the four traditional/basic arithmetical operations are addition, multiplication, subtraction and division. When any of these operations are applied on some quantity of things (expressed as numbers), the quantity of things is seen in a new light (once again expressed as numbers).

The four operations are processes that input and output numbers; therefore, define what we can do with numbers. It may be added that these operations are part of all of math, the most complex math too - all mathematical expressions have to use at least one of these operations, even if implicitly. More details on these lines is well beyond the scope of the book.
For instance, if there is 1 kg of fruits, the following are the only two changes we can expect:

  1. The quantity of fruit increases or
  2. The quantity of fruits decreases

Yes, a given quantity can increase or decrease when we get more of it or use/distribute it. But the increase, as well as decrease in quantity can happen in two ways each:

  1. The increase, or decrease in the quantity can be in a random way (e.g., take away 1 fruit out of 5) – the amount of increase or decrease in quantity bears no intended relationship with the given quantity, or
  2. The increase, or decrease in the quantity can be in a ‘special way’ (e.g., take away 1/2, 1/3, 1/5, etc. of the 1 kg, or add half, twice, thrice, five times of 1 kg) – the amount of increase or decrease in quantity bears a definable relationship with the given quantity

What can happen with numbers?
Thus, there are just 4 ways a given quantity of things, in other words, a given number, can change:

  1. Increase in the quantity in a ‘random way’ (what we call addition)
  2. Decrease in the quantity in a ‘random way’ (what we call subtraction)
  3. Increase in the quantity in a ‘special way’ (what we call multiplication)
  4. Decrease in the quantity in a ‘special way’ (what we call division)

It needs no emphasis that the aforementioned interpretation of the operations is rather simplistic, and only for the purpose of connecting all the four operations together. One other purpose of the above interpretation is to present how the operations are actually connected to daily life, how the operations help us express very real-world events, situations, and outcomes.

Why we invented the arithmetical operations?
In other words, we will get to understand what the arithmetical operations really are, what do they achieve for us!
Given the specific context for exploring the operations, we must take the example way to discreetly experience the four operations.
Let us consider an example of a quantity of money, say INR 1000, and think of how we can respond to the following situations if there were no mathematical operations:

  1. Of the INR 1000, INR 200 was spent. How can we find out the amount of money left?
  2. INR 100 is received over and above the INR 1000 in possession. How can we find out the total money in hand now?
  3. Twice the amount is received over and above the INR 1000. How can we find out the total money in hand now?
  4. INR 1000 is to be split equally among 4 people. How can we find out the money that each of the 4 people will get?

In all the four situations, the only way to find out the changed quantity of money is to count the money again after the change has taken place. Is there any alternative? Not really. A better and easier way of determining the changed quantity of money, without having to count again had to be discovered. Counting again and again after each change in quantity is clearly very tedious and not an option.
The simplest way to find out the changed quantity without counting is by performing mathematical operations on them. The four operations are the special ways of counting, without having to go through the tiresome process of counting again. Whenever we know the original quantity and the quantity of change, the four operations can be used as shortcuts to counting.
Let us broadly see how we can find out the changed quantity in each of the above four situations, without counting after the change has taken place.  Obviously, we will not discuss the 4 mathematical operations in details here as it is well beyond the scope of this book. Here is what is possible:

Situation 1
Of the INR 1000, INR 200 was spent. How can we know the amount of money left?
A possible way of knowing the quantity without having to count what is left, is by using the operation subtraction.
We have two quantities INR 1000 and INR 200, of this the second quantity INR 200 is used up, hence, we subtract it from the first quantity to get the changed amount of INR 800 without counting.

Situation 2
INR 100 is received over and above the INR 1000 in possession. How can we know the total money in hand now?
A possible way of knowing quantity without having to count what is left is by using the operation addition.
We have two quantities INR 1000 and INR 100 of which the second quantity is over and above the first quantity, hence we add the two quantities to get INR 1100 without counting.

Situation 3
Twice the amount is received over and above the INR 1000. How can we know the total money in hand now?
We have two quantities given in the situation – INR 1000 and twice of INR 1000. The two quantities can be used to get the total amount – thrice of INR 1000, by using the operation multiplication, we get INR 3000 without counting the total amount.

Situation 4
INR 1000 is to be split equally among 4 people. How can we know the money that will be in the hands of each of the 4 people?
We have the two quantities given in the situation – INR 1000 and 4 people. We use the operation of division on the two quantities to get the money in the hands of the four people – INR 250 without counting.
Closing this discussion at this point, we have seen how we use quantities at hand to get new, or changed quantities, without counting again. This is why the arithmetic operations were invented and remain the most fundamental ways of working with quantities, or numbers.

Operations are ‘grammar’ of math
You may already have guessed it right – numbers are to math what words are to languages, and arithmetic is to math what grammar is to languages. Hopefully, you also realise that math is actually a language (to express any real, or imaginary situations, using numbers and operations).
More pertinently, the words of math – numbers – are limitless. Thus, at the heart of math is really arithmetic (and reasoning). Arithmetic sets the ground rules for playing with numbers and helps us most effectively use the endless possibilities of numbers. The better we know arithmetic, the better our ability to master math.
Expectedly, we will start our journey of re-learning math with exploration of the mathematical operations (arithmetic, to start with).
You are requested to be patient in your reactions to what you read and learn, each of the learning outcomes discussed herein is critical to understanding and applying the operations correctly in the more complex mathematical operations.

Summary

The 10 mathematical symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called digits. We combine the digits to get numerals. Numbers are mathematical symbols we use to denote the outcome of counting, and measurement. Numbers contain the unit of counting, or measurement, whereas numerals do not.

Excerpted from the book ‘Foundations of Addition (Mathematics as a language)’ by Sandeep Srivastava and Saloni Srivastava