A Brief Introduction of Number System

Published by Ganit Charcha | Category - Math Articles | 2014-09-19 13:17:44

We provide in this note a brief account of our number system. Idea is to introduce different types of numbers and to discuss a few relevant concepts associated with them. Before we start, let us recall what Leopold Kronecker once told - ``God made the integers, all the rest is the work of man''. We therefore start with the introduction of integers.

Let us start at the beginning - with what a kid is introduced with Mathematics. His or her journey starts with Natural Numbers - $1, 2, 3, 4, \ldots$ etc with which he or she learns to count. They are also called Counting Numbers and these numbers served mankind for a long time. The difficulty was how to distinguish between $4$ and $40$. For many years, an empty space was used to depict that there is no digit here and that formed the basis of ancient place-value system until the concept of $0$ as a number is discovered. Today's modern decimal-based place value notation is attributed by ancient Indian mathematician Aryabhata who stated that ``from place to place each is ten times the preceding''.
Therefore, with the introduction of number $0$, we have a new set of numbers $0, 1, 2, 3, 4, \ldots$  along which we can count in one direction. Counting in the opposite direction was meaningless and absurd in ancient days and in fact not known. The concept of ``debt'' or ``loss'' and the rules governing operations involving this ``debt'' with number $0$ is known from the work of ancient Indian mathematician Brahmagupta (A.D 628). So, we tend to learn the concept of negative numbers as anything which is less than 0 and this is how we arrive at the number line.
We therefore have a bigger set of numbers $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ and introduce them as integers. Subsequently, the integers on the right side of $0$ are called positive integers and those on the left side of it are negative integers. Note that number line introduced above is also referred as Integer Line. The operations of addition, substraction, multiplication and division are defined on these numbers as a process of combining a pair of these numbers, what we call today as elementary arithmetic.

Rational Numbers
The word fraction borrowed from the Latin ``fractio'' means break and in ancient days was used to mean a part of a whole thing which is divided into some number of equal parts. In other words, these are unit fractions meaning one part of two ($1/2$), one part of three ($1/3$) and so on. This concept then got extended to a numerator divided by a denominator, as we mean by fraction today as a natural generalization of multiplying a unit fraction with a counting number.
But the history of mathematics is all about asking relevant questions, and seeking the answers. The above extension naturally shows direction for a question, what if an unit fraction is multiplied by any integer. So, we get new form of numbers like $4/3, -15/6$ etc and we need a new name. We therefore, start to call these numbers as Rational Numbers and is defined as a fraction $p/q$ of two integers, where denominator $q$ is not $0$. Surprisingly, we discovered that it generalizes all numbers we have defined so far.

Since integers are generalized by rational numbers, natural question therefore is can integer line be extended to have rational numbers represented geometrically. Answer is yes and to demonstrate this we consider two intersecting integer lines $P$ and $Q$. Let us now consider an integer $q$ on the line $Q.$ We join the point $q$ on $Q$ and point $1$ on $P.$ We then draw a line through $1$ on $Q$ parallel to the joining line and the point of intersection of this parallel line with line $P$ represents the unit fraction $1/q.$ And it is easy to see that this representation is correct, since the distance of this point on $P$ from point $0$ on $P$ is one of $q$ parts of the distance between points $0$ and $1$ on $P$.
Same construction and argument holds true while representing the rational number $p/q,$ where $p$ and $q$ are both integers. Only exception is we need to start the construction by joining $q$ on $Q$ and $p$ on $P$. The number $p/q$ is given by the point of intersection of this parallel line and the line $P.$

Irrational Numbers
A number $a$ multiplied by itself, i.e $a*a = a^2$ is the square of the number and the number $a$ is said to be the square root of $a^2$ and is depicted as $\sqrt{a}$. Pythagorean mathematician Hippasus of Metapontum landed to the problem of finding the length of the hypotenuse of an isosceles right angled triangle whose other two sides are of unit length and it is all about asking what is the square root of 2. We assume that root $\sqrt{2} = p/q$ where $p$ and $q$ have no common factor and that is the most generalized assumption that we could have made given our knowledge of numbers so far. The assumption $p^2 = 2.q^2$ leads us to the fact that both $p$ and $q$ are even, and so have a common factor $2$. This contradicts our initial assumption, and in turn proves the existence of another kind of numbers which are not rational. Mathematicians begin to refer them as Irrational Numbers.

Real Numbers
The rational and irrational numbers together are called Real Numbers and these are the numbers whose values we can visualize in this real world. The set of integers can be thought of as points on a line starting from a point called 0 and stretched to either direction indefinitely. The positive integers are on the right side of 0, whereas negative integers are on its left side. One part of two, i.e, unit fraction $1/2$ is the point on the line which bisects the line segment joining the integers 0 and 1. Since $1 < 2 < 4,$ so $1 < \sqrt{2} < 2$ and therefore irrational $\sqrt{2}$ can also be represented as some point between the points representing the integers $1$ and $2$. And in fact each and every rational and irrational numbers are points on that line and hence we start to refer it as Real Line as shown in the figure below. This will play critical role in understanding different properties and mathematical structures that deal with real numbers.
Complex Numbers
Any number multiplied by itself will always yield a positive number, so the square root of the number introduced above is applicable for positive numbers only. Natural question is therefore, what is the square root of a negative number. It can not be answered from our knowledge of numbers so far, so came the definition of Imaginary Numbers. The square root of $-1$ is defined as the number $i$, assuming it exists, such that $i*i = -1$. Therefore, square root of $-2$ is defined as $\sqrt{2}*i$ and so on. Therefore the definition of $i$ as $\sqrt{-1}$ is consistent and helps us to define square root of all negative numbers and the numbers involving $i$, as for example, $2i$, $\sqrt{2}i$, $3i$, $2i/3$, are called imaginary numbers.
In 1545, Girolamo Cardano in his book Ars Magna solved the equation $x(10 -x) = 40$ as $5 + \sqrt{-15}$ and $5 - \sqrt{-15}$. Later in 1637, Rene Descartes proposed the standard form of numbers involving imaginary numbers as $a + bi$, where $a$ and $b$ are both real numbers. These numbers are referred today as complex numbers. But, both of them did not like the concept of complex numbers and thought it to be as useless. The symbol $i$, though we introduced earlier for the sake of clarity and readability, but actually was introduced much later by L. Euler in 1777. The fact that the algebraic identity $\sqrt{a}\sqrt{b} = \sqrt{ab}$ does hold only when $a$ and $b$ are positive and for negative $a$ and $b$ it lead to inconsistencies since $\sqrt{-1}.\sqrt{-1} = -1$. And this inconsistency led to the use of the symbol $i$ for $\sqrt{-1}$.
In 1806, Jean-Robert Argand in his work titled ``Essay on the Geometrical Interpretation of Imaginary Quantities'' showed how to visualize complex numbers as points in a plane where real numbers are plotted against X-axis and purely imaginary numbers are plotted against Y-axis, aka, imaginary-axis. And $a + bi$ is the point $(a, b)$ in the that plane which is referred today as Argand Plane or Argand Diagram. Carl Friedrich Gauss in 1831, made Argand's idea popular and well accepted and also took Descartes $a + bi$ notation to refer it as Complex Numbers. Complex numbers are the highest generalization of numbers as it exists today, since it includes all real, purely imaginary numbers and any combinations of them.


1. E. T. Bell, Men of Mathematics, New York: Simon and Schuster, 1986, ISBN 0-671-46400-0.
2. Craig Smorynski, History of Mathematics: A Supplement, ISBN 978-0-387-75481-9, Springer New york.
3. B. K. Lahiri and K. C. Roy, Real Analysis, World Press, 1988.
4. Orlando Merino, A Short History of Complex Numbers, http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf


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