The most basic form of mathematics is counting.
Most cultures count (although some don’t). Thus, the simplest, most basic set of numbers is the counting numbers (1, 2, 3, ….) also known as the Natural Numbers, symbolized by a capital N.
An issue arises in algebra if we create an equation using only natural numbers whose solution is not a natural number –
This leads to the creation of the Integers, symbolized by a capital Z (for the German word for number – zahlen) which includes all the positive and negative whole numbers along with zero – (…, -3, -2, -1, 0, 1, 2, 3, …).
Again, a problem arises if we use integers to write an equation whose solution is not an integer.
This leads to the creation of the Rational Numbers (symbolized by a capital Q for quotient), which is the set of all numbers that can be represented as the ratio of two integers. We can see a connection between the operations of multiplication and division and the set of Rational Numbers Q .
Again, we can create an equation using rational numbers whose solution is not a rational number.
This was actually a famous problem for the Pythagoreans, who believed that all numbers were rational and could be represented as the ratio of two whole numbers. However, the diagonal of a square whose sides are length 1 will have a diagonal whose length is the square root of 2. When the Pythagoreans realized that this number was not rational, it caused them great concern.
The existence of irrational numbers requires the existence of a set of numbers that includes both rational and irrational numbers – the Real Numbers. This allows us to include such irrational numbers as square and cube roots as well old favorites like pi and relative newcomer e.
I think that the best way to think of the Real Numbers is on the number line. Every Real Number corresponds to a point on the number line and every point on the number line corresponds to a real number. In fact, even the technical mathematical definition of Real Numbers can be visualized on the number line.
Also, we can see a connection between the processes of finding roots and exponentiation to the need to broaden the conception of numbers to the Real Numbers.
The process of finding roots leads not only to the realm of the irrational numbers, but also to that of the Complex Numbers – symbolized with a capital C.
We’ll look at the complex numbers in the next post…