The set ℕ
of natural, 0,1,2,...
(non-negative integers). The sum and product of two natural numbers are natural numbers.
The set ℤ
of all integers, 0,±1,±2....
This closes under addition, subtraction and multiplication.
The set ℚ
of all rational numbers (fractions) p∕q
like 3∕5,
closes under division as well.
To this we add the irrational numbers, like \sqrt{2},{3}^{1∕3}
And obtain ℝ
the set of real numbers includes 1,2.34,π,4∕5,e = {e}^{1}
etc
There is a lot of subtle mathematics associated with them. Are there more rational numbers than integers? More reals
than rationals? We can also try to solve equations. In physics we usually mean “find a real number that solves the
equation”.
Example 1.1:
Find the zero of the function (polynomial) p(x) = {x}^{2} − 1.
The square root of − 1
is not defined within the real numbers. There is no real zero of
q(x) (look at the
curve q(x) = {x}^{2} + 1).
We define new numbers (complex numbers) so that we can solve any equation of the kind of (1.2).
Complex numbers are defined as z = x + iy
with real x
and real y
and i := +\sqrt{−1}.
The symbol i
is called ‘complex unit’ with the property
In z = x + iy,
x = \text{Re}(z) is called real part and
y = \text{Im}(z) is called imaginary part of the
complex number z. Complex numbers
are elements of a set called ℂ.
So far this may seem a bit artificial. However, we now can solve arbitrary quadratic equations, i.e., find solutions in
ℂ:
In fact, within the complex numbers one can always find the root of a quadratic
equation (and in fact all the roots of an arbitrary polynomial, i.e. solve equations like
{z}^{12} + 4{z}^{4} + 17 = 0 etc.,
although for this last example there is no general formula as in the quadratic case.)
Actually, complex numbers first arose in the 15th century in the solution of cubic equations of the form
{z}^{3} + bz + c = 0. The
general solution of such equations are