This subject covers the foundational concepts of number and quantity, focusing on real numbers, complex numbers, and their applications in various mathematical contexts.
Rational numbers are numbers that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). Irrational numbers can't be written as such fractions; their decimal expansions go on forever without repeating.
Suppose \(\sqrt{2}\) is rational, so \(\sqrt{2} = \frac{a}{b}\). With some algebra, you can show this leads to a contradiction!
Engineers often approximate irrational numbers for calculations, knowing they can't represent them exactly.
\[x = a + bi\]
The number \(0.5\) is rational because it equals \(\frac{1}{2}\).
The number \(\pi\) is irrational because its decimal never ends or repeats.
Rational numbers can be written as fractions; irrational numbers cannot.