Section 8.1 Named properties and special subsets of real numbers
Subsection 8.1.1 Special subsets of the real numbers
For our purposes, the real numbers consists of all possible decimals. These include whole numbers, both positive and negative, fractions, simple decimals like \(1.25\text{,}\) repeating decimals like \(1.333\ldots=\frac43\) and irrational numbers like \(\sqrt2\) and \(\pi\text{.}\)
We have some special notation that is used with the real numbers:
\(\N\text{,}\) the natural numbers, is the set of all nonnegative integers: \(\{0, 1, 2, 3,\ldots\}\text{.}\)
\(\Z\text{,}\) the integers, consists of \(\{0, \pm1, \pm2, \pm3, \ldots\}\text{.}\)
\(\Q\text{,}\) the rationals, is the set of all fractions: \(\frac mn\) where \(m\) and \(n\) are integers with \(n\not=0\text{.}\)
\(\R\) is the set of all real numbers, that is, ordinary decimals.
Subsection 8.1.2 Named properties of real numbers
| Additive properties | Multiplicative properties | |
| Closure: | If \(x\) and \(y\) are real then so is \(x+y\) | If \(x\) and \(y\) are real then so is \(xy\) |
| Associativity: | \(x+(y+z)=(x+y)+z\) | \((xy)z=x(yz)\) |
| Identity: | \(x+0=x\) | \(1x=x\) |
| Inverse: | For any \(x\) there is a \(-x\) so that \(x+(-x)=0\) | For any \(x\not=0\) there is a number \(x^{-1}\) so that \(xx^{-1}=1\) |
| Commutativity: | \(x+y=y+x\) | \(xy=yx\) |
| \(\strut\) | ||
| Distributive: | \(x(y+z)=xy+xz\) and \((x+y)z=xz+yz\) | |