Chapter 27 Representations of numbers
There are some families of numbers which we often want to refer to (integers, even integers, square numbers, multiples of \(3\), etc.). Therefore it is useful to have notation for these families. Also, if we want to use these families in a proof using algebra, having algebraic representations of them is essential.
The table below lists the notation for these families, often in reference to an integer \(k\).
A few examples of members of each family are given, for illustrative purposes. Note that the symbol “…” means “continue this pattern till infinity.”64
| If you want to refer to all possible… | The general form is | The set is |
|---|---|---|
| integers | \(a \in \mathbb{Z}\) | \(\{..., –3, –2, –1, 0, 1, 2, 3, ...\}\) |
| positive integers | \(a \in \mathbb{Z}^+\) | \(\{1, 2, 3, ...\}\) |
| nonnegative integers | \(a \in \mathbb{Z}^*\) | \(\{ 0, 1, 2, 3, ...\}\) |
| negative integers | \(a \in \mathbb{Z}^–\) | \(\{..., –3, –2, –1, 0, 1, 2, 3, ...\}\) |
| even integers | \(a = 2k\), where \(k \in \mathbb{Z}\) |
\(\{..., –6, –4, –2, 0, 2, 4, 6, ...\}\) |
| odd integers | \(a = 2k + 1\), where \(k \in \mathbb{Z}\) |
\(\{..., –5, –3, –1, 1, 3, 5, 7, ...\}\) |
| square numbers | \(a = k^2\), where \(k \in \mathbb{Z}\) |
\(\{0, 1, 4, 9, ...\}\) |
| cube numbers | \(a = k^3\), where \(k \in \mathbb{Z}\) |
\(\{0, 1, 8, 27, ...\}\) |
| multiples of 3 | \(a = 3k\), where \(k \in \mathbb{Z}\) |
\(\{..., –9, –6, –3, 0, 3, 6, 9, ...\}\) |
| multiples of 4 | \(a = 4k\), where \(k \in \mathbb{Z}\) |
\(\{..., –12, –8, –4, 0, 4, 8, 12, ...\}\) |
| multiples of 5 | \(a = 5k\), where \(k \in \mathbb{Z}\) |
\(\{..., –15, –10, –5, 0, 5, 10, 15, ...\}\) |
| rational numbers | \(a = \frac{k}{l}\), where \(k,l \in \mathbb{Z}\) |
There’s no systematic order to write the rational numbers, but some examples are \(\frac{2}{3}, 3, –1, \frac{1}{4}, \frac{5}{6}, \frac{21}{7}, –\frac{101}{2}\) |
| two consecutive integers | \(a\), \(a + 1\), where \(a \in \mathbb{Z}\) |
|
| three consecutive integers | \(a\), \(a + 1\), \(a + 2\), where \(a \in \mathbb{Z}\), or \(a-1\), \(a\), \(a + 1\), where \(a \in \mathbb{Z}\) |
Sometimes the first form will work best, sometimes the second. If struggling with one, try the other! |
Can you think of any families of numbers I have forgotten? Let me know and I can add them to this list.
Unless an endpoint is given, like \(\{1, 2, 3, ..., 99, 100\}\) indicates the set containing the first hundred positive integers.↩︎