2 Sets of Numbers
- Natural numbers, \(\mathbb {N}=\{0,1,2,3,4,\cdots \}\)
- Integers, \(\mathbb {Z}=\{0,\pm 1, \pm 2, \pm 3,\cdots \}\)
- Positive integers, \(\mathbb {Z}^+=\{1, 2, 3, \cdots \}\)
- \(\mathbb {Q}\), the set of rational numbers
- \(\mathbb {R}\), the set of real numbers
- \(\mathbb {C}\), the set of complex numbers
- Prime numbers, \(\{2, 3, 5, 7, 11, 13,\cdots \}\)
- Odd numbers, \(\{1, 3, 5, 7, 9, 11, 13, 15, \cdots \}\)
- Even numbers, \(\{0, 2, 4, 6, 8, 10, \cdots \}\)
These sit inside one another, \[\mathbb {N}\subset \mathbb {Z}\subset \mathbb {Q}\subset \mathbb {R}\subset \mathbb {C},\] and the chain is not a list to be memorised — it is a history. Each system was built because the one before it could not answer a question that had already been asked.
- In \(\mathbb {N}\) you can add and multiply, but \(x+3=1\) has no solution. Inventing negative numbers gives \(\mathbb {Z}\), where subtraction always works.
- In \(\mathbb {Z}\) you can subtract, but \(2x=1\) has no solution. Inventing fractions gives \(\mathbb {Q}\), where division by anything non-zero works.
- In \(\mathbb {Q}\) you can divide, but \(x^{2}=2\) has no solution — proved later in this chapter. Filling the gaps gives \(\mathbb {R}\).
- In \(\mathbb {R}\) you can take roots of positive numbers, but \(x^{2}=-1\) has no solution. Inventing \(i\) gives \(\mathbb {C}\), and there the process stops: every polynomial equation has a solution in \(\mathbb {C}\).
Each step keeps everything that worked before and adds one new capability. That is the pattern worth carrying through the whole course.
Note 2.1. Whether \(0\) belongs to \(\mathbb {N}\) is a convention, not a fact, and books differ. These notes include it. When a result depends on the distinction — and a few do — say which convention you are using rather than assuming the reader shares it. \(\mathbb {Z}^{+}\) is unambiguous when you mean \(1,2,3,\dots \)
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