Definition:Real Number/Operations on Real Numbers
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Definition
Let $\R$ be the set of real numbers.
We interpret the following symbols:
| \((\text R 1)\) | $:$ | Negative | \(\ds \forall a \in \R:\) | \(\ds \exists ! \paren {-a} \in \R: a + \paren {-a} = 0 \) | |||||
| \((\text R 2)\) | $:$ | Minus | \(\ds \forall a, b \in \R:\) | \(\ds a - b = a + \paren {-b} \) | |||||
| \((\text R 3)\) | $:$ | Reciprocal | \(\ds \forall a \in \R \setminus \set 0:\) | \(\ds \exists ! a^{-1} \in \R: a \times \paren {a^{-1} } = 1 = \paren {a^{-1} } \times a \) | it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$ | ||||
| \((\text R 4)\) | $:$ | Divided by | \(\ds \forall a \in \R \setminus \set 0:\) | \(\ds a \div b = \dfrac a b = a / b = a \times \paren {b^{-1} } \) | it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$ |
The validity of all these operations is justified by Real Numbers form Field.
Sources
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers