Field example
\(\mathbb{R}\): set of real numbers.
\(0, 1 \, \in \, \mathbb{R}\): two points on a number line.
\(x\): any point on the number line, distance from \(0\) relative to \(1\).
\(x_1 + x_2\): addition.
\(x_1 \times x_2\): multiplication.
\(x_1 \times (x_2 + x_3) = (x_1 \times x_2) + (x_1 \times x_3)\).
\(1\): identity.
\(x_1 + x_2 = x_2 + x_1\): order doesn’t matter.
\(x_1 \times x_2 = x_2 \times x_1\): order doesn’t matter.
\(-x\): inverse of \(x\) for \(+\).
\(x^{-1}\): inverse of \(x\) for \(\times\).
\(2 \times \frac{1}{2}\): inverses.
Ring example
\(\mathbb{Z}\): set of integers.
\(\mathbb{Z} \subset \mathbb{R}\): integers are a subset of reals.
\(\frac{1}{2}\): not an integer.
\(2\): has no inverse for \(\times\) in \(\mathbb{Z}\).
Field
\(F(+, \times)\): field.
\(F\): a set.
\(+, \times\): binary operations.
\(F(+)\): commutative group, identity \(0\).
\((F \setminus \{ 0 \})(\times)\): commutative group with identity \(1 \neq 0\).
\(f_1 \times (f_2 + f_3) = (f_1 \times f_2) + (f_1 \times f_3)\).
\((f_1 + f_2) \times f_3 = (f_1 \times f_3) + (f_2 \times f_3)\).
Ring
Same as a field except…
\(\times\): might not be commutative, might not have inverses.
Zero divisors
\(r_1, r_2\): rings can have elements where \(r_1 \times r_2 = 0\) and neither is \(0\).
##∑ Other examples
\(\mathbb{Q}\): set of rationals \(\{\frac{a}{b} \mid a, b \in \mathbb{Z} \}\).
\(\mathbb{C}\): set of complex numbers \(\{a + bi \mid a, b \in \mathbb{R} \}\).
\(\mathbb{Z}_7\): integers modulo 7.
\(\mathbb{R} \times \mathbb{R}\): Cartesian plane.