Inequality

[JoeFerri]

lessThan()
greaterThan()
lessThanOrEqualTo()
greaterThanOrEqualTo()

The notation a < b means that a is less than b.
The notation a > b means that a is greater than b.
The notation a ≤ b or a ⩽ b means that a is less than or equal to b (or, equivalently, at most b, or not greater than b).
The notation a ≥ b or a ⩾ b means that a is greater than or equal to b (or, equivalently, at least b, or not less than b).

Complex:

Because ≤ is a total order, for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ −a). In either case 0 ≤ a2; this means that i2 > 0 and 12 > 0; so −1 > 0 and 1 > 0, which means (−1 + 1) > 0; contradiction.

However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b, then a + c ≤ b + c"). Sometimes the lexicographical order definition is used:

• a ≤ b, if
• • Re(a) < Re(b), or
• • Re(a) = Re(b) and Im(a) ≤ Im(b)

It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.

see Complex numbers and inequalities