The complex numbers, denoted as C, form a field, which is a rich algebraic structure with two fundamental operations: addition and multiplication.
Understanding Complex Numbers
A complex number, as defined by the reference, is expressed in the form z = a + ib, where:
- a and b are real numbers.
- i is the imaginary unit, defined as the square root of -1 (i.e., i2 = -1).
- Re z = a represents the real part of the complex number.
- Im z = b represents the imaginary part of the complex number (note: b, not ib is the imaginary part).
Complex Numbers as a Field
To understand the algebraic structure, it's essential to know why complex numbers form a field. A field must satisfy several properties under addition and multiplication:
Addition (+)
- Closure: The sum of two complex numbers is also a complex number. If z1 = a + ib and z2 = c + id, then z1 + z2 = (a + c) + i(b + d), which is also in the form of a complex number.
- Associativity: (z1 + z2) + z3 = z1 + (z2 + z3)
- Commutativity: z1 + z2 = z2 + z1
- Additive Identity: There exists a complex number 0 (0 + 0i) such that z + 0 = z for all complex numbers z.
- Additive Inverse: For every complex number z = a + ib, there exists a complex number -z = -a - ib such that z + (-z) = 0.
Multiplication (×)
- Closure: The product of two complex numbers is also a complex number. If z1 = a + ib and z2 = c + id, then z1 * z2 = (ac - bd) + i(ad + bc), which is a complex number.
- Associativity: (z1 z2) z3 = z1 (z2 z3)
- Commutativity: z1 z2 = z2 z1
- Multiplicative Identity: There exists a complex number 1 (1 + 0i) such that z * 1 = z for all complex numbers z.
- Multiplicative Inverse: For every non-zero complex number z = a + ib, there exists a complex number z-1 such that z z-1 = 1. The inverse is calculated as z-1 = (a/(a2 + b2)) - i(b/(a2 + b2)). This requires that z is non-zero (i.e., a and b* are not both zero), otherwise division by zero would occur.
Distributive Property
- Multiplication distributes over addition: z1 (z2 + z3) = (z1 z2) + (z1 * z3)
Examples and Practical Insights
- Example of Addition: Let z1 = 2 + 3i and z2 = 1 - i. Then z1 + z2 = (2 + 1) + (3 - 1)i = 3 + 2i.
- Example of Multiplication: Using the same z1 and z2, z1 z2 = (2 1 - 3 (-1)) + i(2 (-1) + 3 * 1) = (2 + 3) + i(-2 + 3) = 5 + i.
- Finding the Multiplicative Inverse: Let z = 1 + i. Then z-1 = (1/(12 + 12)) - i(1/(12 + 12)) = (1/2) - (1/2)i. You can verify that (1 + i) * ((1/2) - (1/2)i) = 1.
Summary
The complex numbers, with their defined addition and multiplication, satisfy all the necessary axioms to be classified as a field in abstract algebra. This makes them an incredibly powerful and useful tool in mathematics, physics, and engineering.
