If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are integers, not all equal, and \(\mathrm{w}\) is a cube root of unity \((\mathrm{w} \neq 1)\) Then the minimum value of \(\left|\mathrm{a}+\mathrm{bw}+\mathrm{cw}^{2}\right|\) is (a) 0 (b) 1 (c) \((\sqrt{3} / 2)\) (d) \((1 / 2)\)

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are usually written in the form of a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, satisfying the equation i^2 = -1.

For example, in our cube roots of unity problem, w and w^2 are complex numbers derived from the exponential form using Euler's formula, which connects complex exponentials to trigonometric functions. This formula states that e^(iθ) = cos(θ) + i*sin(θ), thus allowing the representation of the complex roots of unity using trigonometric functions.

When working with complex numbers, especially for finding the absolute value or modulus, we often convert from rectangular form, a + bi, to polar form, r(cosθ + i*sinθ), where r is the absolute value and θ is the argument of the complex number.

Trigonometric functions are fundamental in mathematics, particularly when dealing with complex numbers and geometry. The two primary trigonometric functions are sine (sin) and cosine (cos), which relate an angle to the ratios of sides in a right-angled triangle.

In the context of complex numbers, these functions help to express complex roots of unity. For instance, the cube roots of unity can be depicted using cosine and sine as seen in the problem's solution: w = cos(2π/3) + i*sin(2π/3) and w^2 = cos(4π/3) + i*sin(4π/3). These values correspond to the vertices of an equilateral triangle inscribed in the unit circle of the complex plane, thus proving the geometric relationship between trigonometric values and the roots of unity.

Understanding trigonometric functions is key to comprehending the geometric interpretations of complex numbers and hence, is essential for solving problems involving complex roots of unity.

The absolute value, or modulus, of a complex number a + bi is its distance from the origin in the complex plane. It's denoted as |a + bi| and is equivalent to the square root of the sum of the squares of its real part a and its imaginary part b, mathematically expressed as |a + bi| = √(a^2 + b^2).

In the solution to our exercise, the focus is on finding the minimum value of the absolute value of an expression involving cube roots of unity. The expression's minimum absolute value represents the shortest distance from the origin to any point corresponding to a + bw + cw^2 on the complex plane. Since a, b, and c are integers and not all equal, we deduce that this distance can't be zero as it would imply all terms are equal, which does not satisfy the condition given in the problem. Therefore, the minimum absolute value must be positive, and as the solution step shows, this value is indeed 1.