(a) Write i in phase-modulus form. (b) Use this expression to find √(i), i.e., a complex number

step 1 In this problem, we need to find the complex number where we are given modulus and argument. So,

(a) Write i in phase-modulus form. (b) Use this expression...

(a) Write $\mathrm{i}$ in phase-modulus form. (b) Use this expression to find $\sqrt{\mathrm{i}}$, i.e., a complex number $z$ such that $z^2=\mathrm{i}$. Can you find a second square root? (c) Find explicit formulas for the three third roots and four fourth roots of $i$.

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Key Concepts

Polar Form Representation

The polar form of a complex number expresses it in terms of its modulus (the distance from the origin) and its argument (the angle it makes with the positive real axis). This form, commonly written as r·e^(i?), facilitates operations like multiplication, division, and extraction of roots.

Euler's Formula

Euler's formula, which states that e^(i?) = cos(?) + i·sin(?), provides a bridge between exponential and trigonometric representations of complex numbers. It is essential for rewriting complex numbers in polar form and for applying methods such as De Moivre's Theorem.

De Moivre's Theorem

De Moivre's Theorem is used to raise complex numbers in polar form to a power or to extract roots. It states that (r·e^(i?))^(1/n) = r^(1/n)·e^(i(? + 2?k)/n) for k = 0, 1, …, n-1, thereby providing all n distinct nth roots when solving equations involving complex numbers.

Periodicity of the Complex Exponential

The periodicity of the complex exponential function, where e^(i(? + 2?)) = e^(i?), is key to understanding why multiple solutions exist when extracting roots. This periodic behavior allows for an infinite number of representations of the same complex number, thus yielding distinct values for different choices of k in formulas for nth roots.

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