COMPLEX NUMBERS a) Express the complex numbers \(1+2 j .-1+2 j,-1-2 j, 1-2 j\) in polar form. b) Simplify the following expressions, leaving them in Cartesian coordinates: $$ \begin{array}{ll} (1+2 j)(1-2 j) & (1+2 j)^{2} \\ 1 /(1+2 j)^{2}, & (1+2 j) /(1-2 j) \end{array} $$

The concept of Cartesian coordinates is a cornerstone of analytical geometry, which allows us to describe the position of points in a two-dimensional plane using two numbers, usually referred to as the x-coordinate (horizontal) and the y-coordinate (vertical). When dealing with complex numbers, these coordinates serve a similar purpose with a slight twist: the x-coordinate represents the real part of the complex number, denoted as 'a', and the y-coordinate represents the imaginary part, denoted as 'b' using the imaginary unit 'j' (or 'i' in some texts).

So, the expression of a complex number like 1+2j directly corresponds to a point (1,2) in the Cartesian plane, where 1 is the real part and 2 is the imaginary part. This representation is invaluable because it allows us to visualize complex numbers just as we would any point or vector in a two-dimensional space, making abstract problems more concrete and easier to conceptualize.

When we interact with complex numbers, we often want to determine their magnitude (also known as modulus or absolute value) and angle (also known as argument or phase). The magnitude of a complex number tells us how 'far' the number is from the origin, while the angle tells us the 'direction' from the positive real axis to the line connecting the origin with the complex number.

To calculate the magnitude, we use the formula \( r = \sqrt{a^2 + b^2} \) where 'a' is the real part and 'b' is the imaginary part of the complex number. For example, the magnitude of 1+2j is \( \sqrt{1^2 + 2^2} = \sqrt{5} \).

The angle, denoted by \( \theta \), can be found using the arctangent function of the ratio \( \frac{b}{a} \). This calculation must consider the signs of 'a' and 'b' to determine the correct quadrant in which the complex number lies. To express this more contextually, for the complex number 1+2j, the angle would be calculated using \( \theta = \arctan(\frac{2}{1}) \) ensuring we identify that the point lies in the first quadrant.

Algebraic identities are the tools that help us simplify complex expressions and solve equations more efficiently. They are like shortcuts in mathematics, giving us standard forms to transform the expressions without having to expand each term manually every time.

Two key identities often used with complex numbers are the difference of squares \( (a+b)(a-b) = a^2 - b^2 \) and the binomial expansion \( (a+b)^2 = a^2 + 2ab + b^2 \). These become particularly handy when simplifying complex expressions in Cartesian coordinates. For instance, when multiplying a complex number by its conjugate, as in \( (1+2j)(1-2j) \) we apply the difference of squares identity. The result of this will have a real expression \( a^2 - b^2 \) and since \( b \) is the imaginary part, \( b^2 \) will be negative which simplifies to a positive real number.

Understanding and using these identities correctly allows us to carry out complex calculations with ease, providing a clear path through potentially dense algebraic thicket, translating to better mastery of complex number manipulations.