All linear equations can be expressed as $ax+b =0$, where $a,b \in \mathbb{R}$ and $a \neq0$.
All linear equations can be solved with the solution being $x = -\frac{b}{a}$. This means for real coefficients, all solutions are also real : $x \in \mathbb{R}$.
This ceases to be the case once we go to higher powers. Take, for example, the quadratic equation
$$ a x^2 + bx + c = 0 \;. $$
This can be solved once again in a closed form
$$ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $$
However, this solution is real if and only if the term inside the square root (discriminant) is non-negative.
$$ x \in \mathbb{R} \quad, \; iff \qquad b^2 - 4ac \geq 0. $$
Specifically, consider the case when $a=1,b=0, c=1$, i.e. $x^2+1=0.$ Clearly this cannot have a real solution since the equation itself states $x^2 = -1 <0$.
We can say that these equations do not have a solution, so we do not need to worry about them. On the other hand, recalling that numbers, after all, are abstract entities, one can wonder if we can meaningfully enlarge the set of real numbers to include numbers such that their square can also be negative? Of course, the answer is yes, and we call such numbers complex numbers.
<aside> 🛠 1.1 Suppose you are given the general quadratic equation $a x^2 + bx + c = 0$. Define a new variable $y^2=\frac{(2ax+b)^2}{4ac-b^2}$ (assuming the discriminant $b^2 -4ac \neq 0$) and show that $y$ satisfies the quadratic equation $y^2 +1=0$.
For the case when discriminant vanishes, show that the correct variable change is given by $y^2 = - \left( \frac{2ax}{b} \right)^2$.
This shows that any quadratic equation can be cast into the form $x^2+1=0$.
</aside>
<aside> 🛠 1.2 We argued the equation $x^2+1=0$ has no real solution. We also argued any quadratic equation could be cast as an equation of the form $x^2+1=0$. So no quadratic equation should have a real solution! On the other hand, we know that infinitely many quadratic equations have real solutions! Resolve this apparent conflict and show that the statement proved in the above exercise is consistent with what we know about quadratic equations with real solutions.
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Definition:
A complex number is an ordered ****pair of real numbers that satisfies a specific set of properties. We will list these properties below.
Complex numbers can be expressed in the form $z=a+ib$, where $a, b \in \mathbb {R}$ and $i \stackrel{def}{=} \sqrt {-1}$ , also called the imaginary unit. This is called the algebraic form of complex numbers.
Complex numbers can be represented graphically on the complex plane. The real part of the complex number $z=a+ib$, denoted by $Re(z)$, is defined to be $a$. This serves as the x-coordinate on the plane. Similarly, the imaginary part, denoted by $Im(z)$, is defined as $b$ , and it serves as the y-coordinate on the plane.
For example: $z=2+3i$, then $Re(z)=2, \;Im(z)=3$.
<aside> ⚠️ Note that the imaginary part of a complex number is defined without the factor of $i$, so it is actually a real number.
</aside>
Complex plane
Examples of complex numbers as points on the complex plane
Even though we can represent complex numbers as points on a plane, we will see later that complex numbers have many more features than geometric points on a plane. In many ways, the algebraic form of complex numbers makes many of their properties seemingly obvious based on our knowledge of the properties of real numbers. But it is good to keep in mind that the algebraic form is just a notation (we will, in fact, study another way of representing complex numbers). The true definition of the complex numbers is given in a set of axioms or properties that we will summarize in a later section below. For now, we explore the various properties that will intuitively follow from the algebraic form.