Solving inequalities with negative square roots

Question

Let's say you have an inequality: $$ a^2 > b \cdot c \tag{1} $$ Let's say $a = 4$, $b = 1$, and $c = -3$ we can compute the result $$ 4^2 > 1 \cdot (-3) \tag{2} $$ so $$ 16 > -3 \tag{3} $$

Which is true so these values are solutions to the equation at (1). Now lets say we took the square root of both sides $$ a > \sqrt{b \cdot c} \tag{4} $$

Trying to put in the values for $a$, $b$, and $c$ we stated above we get $$ 4 > \sqrt{-3} \tag{5} $$

So I'm not really sure where we go from here. The square root of the negative makes me think this is a complex number like seen below $$ 4 > \sqrt{3}i \tag{6} $$

But now we have an inequality between a real number and a complex number (which we know must be true from (3)). How do we know that this is a true statement, and in general if you have $x \in \mathbb{R}$ and $y \in \mathbb{C}$ how do you determine if $x > y$ is true, $x < y$ is true, and $x = y$ is true? Thanks.

Answer 1

There is no direct notion of "greater" and "lesser" in complex numbers in the same way as it exists in real numbers, where we can say that one real number is greater or lesser than another using the number line. This is because complex numbers cannot be easily represented on a single number line, as is the case with real numbers.

Complex numbers consist of two parts: the real part and the imaginary part. They are typically represented in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part, and $i$ is the imaginary unit, defined as $\sqrt{-1}$.

When comparing complex numbers, various properties such as magnitude and argument (angle) of the complex number are considered. The magnitude of a complex number $a + bi$ is given by $\sqrt{a^2 + b^2}$, and this provides a measure of its size in the complex plane. The argument of a complex number is the angle it forms with the positive real axis.

To compare two complex numbers in terms of magnitude, we can say that $z_1$ is "greater" than $z_2$ if the magnitude of $z_1$ is greater than the magnitude of $z_2$. However, this comparison is based solely on the size of the complex numbers and does not take into account their direction.

In summary, while you can compare complex numbers based on their magnitude, it is not as straightforward as comparing real numbers on a number line. The notion of "greater" and "lesser" in complex numbers is typically related to magnitude and argument, taking into consideration both size and direction in the complex plane.