Solving the Complex Equation $z^2 + |z| = 0$

We are asked to find the number of distinct solutions for the equation $z^2 + |z| = 0$, where $z$ is a complex number.

Representing the Complex Number $z$

Let the complex number $z$ be represented in the rectangular form:

$\quad z = x + iy$

where $x$ and $y$ are real numbers. The modulus of $z$, denoted by $|z|$, is given by:

$\quad |z| = \sqrt{x^2 + y^2}$

Setting up Real and Imaginary Part Equations

Substitute $z = x + iy$ and $|z| = \sqrt{x^2 + y^2}$ into the given equation:

$\quad (x + iy)^2 + \sqrt{x^2 + y^2} = 0$

Expand the term $(x + iy)^2$:

$\quad (x + iy)^2 = x^2 + 2ixy + (iy)^2 = x^2 + 2ixy - y^2 = (x^2 - y^2) + i(2xy)$

So, the equation becomes:

$\quad (x^2 - y^2) + i(2xy) + \sqrt{x^2 + y^2} = 0$

For a complex number to be equal to zero, both its real part and its imaginary part must be zero. The real part of the equation is the sum of the real terms, and the imaginary part is the coefficient of $i$.

Real part: $\quad x^2 - y^2 + \sqrt{x^2 + y^2} = 0 \quad$ (Equation 1)

Imaginary part: $\quad 2xy = 0 \quad$ (Equation 2)

Solving the Imaginary Part Equation: Case Analysis

Equation 2, $2xy = 0$, implies that either $x = 0$ or $y = 0$ (or both). We will consider these two cases separately.

Case 1: $x = 0$

Substitute $x = 0$ into Equation 1:

$\quad 0^2 - y^2 + \sqrt{0^2 + y^2} = 0$

$\quad -y^2 + \sqrt{y^2} = 0$

Since $\sqrt{y^2} = |y|$, the equation is:

$\quad -y^2 + |y| = 0$

$\quad |y| = y^2$

We need to solve $|y| = y^2$ for $y$.

• If $y \ge 0$, then $|y| = y$. The equation becomes $y = y^2$, which is $y^2 - y = 0$. Factoring gives $y(y - 1) = 0$. The solutions are $y = 0$ or $y = 1$. Both satisfy $y \ge 0$.
• If $y < 0$, then $|y| = -y$. The equation becomes $-y = y^2$, which is $y^2 + y = 0$. Factoring gives $y(y + 1) = 0$. The solutions are $y = 0$ or $y = -1$. Only $y = -1$ satisfies $y < 0$. (The solution $y=0$ was covered in the $y \ge 0$ case).

So, for $x = 0$, the possible values for $y$ are $0, 1, -1$. This gives the following solutions for $z = x + iy$:

• If $x=0, y=0$, then $z = 0 + i(0) = 0$.
• If $x=0, y=1$, then $z = 0 + i(1) = i$.
• If $x=0, y=-1$, then $z = 0 + i(-1) = -i$.

Case 2: $y = 0$

Substitute $y = 0$ into Equation 1:

$\quad x^2 - 0^2 + \sqrt{x^2 + 0^2} = 0$

$\quad x^2 + \sqrt{x^2} = 0$

Since $\sqrt{x^2} = |x|$, the equation is:

$\quad x^2 + |x| = 0$

We need to solve $x^2 + |x| = 0$ for $x$.

• If $x \ge 0$, then $|x| = x$. The equation becomes $x^2 + x = 0$, which is $x(x + 1) = 0$. The solutions are $x = 0$ or $x = -1$. Only $x = 0$ satisfies $x \ge 0$.
• If $x < 0$, then $|x| = -x$. The equation becomes $x^2 - x = 0$, which is $x(x - 1) = 0$. The solutions are $x = 0$ or $x = 1$. Neither satisfies $x < 0$.

So, for $y = 0$, the only possible value for $x$ is $0$. This gives the following solution for $z = x + iy$:

• If $x=0, y=0$, then $z = 0 + i(0) = 0$.

This solution $z=0$ is the same as one of the solutions found in Case 1.

Listing Distinct Solutions

Combining the distinct solutions found from both cases, we have:

• $z = 0$
• $z = i$
• $z = -i$

These are the three distinct complex numbers that satisfy the equation $z^2 + |z| = 0$.

Verification of Solutions for $z^2 + |z| = 0$

• For $z = 0$: $0^2 + |0| = 0 + 0 = 0$. The equation is satisfied.
• For $z = i$: $i^2 + |i| = -1 + \sqrt{0^2 + 1^2} = -1 + \sqrt{1} = -1 + 1 = 0$. The equation is satisfied.
• For $z = -i$: $(-i)^2 + |-i| = (-1)^2 i^2 + \sqrt{0^2 + (-1)^2} = (1)(-1) + \sqrt{1} = -1 + 1 = 0$. The equation is satisfied.

All three solutions satisfy the given equation.

Final Count of Distinct Solutions

Based on our analysis, there are three distinct complex solutions to the equation $z^2 + |z| = 0$.

Revision Table: Solving Complex Equations

Additional Information: Modulus of a Complex Number

The modulus of a complex number $z = x + iy$, denoted as $|z|$, represents the distance of the point $(x, y)$ from the origin $(0, 0)$ in the complex plane. It is calculated as $|z| = \sqrt{x^2 + y^2}$. The modulus is always a non-negative real number.

Properties of the modulus include:

• $|z| \ge 0$ for all $z \in \mathbb{C}$.
• $|z| = 0$ if and only if $z = 0$.
• $|z_1 z_2| = |z_1| |z_2|$.
• $|z_1 / z_2| = |z_1| / |z_2|$ (where $z_2 \ne 0$).
• $|z^n| = |z|^n$ for any integer $n$.
• $|z_1 + z_2| \le |z_1| + |z_2|$ (Triangle inequality).
• $|z_1 - z_2| \ge ||z_1| - |z_2||$.

In the equation $z^2 + |z| = 0$, we see a mix of algebraic terms ($z^2$) and terms involving the modulus ($|z|$), which is a real number derived from $z$. Solving such equations often involves converting the complex equation into a system of real equations by considering the real and imaginary parts, as demonstrated in the solution above.