Problem 70 - Precalculus

Question

Determine whether the statement is true or false. There exists a complex number for which there is no complex square root.

Step-by-Step Solution

Answer

False, every complex number has a complex square root.

Step 1: Understanding the Properties of Complex Numbers

Complex numbers are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, satisfying \(i^2 = -1\). Any complex number can be expressed in this form.

Step 2: What is a Complex Square Root?

A complex square root of a complex number \(z\) is another complex number \(w\) such that \(w^2 = z\). This means we need to find \(w = a + bi\) satisfying \((a + bi)^2 = z\).

Step 3: Existence of Square Roots for Complex Numbers

For any complex number \(z = a + bi\), we can generally find \(w = u + vi\) such that \((u + vi)^2 = z\). Expanding, we have \((u + vi)^2 = u^2 - v^2 + 2uvi\). By comparing parts, for any \(z\), we can find suitable \(u\) and \(v\) to satisfy this equation.

Step 4: Proving the Existence for All Complex Numbers

It is a known mathematical result that every non-zero complex number has two complex square roots. Even zero, a special case, has a square root (since 0 squared is zero). Therefore, there cannot be a complex number without a square root.

Key Concepts

Complex Square Root

A complex square root refers to finding another complex number that, when squared, gives the original complex number. Suppose we have a complex number say, \(z = a + bi\). We need to find \(w = u + vi\) such that \((u + vi)^2 = z\). This results in solving a system of equations:

• \(u^2 - v^2 = a\)
• \(2uv = b\)

By solving these equations, we can determine the square roots of \(z\).
The formula for the square root of a complex number involves calculations that aid in finding \(u\) and \(v\).
This includes the modulus and argument of the complex number.
Moreover, any non-zero complex number will have two square roots. Each root is equally valid and is generally a reflection of one another in the complex plane.

Imaginary Unit

The imaginary unit is a fundamental concept in complex numbers represented by \(i\). It is defined by the equation \(i^2 = -1\). This might seem unconventional since the square of any real number is never negative.
With the introduction of \(i\), mathematicians extended the real number system to the complex number system, allowing for solutions to equations like \(x^2 + 1 = 0\). Here, \(x = i\) and \(x = -i\) are solutions.

The imaginary unit acts as a building block for all complex numbers, facilitating combinations with real numbers to form expressions such as \(a + bi\). It enables more comprehensive analysis and problem-solving beyond real number limitations.

Complex Arithmetic

Complex arithmetic involves operations such as addition, subtraction, multiplication, and division on complex numbers. Make sure to remember that complex numbers are generally written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

• Addition and Subtraction: Combine like terms, real with real and imaginary with imaginary. For instance, \((a + bi) + (c + di) = (a + c) + (b + d)i\).
• Multiplication: Use distributive properties and simplify \(i^2 = -1\). For example, \((a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i\).
• Division: Multiply the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary component in the denominator.

Complex arithmetic may first appear daunting but relies on basic algebraic rules.
Mastering these operations is crucial for deeper mathematical exploration with complex numbers.