Question

Write the complex number in standard form and find its complex conjugate. $$(\sqrt{-4})^{2}-5$$

Short answer

The complex number in standard form is -9, and its complex conjugate is also -9.

Step-by-step solution

Simplify the Expressions in Square Roots

First, we need to simplify the expression under the square root. Here, we are given \(\sqrt{-4}\), where -4 is a negative number. Remember that the square root of a negative number is defined in terms of \(i\) where \(i\) is the imaginary unit with the property that \(i^{2} = -1\). Therefore, \(\sqrt{-4} = 2i\). Thus, our expression becomes \((2i)^{2} - 5\).

Simplify the Entire Expression

Next, we need to calculate \((2i)^{2} - 5\). Recall that \((2i)^{2}\) gives \(4i^{2}\), and since \(i^{2}\) is equal to -1, the result is \(4*(-1) = -4\). Thus, the simplified expression becomes \(-4 - 5\). Simplifying this further gives the result -9.

Find the Complex Conjugate

The complex conjugate of a complex number is found by changing the sign of its imaginary part. In this case however, our number is -9, which is a real number. In the complex plane, the real numbers are situated on the real axis where the imaginary part is zero. The conjugate of 0 is simply 0, hence the complex conjugate of -9 is -9 itself as there is no imaginary part to switch the sign of.

Key concepts

Complex Conjugate

A complex conjugate is a special kind of pair for every complex number. This is especially helpful when you want to do operations like division involving complex numbers.
The complex conjugate mirrors the original complex number across the real axis of the complex plane.To find the conjugate of a complex number, you simply change the sign of the imaginary part. For example, if you have a complex number expressed as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, then the conjugate would be \( a - bi \).

• It provides a way to neutralize the imaginary component by computing \( (a + bi)(a - bi) = a^2 + b^2 \), eliminating the imaginary unit \( i \).
• In purely real numbers, such as \(-9\), the conjugate is the same number since there is no imaginary part.

Getting familiar with the concept of conjugates can aid in simplifying complex expressions and enhance your understanding of complex numbers.

Imaginary Unit

The imaginary unit is denoted by \( i \) and is essential for understanding more about complex numbers. It opens a new dimension in mathematics, particularly when dealing with numbers involved in the square root of a negative number.The imaginary unit \( i \) is defined such that \( i^2 = -1 \). This is an extraordinary concept since traditionally, the square of any real number is non-negative.
This definition of \( i \) parallels the realization that every number previously considered 'impossible' is made possible.

• Any negative square root can be expressed with the help of \( i \). For example, \( \sqrt{-4} \) can be rewritten as \( \sqrt{4} \times \sqrt{-1} = 2i \).
• Understanding this concept helps you express and compute expressions involving negative square roots effectively.

The imaginary unit is a foundation for complex numbers and is crucial for deeper mathematical studies.

Real and Imaginary Parts

A complex number consists of two parts: the real part and the imaginary part. Understanding these parts is essential for working with complex numbers effectively.Complex numbers are typically expressed in the form \( a + bi \), where:

• \( a \) is the real part, which is the component without the imaginary unit \( i \).
• \( bi \) is the imaginary part, and \( b \) is the coefficient of the imaginary unit \( i \).

In the context of your exercise, the number \(-9\) is actually a complex number where the real part is \(-9\) and the imaginary part is \(0i\).

• Real numbers can be viewed as complex numbers where the imaginary part is zero.
• This allows us to utilize techniques like the complex conjugate even on real-number expressions.

Grasping this fundamental layout of complex numbers allows you to seamlessly navigate their real and imaginary components.