Question

Write the complex number in standard form and find its complex conjugate. $$(\sqrt{-6})^{2}+3$$

Short answer

The standard form of the given expression is \(-3 + 0i\). The complex conjugate of this number is \(-3\).

Step-by-step solution

Simplify the expression

First, simplify \((\sqrt{-6})^{2}+3\). The square root of \(-6\) is \(0 + \sqrt{6}i\). So \((\sqrt{-6})^{2}+3\) simplifies to \((0 + \sqrt{6}i)^{2}+3 = -6+3 = -3\).

Write it in standard form

The number \(-3\) in the standard form of a complex number is \(-3 + 0i\). This is because a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Here, the real part is -3 and the imaginary part is 0.

Find the complex conjugate

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). For the number \(-3 + 0i\), its complex conjugate is \(-3 - 0i = -3\).

Key concepts

Standard Form of a Complex Number

Complex numbers are a fundamental part of mathematics and engineering. They consist of two parts: a real part and an imaginary part. The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part with \(i\) being the imaginary unit, satisfying \(i^2 = -1\). Understanding the standard form is crucial as it is the foundation for addition, subtraction, and multiplication of complex numbers.

Take the complex number from the exercise, \(-3 + 0i\), which is already in standard form. Here, \(-3\) represents the real part and \(0i\) represents the imaginary part, indicating that the number is purely real without any imaginary component.

Complex Conjugate

The complex conjugate of a complex number is a reflection across the real axis on the complex plane. In essence, if you have a complex number \(a + bi\), its complex conjugate would be \(a - bi\). The conjugate is extremely useful when we need to eliminate imaginary parts from a denominator of a fraction or when finding the magnitude of a complex number.

According to the solution provided, the complex conjugate of \(-3 + 0i\) is \(-3 - 0i\), which simplifies to just \(-3\), as the imaginary part is zero. This tells us that the conjugate of a purely real number is the number itself, as there is no imaginary component to negate.

Imaginary Numbers

Imaginary numbers are numbers that when squared give a negative result, which is not possible in the set of real numbers. The unit imaginary number is \(i\), defined by the property that \(i^2 = -1\). Any multiple of the imaginary unit \(i\) is an imaginary number. For example, \(\sqrt{6}i\) is an imaginary number. It is part of the complex number system and can be added to a real number to form a complex number.

Imaginary numbers are essential when dealing with square roots of negative numbers, as shown in the exercise where \((\textbackslash sqrt\textbraceleft -6\textbraceright)^{2}\) has to be interpreted as \(6i^2\), which evaluates to \(-6\), integrating with real numbers to form complex ones.

Simplifying Complex Numbers

Simplifying complex numbers often involves operations such as addition, subtraction, multiplication, and finding powers. The key to simplification is understanding the property of the imaginary unit \(i\) and performing algebraic manipulations according to standard algebraic rules.

In the given exercise, we simplify by squaring \(\sqrt{-6}\), which equals \(-6\) when considering the properties of \(i\). Thus, we obtain the simplified form -3 by adding 3, demonstrating the simplification of a complex expression into a real number, in this case. It's significant to note here that simplification sometimes results in a complex number and other times, as with this exercise, may result in a real number.