Question

Write the conjugate of each complex number. $$-5-7 i$$

Short answer

\(-5+7i\)

Step-by-step solution

Understanding the Conjugate of a Complex Number

The conjugate of a complex number is found by changing the sign of the imaginary component. If the complex number is written as \(a + bi\), its conjugate is \(a - bi\).

Identifying the Real and Imaginary Parts

In the complex number \(-5-7i\), the real part is \(-5\) and the imaginary part is \(-7i\).

Writing the Conjugate

Change the sign of the imaginary part to find the conjugate. The conjugate of \(-5-7i\) is \(-5+7i\).

Key concepts

Complex Numbers

Complex numbers are a fundamental element in mathematics and engineering, forming the cornerstone of complex analysis. A complex number is composed of two parts: a real part and an imaginary part. It's generally represented as a + bi, where a is the real part, and bi is the imaginary part, with i representing the square root of -1.

For instance, in the complex number -5-7i, -5 is the real component, and -7i is the imaginary component. The operations involving complex numbers are based on rectilinear algebra and require special rules such as those for addition, subtraction, multiplication, and finding the conjugate, as is the case in our exercise.

Understanding complex numbers can seem daunting at first, but remembering that they simply extend our concept of what numbers can do — by allowing for the existence of a square root of negative one — can help in grasping their role in extending the number system and solving problems that would otherwise be impossible.

Imaginary Numbers

Imaginary numbers may sound like an abstract concept, but they are incredibly practical and have real-world applications in fields like electrical engineering and quantum mechanics. An imaginary number is defined as a number that gives a negative result when squared. It is always a real number multiplied by imaginary unit i, which is defined as the square root of -1.

The term 'imaginary' can be misleading; while they may not represent a quantity along the traditional number line of real numbers, they do exist in a sense that enables us to solve equations that have no real solutions. When you encounter an expression like -7i, it's not just a figment of mathematical imagination but instead represents a quantity that can be manipulated, combined with real numbers, and has effects on the real world. For example, the alternating current in electrical circuits can be represented using imaginary numbers.

Real Numbers

In contrast to imaginary numbers, real numbers are all the numbers on the infinite continuum that includes rational and irrational numbers, integers, and whole numbers. These are the numbers we encounter in everyday life, such as -5, 0, 0.75, and π (pi). A real number can be positive, negative, or zero, and it does not include the imaginary unit i.

The real number system is essential for measuring, counting, and describing the physical world. When we discuss the real part of a complex number, such as -5 in -5-7i, we're referring to a quantity that can be visually represented on the number line and utilized in calculations involving tangible measurements. The real numbers are characterized by their completeness; any point on the number line corresponds to a real number, making the system perfect for geometric representation.