Question

Add or subtract. Write the answer in standard form. \((16-9 i)-(2-9 i)\)

Short answer

The answer is 14

Step-by-step solution

Separate real and imaginary parts

Separate the real and the imaginary parts: \(16 - 2\) and \(-9i - (-9i)\)

Calculate real part

Subtract the real parts: \(16 - 2 = 14\)

Calculate imaginary part

Subtract the imaginary parts: \(-9i - -9i = -9i + 9i = 0i\)

Write in Standard Form

The result of the subtraction, in standard form, is \(14 + 0i\). However, if the imaginary part is zero, it's customary to write it as just the real part

Key concepts

Standard Form

In the context of complex numbers, the standard form refers to expressing a complex number as a combination of a real part and an imaginary part, represented as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, and \((i\)) is the unit imaginary number, the square root of \-1. The standard form is helpful as it provides a uniform way to write complex numbers for easy addition, subtraction, multiplication, and division.

In the given exercise, the subtraction of two complex numbers is simplified to \(14 + 0i\), and while this is correct, conventionally we drop the zero imaginary part and simply write \(14\), focusing on the real part alone when the imaginary part is equivalent to zero.

Imaginary Numbers

Imaginary numbers are built on the foundation of the imaginary unit \(i\), which is defined by the property that \((i^2 = -1\)). These numbers are not 'imaginary' in the sense of being unreal; they are a distinct type of number used to extend the real number system. Imaginary numbers allow the solution of certain equations that have no solutions in the realm of real numbers, such as \((x^2 + 1 = 0\)).

In practical applications, imaginary numbers are used in advanced mathematics, engineering, and physics. In the subtraction exercise, the term \(9i\) represents the imaginary parts from each complex number. Here, when the same imaginary parts are subtracted, they cancel each other out, leading to a result with no imaginary component.

Algebraic Operations

Algebraic operations are the foundation of manipulating expressions and equations in mathematics. These include addition, subtraction, multiplication, division, and exponentiation among others. When working with complex numbers, these operations follow specific rules, especially due to the presence of the imaginary unit \(i\).

It is crucial to treat the real and imaginary parts separately during these operations. In the presented exercise, subtraction is performed by separately subtracting the real parts \(16-2\) and the imaginary parts \(9i - 9i\), illustrating the concept of corresponding parts in algebraic operations. This step-by-step approach ensures that each part is handled correctly according to algebraic rules.

Real Numbers

Real numbers include the set of all rational and irrational numbers; they can be found on the number line and do not involve the imaginary unit \(i\). In simpler terms, they include all the numbers we use for counting, measuring, and labeling, and they do not have an \(i\) component.

During operations involving complex numbers, like in the subtraction exercise, the real numbers operate independently of the imaginary numbers. The solution process involves handling the real parts distinctly and combining them according to the rules of arithmetic. In the subtraction \(16 - 2\), we are solely dealing with real numbers which simplifies down to \(14\), showcasing that even within the realm of complex numbers, the real components maintain their conventional arithmetic properties.