Problem 18 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ 6 i-(4-i) $$

Step-by-Step Solution

Answer

The result is \(-4 + 7i\).

Step 1: Distribute the Negative Sign

The problem has a subtraction, and we need to distribute the negative sign through the parentheses. Replace \(-i\) in the expression \((4-i)\) with \(-4+i\), so the expression becomes:\[6i - 4 + i\]

Step 2: Combine Like Terms

In the expression \(6i - 4 + i\), we can combine like terms. The real numbers are \(-4\) and the imaginary numbers are \(6i\) and \(+i\). Adding the imaginary numbers gives \(7i\), resulting in the expression:\[-4 + 7i\]

Step 3: Write the Expression in the Form \(a+bi\)

The expression \(-4 + 7i\) is already in the form \(a+bi\), where \(a = -4\) and \(b = 7\). Hence, we don't need further modifications.

Key Concepts

Imaginary Numbers

Imaginary numbers are an important concept in mathematics, widely used in various fields like engineering and physics. So, what exactly is an imaginary number? To understand this, we begin with the square root of negative numbers. Typically, you cannot have a real number that satisfies equations like \[x^2 = -1\]. However, the invention of imaginary numbers solves this problem by introducing 'i', defined as:

• \[i = \sqrt{-1}\]

The imaginary unit 'i' enables us to work with the square roots of negative numbers, transforming them into something usable.
In many expressions, imaginary numbers appear alongside real numbers. A number with a real and an imaginary part is called a "complex number". For example, in the expression given in this exercise, the components \[6i, -4, \text{ and } i\] identify imaginary and real portions that build the solution.
Imaginary numbers are termed so because they don't exist on the regular number line. Instead, they form a different axis, allowing complex numbers to occupy their unique space in the math world.

Combining Like Terms

When solving problems involving complex numbers, combining like terms is a key skill. Like terms in algebra involve combining constants with constants and variables with the same power or base with their counterparts. When you have expressions containing both real and imaginary terms, you need to identify and group them separately.

• Real parts get bundled together. For this exercise, the number \[-4\] stands alone as the real part.
• Imaginary parts require grouping and simplification. In our given expression, \[6i\] and \[+i\] together form the imaginary component.

By adding the coefficients of these imaginary terms, we arrive at \[7i\]. This whole process emphasizes simplifying expressions by gathering together like terms, ensuring calculations can be made easily.
Remembering this approach aids in understanding and tackling more complicated algebraic problems involving not just imaginary numbers but real numbers too.

Complex Number Notation

Complex number notation involves the combination of real and imaginary numbers, making expressions like \(a + bi\) vital. Here, 'a' represents the real part, and 'bi' represents the imaginary part with 'b' as the coefficient of 'i'.
This form is standardized to ensure consistency and clarity when performing operations with complex numbers. For the exercise given, our result \[-4 + 7i\] is already in this standardized complex number notation.
Complex number notation provides several advantages:

• It simplifies arithmetic operations. Having a clear distinction of the real and imaginary parts is beneficial when adding, subtracting, multiplying, or dividing complex numbers.
• It aids in visualizing complex numbers on the complex plane, where the horizontal axis represents real numbers, and the vertical axis corresponds to imaginary numbers.

Using complex number notation, mathematicians and scientists can effectively work with phenomena that involve two dimensions, encompassing both real and imaginary realms.