Problem 6 - Advanced Engineering Mathematics

Question

Write the given number in the form \(a+i b\). $$ 3(4-i)-3(5+2 i) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-3 - 9i\).

Step 1: Distribute the first term

Start by distributing the number 3 across the terms inside the first parenthesis. The expression becomes: \[3 \times 4 + 3 \times (-i)\]This simplifies to:\[12 - 3i\]

Step 2: Distribute the second term

Now, distribute the number 3 across the terms inside the second parenthesis. The expression is:\[3 \times 5 + 3 \times 2i\]Simplifying this gives:\[15 + 6i\]

Step 3: Subtract the two expressions

Combine the results from Step 1 and Step 2 by subtracting the terms:\[(12 - 3i) - (15 + 6i)\]Simplify by distributing the negative sign:\[12 - 3i - 15 - 6i\]

Step 4: Combine like terms

Add the real parts and the imaginary parts:Real part: \[12 - 15 = -3\]Imaginary part: \[-3i - 6i = -9i\]This results in:\[-3 - 9i\]

Key Concepts

Complex Number Operations

Complex numbers blend real numbers and imaginary components, represented in the form \(a + ib\) where \(i\) is the imaginary unit. To efficiently work with these numbers, you will use addition, subtraction, multiplication, and division. These operations often involve both the real and imaginary parts separately.

• When adding or subtracting complex numbers, combine the real parts and the imaginary parts independently.
• Multiplying involves distributing each term, much like polynomials, and applying the rule \(i^2 = -1\).
• For division, it's common to multiply by the conjugate to remove the imaginary part from the denominator.

Understanding these operations allows you to manipulate complex expressions and solve equations involving complex numbers.

Distributive Property

The distributive property is a fundamental algebraic principle that allows you to multiply a single term by each term within a parenthesis. In complex number operations, the distributive property helps simplify expressions more efficiently.
When you encounter expressions like \(3(4-i)\), the distributive property lets you distribute 3 across each term:

• Multiply 3 by 4, resulting in 12.
• Multiply 3 by \(-i\), resulting in \(-3i\).

This property ensures you correctly expand and simplify expressions, which is crucial when you deal with complex numbers in algebra.

Combining Like Terms

Combining like terms is key to simplifying complex expressions into a more manageable form. It involves adding or subtracting terms that share the same variable component. In the context of complex numbers, this means combining terms with \(i\) separately from those without.
For example, when simplifying the expression \(12 - 3i - 15 - 6i\), you'll:

• Identify the real parts: 12 and \(-15\).
• Identify the imaginary parts: \(-3i\) and \(-6i\).
• Add or subtract the real parts (\(12 - 15 = -3\)).
• Add or subtract the imaginary parts (\(-3i - 6i = -9i\)).

Using combining like terms, you finalize the expression and clearly present complex equations, simplifying mathematical problems efficiently.

Imaginary Unit

The imaginary unit \(i\) is a cornerstone of complex numbers, defined by the equation \(i^2 = -1\). This intriguing property lets us represent numbers that don't have real roots, such as the square root of negative one.
Complex numbers involve this imaginary unit to extend traditional number systems and solve equations that real numbers alone can't.

• The imaginary part of a complex number is usually denoted as \(ib\), where \(b\) is the coefficient of \(i\).
• Understanding \(i\)'s role allows manipulation of complex numbers using usual algebraic rules while factoring in the unique property \(i^2 = -1\).

Grasping this concept helps decode expressions and equations that appear daunting, enhancing problem-solving skills across mathematics.