Problem 31 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ (1+i)(2-3 i) $$

Step-by-Step Solution

Answer

The expression in standard form is \(5 - i\).

Step 1: Multiply the Real Parts

Start by multiplying the real parts of the complex numbers: \(1 \cdot 2 = 2\)

Step 2: Multiply the Outer Parts

Next, multiply the outer parts of the complex numbers: \(1 \cdot (-3i) = -3i\)

Step 3: Multiply the Inner Parts

Multiply the inner parts of the complex numbers: \(i \cdot 2 = 2i\)

Step 4: Multiply the Imaginary Parts

Now, multiply the imaginary parts: \(i \cdot (-3i) = -3i^2\)Remember that \(i^2 = -1\) so, \(-3i^2 = -3(-1) = 3\).

Step 5: Combine the Results

Add the results from steps 1 to 4: \(2 + (-3i) + 2i + 3\).Combine like terms: \((2 + 3) + (-3i + 2i) = 5 - i\).

Step 6: Write in Standard Form

The standard form of a complex number is \(a + bi\).The expression \(5 - i\) is already in standard form, where \(a = 5\) and \(b = -1\).

Key Concepts

Standard Form of a Complex Number

When dealing with complex numbers, it's important to express them in a format that's easy to understand and work with. This is known as the "standard form". The standard form of a complex number is written as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. Here, \( a \) is called the "real part", and \( b \) is the "imaginary part". This form is much like the way we write coordinates or linear equations, providing a clear guideline to handle operations.
To form this expression, all you need to do is combine the real and imaginary components. Separate them neatly using a plus sign, adjusting the sign of \( b \) as needed based on calculations. For example, if calculating gives a \(-b\), you write \( a - bi \). In our specific exercise, the expression \( 5 - i \) is the result, with \( a = 5 \) and \( b = -1 \). Not only does this clarity help in performing operations like addition or multiplication, but it is also the most efficient way to present your solutions.

The Imaginary Unit (i)

In the realm of complex numbers, the imaginary unit \( i \) plays a crucial role. It is defined by the property that \( i^2 = -1 \). This is what sets it apart from real numbers and makes it the cornerstone of complex calculations.
Understanding \( i \) helps in numerous ways:

• Using \( i \), you can express numbers that are otherwise impossible to describe within the real number system. For instance, the square root of negative numbers.
• The manipulation of \( i \) involves simple rules. Multiplying \( i \) with itself results in \(-1\) — key in converting complex results to real numbers during multiplication.

Whenever you have to deal with a multiplication resulting in \( i^2 \), remember to replace it with \(-1\). In the given exercise, this aspect was pivotal in converting \(-3i^2\), using \( i^2 = -1 \), to the real number \(+3\). The understanding of \( i \) and its algebraic properties is essential for further maths and engineering applications.

Complex Multiplication

Complex multiplication involves multiplying two complex numbers. It expands just like binomial expansion and uses the distributive property. Suppose we have two complex numbers, \( (a + bi) \) and \( (c + di) \). When multiplying these, you should follow these steps:

• Multiply the real parts \( a \cdot c \).
• Multiply the outer parts (cross terms) \( a \cdot di \).
• Multiply the inner parts (cross terms) \( bi \cdot c \).
• Multiply the imaginary parts \( bi \cdot di \). Remember here \( i^2 = -1 \), so the term becomes \( -bd \).

Finally, combine all these results using addition or subtraction, as dictated by their signs. This combination gives you the resulting complex number in standard form. In our exercise \((1+i)(2-3i)\), the result is \(5 - i\). Here, understanding the influence of each term on the outcome helps solidify your grasp of complex arithmetic. Practice ensures that these steps become second nature, making future calculations quicker and more accurate.