Problem 19 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ (7-t i)-\left(5+\frac{1}{2} i\right) $$

Step-by-Step Solution

Answer

The result is \(2 - (t + \frac{1}{2})i\).

Step 1: Distribute the Negative Sign

To evaluate the expression and combine like terms, start by distributing the negative sign in front of the second term: \[(7 - ti) - (5 + \frac{1}{2}i) = 7 - ti - 5 - \frac{1}{2}i\]

Step 2: Combine Real Parts

Combine the real parts of the expression. The constants 7 and 5 are the real parts:\[(7 - 5) + (-ti - \frac{1}{2}i) = 2 - ti - \frac{1}{2}i\]

Step 3: Combine Imaginary Parts

Next, combine the imaginary parts. These are the terms that have "i".The imaginary parts are \[-ti - \frac{1}{2}i\].Factor out the common "i":\[-(t + \frac{1}{2})i\]

Step 4: Write in the form a + bi

Now, place the results together as a single expression. Ensure to express it in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part:\[2 - (t + \frac{1}{2})i\] This follows the required form with \(a = 2\) and \(b = -(t + \frac{1}{2})\).

Key Concepts

Distributing Negative Signs in Complex Expressions

When dealing with complex numbers, it is essential to be careful about negative signs, especially when subtraction is involved. To "distribute" a negative sign means to apply it to each term within a given expression.

Suppose you have the expression \((7 - ti) - (5 + \frac{1}{2}i)\). Here, the negative sign in front of the parentheses means that both terms inside need to be negated. So:

• - becomes: \(-5\)
• and \(+ \frac{1}{2}i\) becomes: \(- \frac{1}{2}i\)

This is analogous to multiplying each term by -1. By doing this, you get each part ready for combining with terms from other parts of the expression. Getting this part right ensures accuracy in the final result.

Combining Like Terms in Complex Expressions

Once the negative signs are properly distributed, the next step in simplifying an expression with complex numbers is to combine like terms. "Like terms" are those that share the same variable. In complex numbers, you typically have "real" parts and "imaginary" parts.

Continuing with our example, the expression after distributing the negative signs becomes:

• Real Parts: 7 and -5
• Imaginary Parts: \(-ti\) and \(-\frac{1}{2}i\)

Combine each set:

• Real parts combine as: \(7 - 5 = 2\)
• Imaginary parts combine as: \(-ti - \frac{1}{2}i = -(t + \frac{1}{2})i\)

With the terms combined, you prepare your expression for simplification into the form you need, which is typically \(a + bi\).

Understanding Real and Imaginary Parts

In complex numbers, understanding the distinction between real and imaginary parts is crucial. A complex number usually takes the form \(a + bi\), where:

• \(a\) is the real part
• \(bi\) is the imaginary part, \(b\) being the coefficient of \(i\)

In our final expression: \(2 - (t + \frac{1}{2})i\), the separation is clear:

• Real Part, \(a = 2\)
• Imaginary Part, \(b = -(t + \frac{1}{2})\)

Both these parts are independent but together form the complex number. The real part appears as the ordinary number, while the imaginary part includes the unit \(i\), representing the square root of -1. Mastery of distinguishing between these parts aids in manipulating and working with complex numbers effectively.