Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{5}{2-7 i} $$

Step-by-Step Solution

Answer

The expression \( \frac{5}{2-7i} \) is equivalent to \( \frac{10}{53} + \frac{35}{53}i \).

Step 1: Understand the Expression

We are given the expression \( \frac{5}{2-7i} \) and need to express it in the form \( a + bi \), where \( a \) and \( b \) are real numbers. This involves removing the imaginary number from the denominator.

Step 2: Multiply by the Conjugate

We multiply both the numerator and the denominator by the conjugate of the denominator, \( 2+7i \), to eliminate the imaginary unit from the denominator.\[\frac{5}{2-7i} \times \frac{2+7i}{2+7i}\]

Step 3: Apply Distributive Property

Multiply the numerators and the denominators:\[\text{Numerator: } 5 \times (2 + 7i) = 10 + 35i\]\[\text{Denominator: } (2 - 7i)(2 + 7i) = 2^2 - (7i)^2 = 4 + 49 = 53\]

Step 4: Simplify the Expression

We now have the fraction as:\[\frac{10 + 35i}{53}\]Separate the real and imaginary parts:\[= \frac{10}{53} + \frac{35}{53}i\]

Step 5: Finalize the Result

Now, the expression is in the form \( a + bi \):\[\frac{5}{2-7i} = \frac{10}{53} + \frac{35}{53}i\]

Key Concepts

Conjugate

In the context of complex numbers, the conjugate plays a crucial role, especially when simplifying expressions that involve division of complex numbers. Given a complex number in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit, the conjugate \( \overline{z} \) of \( z \) is found by changing the sign of the imaginary part. Thus, the conjugate is \( a - bi \).

Using the conjugate helps when rationalizing denominators in complex numbers, a process we'll explore further. When a complex number is multiplied by its conjugate, it results in a real number: \((a+bi)(a-bi) = a^2 + b^2 \). This is because the \( i^2 \) term becomes \(-1\), canceling out the imaginary component. This property is especially useful when performing algebraic operations that involve complex fractions.

Imaginary Numbers

Imaginary numbers extend the real number system and are denoted by the symbol \( i \), where \( i = \sqrt{-1} \). This means \( i^2 = -1 \), which is the defining property of imaginary numbers. Complex numbers arise when real numbers combine with imaginary numbers in expressions of the form \( a + bi \), where \( a \) is the real component and \( bi \) is the imaginary component.

Imaginary numbers are used to perform computations in complex domains and play a significant role in fields like engineering and physics. They help us understand phenomena that cannot be described using just real numbers. In mathematical operations, handling imaginary numbers properly often involves converting expressions to the form \( a + bi \), which is essential for calculations, as seen in complex arithmetic operations.

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating any imaginary components by transforming the denominator of a fraction into a real number. This algebraic manipulation simplifies expressions and eases further calculations.

Take the expression \( \frac{5}{2-7i} \). To rationalize the denominator, you multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2-7i \) is \( 2+7i \). This results in:\[ \frac{5}{2-7i} \times \frac{2+7i}{2+7i} \] The denominator transformation comes from multiplying \( (2-7i)(2+7i) \), and exploiting the identity that \( (a + bi)(a - bi) = a^2 + b^2 \). Thus, the denominator becomes \( 2^2 + 7^2 = 53 \), a real number.

This calculation enables us to express the original fraction in the simplified \( a + bi \) form, which settles computations involving complex numbers into a form that's easier to understand and use.