Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the quotient in standard form. $$\frac{8-7 i}{1-2 i}$$

Step-by-Step Solution

Answer

The standard form of the quotient is \(2.8+1.8i\).

Step 1: Identify the numerator and the denominator

The numerator is \(8-7i\) and the denominator is \(1-2i\).

Step 2: Calculate the conjugate of the denominator

The conjugate of \(1-2i\) is \(1+2i\).

Step 3: Multiply the numerator and the denominator by the conjugate of the denominator

The equation becomes \(\frac{(8-7i)(1+2i)}{(1-2i)(1+2i)}\).

Step 4: Distribute and multiply

Carry out the multiplication in the numerator and the denominator. In the numerator, the result is \(8+16i-7i-14i^2\). In the denominator, we get a difference of squares: \((1)^2-(2i)^2\).

Step 5: Simplify

Simplify the numerator and the denominator. \(i^2\) is \(-1\), so \(-14i^2\) becomes \(14\). So, the numerator is \(14+9i\), and the denominator is \(1-(-4)\), which simplifies to 5. Therefore, the complex number in standard form is \(\frac{14+9i}{5}\).

Step 6: Divide the real and imaginary parts by the denominator

Divide the real part 14 by 5 and the imaginary part 9 by 5 to get \(2.8+1.8i\).

Key Concepts

Conjugate of Complex Numbers

The 'conjugate' of a complex number is an important concept to understand when working with complex arithmetic. Given a complex number in the form of \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) represents the imaginary unit, the conjugate of this number is \(a - bi\). Essentially, you just change the sign of the imaginary part.

For example, if we take the complex number \(1 - 2i\), its conjugate is \(1 + 2i\). This operation is useful because multiplying a complex number by its conjugate gives a real number, a trick often used to simplify the division of complex numbers as seen in the textbook exercise. Multiplying both the numerator and the denominator of a complex fraction by the conjugate of the denominator removes the imaginary part from the denominator, making further simplification possible.

Distributive Property in Algebra

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term within a parenthesis in an expression. It is expressed mathematically as \(a(b + c) = ab + ac\). This property makes it possible to expand expressions and is utilized to perform operations like multiplication of complex numbers.

In the context of complex numbers, when you distribute to multiply, you must consider the imaginary unit \(i\) as well, which has the property that \(i^2 = -1\). This is why, during multiplication, when we encounter terms like \(-7i \times 2i\), we end up with \(-14i^2\), which simplifies to \(+14\) because of this unique property of the imaginary unit.

Simplifying Complex Expressions

When simplifying complex expressions, especially after using the distributive property to expand them, we combine like terms and simplify using \(i^2 = -1\). In our example, \(8+16i-7i-14i^2\) becomes \(8 - 14 + (16i - 7i)\), simplifying to \(14 + 9i\) after applying the property of the imaginary unit to the \(-14i^2\) term.

Solutions should present complex numbers in standard form, which is \(a + bi\). If we have a quotient, like in this exercise, we divide both the real and the imaginary parts by the denominator. It's important, as shown in Step 6 of the solution, to individually divide and simplify the real part (14 divided by 5 equals 2.8) and the imaginary part (9 divided by 5 equals 1.8), resulting in the standard form of \(2.8 + 1.8i\).