Problem 79 - Intermediate Algebra

Question

Divide. Write all answers in the form a \(+b i.\) $$ \frac{1-3 i}{3+i} $$

Step-by-Step Solution

Answer

The answer is \(-i\).

Step 1: Identify Conjugate of the Denominator

The denominator of the expression is \(3+i\). To rationalize it, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number \(a+bi\) is \(a-bi\). Thus, the conjugate of \(3+i\) is \(3-i\).

Step 2: Multiply Numerator and Denominator by Conjugate

Multiply both numerator \((1-3i)\) and denominator \((3+i)\) by the conjugate of the denominator \((3-i)\):\[\frac{(1-3i)(3-i)}{(3+i)(3-i)}\]

Step 3: Simplify Denominator Using Difference of Squares

The denominator becomes a difference of squares when multiplied by its conjugate:\[(3+i)(3-i) = 3^2 - i^2 = 9 - (-1) = 10\]So, the denominator simplifies to 10.

Step 4: Expand the Numerator

Expand the numerator \((1-3i)(3-i)\):\[1 \cdot 3 + 1 \cdot (-i) - 3i \cdot 3 - 3i \cdot (-i) = 3 - i - 9i + 3i^2\]Since \(i^2 = -1\), this becomes:\[3 - i - 9i + 3(-1) = 3 - i - 9i - 3 = -10i\]

Step 5: Combine Terms in the Numerator

Combine the real and imaginary parts:\[0 + (-10i) = -10i\]Thus, the numerator simplifies to \(-10i\).

Step 6: Simplify the Fraction

Now divide the simplified numerator \(-10i\) by the simplified denominator 10:\[\frac{-10i}{10} = -i\]

Step 7: Write in Standard Form

Express \(-i\) in the standard form \(a + bi\). Since there is no real part, \(a = 0\) and \(b = -1\), thus,\[0 - i\] or simply \(-i\).

Key Concepts

Complex Conjugate

The complex conjugate of a complex number is a vital concept in simplifying divisions involving complex numbers. If you have a complex number in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, its complex conjugate is \(a - bi\). This means you simply change the sign of the imaginary part to find the conjugate.

When you multiply a complex number by its conjugate, the result is a real number. This is due to the difference of squares formula:

• \((a + bi)(a - bi) = a^2 + b^2\)

This formula is actually the key to getting rid of the imaginary unit in the denominator, making it easier to deal with complex divisions. So, whenever you are dividing by a complex number, using the conjugate as a tool helps us simplify the situation. This is why the division starts by multiplying the numerator and the denominator by the conjugate of the denominator.

Imaginary Unit

The imaginary unit, denoted by \(i\), is defined by the property that \(i^2 = -1\). It is a building block of complex numbers, which are expressed in the form \(a + bi\).

Imaginary numbers allow us to extend the concept of numbers to include solutions to equations that do not have real solutions. A classic example is \(x^2 = -1\), for which \(x\) would equal \(i\).

In the context of complex arithmetic:

• The real part of \(a + bi\) is \(a\), and the imaginary part is \(b\) (not \(bi\), just \(b\)).
• For operations: multiplying \(i\) by itself results in \(-1\), so expressions with higher powers of \(i\) (like \(i^3\) or \(i^4\)) can be simplified accordingly. For example, \(i^3 = i^2 \cdot i = -i\).

Being familiar with these properties helps simplify expressions involving the imaginary unit and is essential in simplifying complex number divisions.

Standard Form of Complex Numbers

Expressing complex numbers in standard form is crucial for clarity and simplicity. The standard form is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. When you divide complex numbers, the goal is to simplify the expression into this standard form.

To do this:

• First, clear any complex terms from the denominator by multiplying by the conjugate.
• Second, simplify the resulting expression to separate the real and imaginary components clearly.

For example, when dividing \(\frac{1 - 3i}{3 + i}\), multiplying by the conjugate \(3 - i\) creates a real number in the denominator, while the numerator is simplified. The result, after all calculations, is rewritten as \(-i\), which in standard form is \(0 - i\). Expressing numbers in this way makes it easier to understand and further manipulate them mathematically.