Problem 16 - Advanced Engineering Mathematics

Question

In Problems 1-26, write the given number in the form \(a+i b\). $$ \frac{10-5 i}{6+2 i} $$

Step-by-Step Solution

Answer

The answer is \(\frac{5}{4} - \frac{5}{4}i\).

Step 1: Define the Objective

The goal is to convert the complex number \(\frac{10-5i}{6+2i}\) into the standard form \(a + ib\), where \(a\) is the real part and \(b\) is the imaginary part.

Step 2: Multiply by the Conjugate

To simplify this expression, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(6 + 2i\) is \(6 - 2i\). So, multiply:\[\frac{10-5i}{6+2i} \times \frac{6-2i}{6-2i} = \frac{(10-5i)(6-2i)}{(6+2i)(6-2i)}\]

Step 3: Simplify the Denominator

The denominator \((6+2i)(6-2i)\) simplifies using the formula \((a+bi)(a-bi) = a^2 + b^2\):\[(6)^2 - (2)^2i^2 = 36 - 4(-1) = 36 + 4 = 40\]

Step 4: Expand the Numerator

Distribute the terms in the numerator:\[(10-5i)(6-2i) = 10 \times 6 + 10 \times (-2i) - 5i \times 6 - 5i \times (-2i)\]Calculate the terms:\[= 60 - 20i - 30i + 10i^2\]

Step 5: Simplify the Numerator

Combine like terms from the expanded numerator and substitute \(i^2 = -1\):\[60 - 50i + 10(-1) = 60 - 50i - 10 = 50 - 50i\]

Step 6: Divide Each Term by the Denominator

Divide each term in the numerator by the denominator to get the standard form:\[\frac{50 - 50i}{40} = \frac{50}{40} - \frac{50i}{40} = \frac{5}{4} - \frac{5}{4}i\]

Step 7: Conclusion

Thus, the complex number \(\frac{10-5i}{6+2i}\) is expressed in the form \(a + ib\) as \(\frac{5}{4} - \frac{5}{4}i\).

Key Concepts

Complex Conjugate

In complex numbers, each number has a special partner known as the complex conjugate. The conjugate of a complex number \(a + bi\) is \(a - bi\). Here, the conjugate is formed by changing the sign of the imaginary part.

• For the complex number \(6+2i\), its conjugate is \(6-2i\).
• Using the conjugate helps simplify division of complex numbers.

By multiplying the numerator and the denominator of a division problem by the conjugate of the denominator, the imaginary components can cancel out. This is because multiplying a complex number by its conjugate gives us a real number: \[(a+bi)(a-bi) = a^2 - (bi)^2 = a^2 + b^2\] This process is important because it transforms a complex division into a simpler operation with real numbers.

Standard Form of Complex Numbers

The standard form of a complex number is written as \(a + ib\), where \(a\) represents the real part and \(b\) represents the imaginary part. Each complex number can always be expressed in this format.

• For example, \(\frac{5}{4} - \frac{5}{4}i\) is in the standard form.
• The coefficients before and after the imaginary unit \(i\) indicate the components.

To express a complex number in standard form, begin by simplifying each component separately, often through division or multiplication as seen in our original problem. Converting the expression \(\frac{10-5i}{6+2i}\) involved using the complex conjugate and arithmetic operations to get the result into the form \(a + ib\). This simplification is key to handling complex numbers efficiently and making them easier to use in mathematical problems.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept used to handle negative square roots. By definition, \(i\) is equal to the square root of \(-1\).

• The equation \(i^2 = -1\) is central to calculations involving complex numbers.
• In practice, when \(i^2\) appears in calculations, it is replaced with \(-1\).

In our exercise, dealing with expressions like \(-5i \times (-2i)\) resulted in \(10i^2\). Knowing that \(i^2 = -1\), this simplifies to \(10(-1)\) or \(-10\). This step ensures the proper conversion of additional imaginary components into easily combinable real numbers, paving the way for clearer results and simpler manipulation of complex terms.