Problem 11 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$i(3+4 i)^{2}$$

Step-by-Step Solution

Answer

The expression in the form \(a+bi\) is \(-24 - 7i\).

Step 1: Expand the Expression

The given expression is \(i(3+4i)^2\). First, focus on expanding \((3+4i)^2\). Use the formula \((a+b)^2 = a^2 + 2ab + b^2\) where \(a = 3\) and \(b = 4i\). This gives:\[(3+4i)^2 = 3^2 + 2(3)(4i) + (4i)^2\]Calculate each part:- \(3^2 = 9\)- \(2 \times 3 \times 4i = 24i\)- \((4i)^2 = 16i^2 = 16(-1) = -16\)So, \((3+4i)^2 = 9 + 24i - 16\). This simplifies to \(-7 + 24i\).

Step 2: Multiply by i

Now, take the result from Step 1 and multiply it by \(i\). We have \(i(-7 + 24i)\). Distribute \(i\) over the sum:\[i(-7 + 24i) = i(-7) + i(24i)\]Calculate each term:- \(i(-7) = -7i\)- \(i(24i) = 24i^2 = 24(-1) = -24\)Combine these: \(-7i - 24\). This can be written in the form \(a + bi\) as \(-24 - 7i\).

Step 3: Identify a and b

The expression \(-24 - 7i\) is already in the form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a = -24\) and \(b = -7\).

Key Concepts

Imaginary Unit

The imaginary unit, often represented as \(i\), is a fundamental concept in complex numbers. It is defined as the square root of -1, this makes complex calculations possible especially where dealing with negative roots.

• Definition: \(i^2 = -1\)
• Purpose: Imaginary numbers allow us to solve equations that otherwise wouldn't have solutions in the real number system. For example, the equation \(x^2 + 1 = 0\) does not have real solutions, but in the complex number system, the solutions involve \(i\).

Whenever an imaginary unit \(i\) is squared, it simplifies to -1. This property is crucial when performing arithmetic with complex numbers as evidenced by the simplification of \((4i)^2\) in the original exercise.

Complex Conjugate

A complex conjugate involves changing the sign of the imaginary part of a complex number. If you have a complex number \(a + bi\), its complex conjugate is \(a - bi\). This concept is especially useful in operations involving division or finding magnitudes of complex numbers.

• Definition: Complex conjugate of \(a + bi\) is \(a - bi\)
• Application: When a complex number is multiplied by its conjugate, the result is a real number.

Using conjugates can help simplify complex fraction or division problems. Although conjugates weren't directly applied in the step-by-step solution, understanding them helps gain a broader understanding of manipulating complex numbers.

Real and Imaginary Parts

The real and imaginary parts of a complex number distinguish it from regular numbers. A complex number can be expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

• Real Part: In \(a + bi\), \(a\) is the real part. It's the component without the imaginary unit \(i\).
• Imaginary Part: In \(a + bi\), \(bi\) is the imaginary part. It includes the imaginary unit \(i\), and \(b\) is a real number.

The original exercise concludes with identifying the real and imaginary parts of the complex expression \(-24 - 7i\), where \(-24\) is the real part and \(-7\) is the imaginary part. Recognizing these components is crucial for complex arithmetic as it allows easy interpretation and manipulation of complex numbers.