Problem 103 - Intermediate Algebra

Question

Write in the form \(a+b i\). $$ i^{3}-i^{4} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-1 - i\).

Step 1: Simplify the Power of i

Recall that the powers of the imaginary unit, i, repeat in a cycle: \[ i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 \] Using this pattern, replace the powers of \(i\) with their equivalent:\[ i^3 = -i \]\[ i^4 = 1 \]

Step 2: Substitute the Simplified Terms

Substitute the simplified expressions for \(i^3\) and \(i^4\) back into the expression:\[ i^3 - i^4 = -i - 1 \]

Step 3: Write in the Form a + bi

The expression \(-i - 1\) can be rewritten in the form \(a + bi\) as:\[ a = -1 \]\[ b = -1 \]So, the expression is \(-1 - i\) which is the required form.

Key Concepts

Imaginary Unit

In the world of complex numbers, the imaginary unit is represented by the symbol \(i\). It is defined as a number that satisfies the equation \(i^2 = -1\). This particular characteristic gives \(i\) a special place in mathematics, as it allows us to construct imaginary numbers, which are numbers that, when squared, provide a negative result.

Imaginary numbers are crucial in various fields of science and engineering. They enable solutions to problems that cannot be solved using only real numbers.

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

These powers of \(i\) form a repetitive cycle, and understanding this cycle is key to simplifying expressions involving powers of \(i\).

Powers of i

The powers of \(i\) follow a predictable cycle of four values: \(i\), \(-1\), \(-i\), and \(1\). This cyclic nature is essential when simplifying expressions with higher powers of \(i\). For instance, \(i^3\) simplifies to \(-i\) and \(i^4\) simplifies to \(1\).

If you continue to multiply \(i\), the cycle will repeat every four steps:

• \(i^5 = i^1 = i\)
• \(i^6 = i^2 = -1\)
• \(i^7 = i^3 = -i\)
• \(i^8 = i^4 = 1\)

Each power can thus be reduced to one of these four core results, making complex calculations faster and more straightforward. Knowing this cycle allows you to quickly determine the value of any power of \(i\) by focusing just on the remainder of the power divided by 4.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and arithmetic operations such as addition and subtraction. When working with complex numbers, these expressions can also include the imaginary unit \(i\).

For example, consider the expression \(i^3 - i^4\). Simplifying this requires replacing each power of \(i\) with its equivalent from the cycle of powers. Thus, \(i^3 = -i\) and \(i^4 = 1\). This turns our original expression into \(-i - 1\).

Handling algebraic expressions with complex numbers involves recognizing and correctly applying these simplifications. Doing so transforms seemingly complicated expressions into manageable forms.

Standard Form a+bi

The standard form for a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers. The term \(a\) represents the real part, while \(b\times i\) represents the imaginary part.

In the exercise, we have the expression \(-i - 1\). To convert this into the standard form, notice it corresponds directly to \(a = -1\) and \(b = -1\). This means the expression can be rewritten as \(-1 + (-1)i\) or simply \(-1 - i\).

• Ensure all complex numbers are expressed in the form \(a + bi\).
• Be aware that \(a\) and \(b\) can be positive, negative, or zero.
• Recognize that understanding this form helps compare and perform operations with complex numbers.

Presenting complex numbers in their standard form makes them easier to work with, especially when dealing with addition, subtraction, or further algebraic manipulation.