This stack introduces the fundamental concept of the imaginary unit 'i', defined as the square root of -1. It covers simplifying powers of 'i' and using it to find the square root of negative numbers.
What is the definition of the imaginary unit, denoted by the symbol \(i\)?
The imaginary unit \(i\) is defined as the principal square root of negative one. $$ i = \sqrt{-1} $$
Based on the definition \(i = \sqrt{-1}\), what is the value of \(i^2\)?
Squaring both sides of the definition \(i = \sqrt{-1}\) gives \(i^2 = -1\). This is a fundamental identity in complex numbers.
What is the value of \(i^3\)?
\(i^3\) can be broken down into \(i^2 \cdot i\). Since \(i^2 = -1\), the result is \(-1 \cdot i = -i\).
What is the value of \(i^4\)?
\(i^4\) can be seen as \((i^2)^2\). Since \(i^2 = -1\), the result is \((-1)^2 = 1\).
Describe the cyclical pattern of the powers of \(i\) (\(i^1, i^2, i^3, i^4, i^5, ...\)).
The powers of \(i\) follow a repeating four-term cycle: \(i, -1, -i, 1\). For example, \(i^5 = i^4 \cdot i = 1 \cdot i = i\), starting the cycle over.
What is the general method for simplifying \(i^n\) where \(n\) is any positive integer?
Divide the exponent \(n\) by \(4\). The value of \(i^n\) is determined by the remainder \(r\) of this division (\(n \div 4\)). The result will be \(i^r\). A remainder of \(0\) is equivalent to \(i^4\).
Simplify \(i^{39}\).
Divide \(39\) by \(4\), which gives \(9\) with a remainder of \(3\). Therefore, \(i^{39} = i^3 = -i\).
Simplify \(i^{100}\).
Divide \(100\) by \(4\), which gives \(25\) with a remainder of \(0\). A remainder of \(0\) corresponds to \(i^4\). Therefore, \(i^{100} = i^4 = 1\).
How do you express the square root of a negative number, such as \(\sqrt{-a}\) where \(a > 0\), using the imaginary unit \(i\)?
You can separate the number into \(\sqrt{-1 \cdot a}\), which becomes \(\sqrt{-1} \cdot \sqrt{a}\). This simplifies to \(i\sqrt{a}\).
