Table of contents | |
Introduction | |
Power of i | |
Square of Iota | |
Square Root of Iota | |
Higher Power of i | |
Value of Iota for Negative Power |
Iota is an imaginary unit number that is denoted by i and the value of iota is √-1 i.e., i = √−1. While solving quadratic equations, you might have come across situations where the discriminant is negative. For example, consider the quadratic equation x2 + x + 1 = 0. If we use the quadratic formula to solve this, we get the discriminant (the part inside the square root) as a negative value.
In such cases, we write √−3 as √−3 = √−1 × √3. This would give the solution of the above quadratic equation to be: x = (−1 ± √3i)/2. Hence, the value of iota is helpful in solving square roots with negative values.
Thus, the value of iota is, i = √−1.
The powers of i, i repeat in a certain pattern in a cycle. Let us start with calculating the value of powers of i for general cases and try to figure out the pattern.
We know that the value of iota, i is defined as, i = √−1. If we square both sides of the above equation, we get: i2 = -1 i.e., the value of the square of iota is -1. Therefore, the square of iota is, i2 = −1.
Iota has two square roots, just like all non-zero complex numbers. The value of the square root of iota, given as √i, can be calculated using De Moivre's Theorem.
We know that, i = cos(π/2) + isin(π/2)
= cos(π/2 + 2nπ) + isin(π/2 + 2nπ), n = 0, 1
= cos[(π + 4nπ)/2] + i sin[(π + 4nπ)/2]
Here, we took n = 0, 1 as we need 2 solutions. But we have to find √i = (i)1/2. Let us raise the exponent to 1/2 on both sides. So we get: √i = [cos{(π + 4nπ)/2} + isin{(π + 4nπ)/2}]1/2 = cos[(π + 4nπ)/4] + isin[(π + 4nπ)/4], n = 0, 1
√i = √2/2 + i√2/2 = −√2/2 − i√2/2
Let us see how to calculate some other powers of i.
From the above calculations, we can observe that the values of iota repeat in certain pattern. The following figure represents the values for various powers of i in the form of a continuous circle.
This signifies that i repeats its values after every 4th power. We can generalize this fact to represent this pattern (where n is any integer), as,
Higher powers of iota can be calculated by decomposing the higher exponents i into smaller ones and thus evaluating the expression. Finding value if the power of i is a larger number using the previous procedure, will take quite some time and effort. If we observe all the powers of i and the pattern in which it repeats its values in the above equations, we can calculate the value of iota for higher powers as given below,
Example: Find the value of i20296.
We just have to remember that i2 = -1 and i3 = -i. We will find some other higher powers of i using these and the above rules.
The value of iota for negative power can be calculated following few steps. We first convert it into a positive exponent using the negative exponent law and then we apply the rule: 1/i = -i. This is because:1/i = 1/i • i/i = i/i2 = i/(-1) = -i
Example: Find the value of i-3927
i-3927 = 1/i3927
∵ a-m = 1/am
= 1/i3
∵ Remainder of 3927 divided by 4 is 3
= 1/-i
∵ i3 = -i
= --i
∵ 1/i = -i
= i
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