Powers of Iota

Powers of Iota

Edited By Komal Miglani | Updated on Jul 02, 2025 07:35 PM IST

The term iota represented by i which is widely used in complex numbers. It is represented by $\sqrt{-1}$ or $i$. In real life, this can be helped in solving some challenging problems of complex numbers. Iota is an imaginary unit number to express complex numbers, where i is defined as imaginary or unit imaginary. The value of the imaginary unit number, i is generated, when there is a negative number inside the square root. Thus, the value of i is given as $\sqrt{-1}$.

Powers of Iota
Powers of Iota

In this article, we will cover the concept of iota and its behavior by increasing its power. This concept falls under the broader category of complex numbers. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of two questions have been asked on this concept, including one in 2019, and one in 2020.

What is iota?

Iota is an imaginary unit number that is denoted by i and the value of iota is $\sqrt{-1} \text { i.e., } i=\sqrt{ -1}$. When we solve quadratic equations, we might have come across situations where the discriminant is negative. For example, consider the quadratic equation $x^2+x+1=0$. If we use the quadratic formula to solve this, we get the discriminant (the part inside the square root) as a negative value. Due to this the term iota is introduced.

For analyzing the powers of iota, we calculate the first few powers of iota. i.e $i=\sqrt{-1}$, $i^2=-1$, $i^3=-\sqrt{-1}$, $i^4=1$

Value of iota ($i$)

The square of any real number, whether it is positive or negative or zero is always non-negative, i.e. $x^2 \geq 0$ for all x ∈ R.

Hence, the equation $x^2+x+1=0$ is not satisfied for any real value of x or not solvable in a real number system.

Thus, the equation $x^2+x+1=0$ has an imaginary solution. ‘Eular’ was the first mathematician to introduce the symbol i (read as ‘iota’). The imaginary number i is defined as the square root of −1.

Hence, the equation
$
\begin{aligned}
& x^2+1=0 \\
& \Rightarrow \quad x^2=-1
\end{aligned}
$
$
\text { or, } \quad \mathrm{x}= \pm \sqrt{-1}= \pm i
$
Equation, $\mathrm{x}^2+1=0$ has two solution, $\mathrm{x}=i$ and $\mathrm{x}=-i$.

$
\begin{aligned}
\sqrt{-1} & =\mathrm{i} \\
1^2 & =(\sqrt{-1})^2=-1
\end{aligned}
$
We can write the square root of any negative number as a multiple of i. Consider the square root of -25

$
\begin{aligned}
\sqrt{-25} & =\sqrt{25(-1)} \\
& =\sqrt{25} \sqrt{-1} \\
& =5 \mathrm{i}
\end{aligned}
$

Value of power of iota

1) If the power of iota is the whole number

$
\begin{aligned}
& \mathrm{i}^0=1, \mathrm{i}^1=\mathrm{i}, \mathrm{i}^2=(\sqrt{-1})^2=-1 \\
& \mathrm{i}^3=\mathrm{i} \cdot \mathrm{i}^2=\mathrm{i} \times-1=-\mathrm{i} \\
& \mathrm{i}^4=\mathrm{i}^2 \cdot \mathrm{i}^2=-1 \times-1=1 \\
& \mathrm{i}^5=\mathrm{i} \cdot \mathrm{i}^4=\mathrm{i} \times 1=\mathrm{i}
\end{aligned}
$

In general,
$
i^{4 \mathbf{n}}=1, \quad i^{4 \mathbf{n}+\mathbf{1}}=i, \quad i^{4 \mathrm{n}+2}=-1, \quad i^{4 \mathrm{n}+\mathbf{3}}=-1
$

2) If the power of iota is the negative integer

\begin{aligned}

$\begin{aligned} & i^{-1}=\frac{1}{i}=\frac{i}{i^2}=\frac{i}{-1}=-i \\ & i^{-2}=\frac{1}{i^2}=-1 \\ & i^{-3}=\frac{1}{i^3}=\frac{i}{i^4}=i \\ & i^{-4}=\frac{1}{i^4}=\frac{1}{1}=1\end{aligned}$

The sum of four consecutive powers of iota (i) is zero

$
\begin{aligned}
& \mathrm{n} \in \mathbb{I} \text { and } \mathrm{i}=\sqrt{-1} \text {, then } \\
& \begin{aligned}
i^{\mathrm{n}}+i^{\mathrm{n}+1}+i^{\mathrm{n}+2}+i^{\mathrm{n}+3} & =i^{\mathrm{n}}\left(1+i+i^2+i^3\right) \\
& =i(1+i-1-i)=0
\end{aligned}
\end{aligned}
$

Recommended Video Based on Powers of Iota:

Solved Examples Based on Powers of Iota

Example 1: If $a<0$ and $b<0$, then $\sqrt{a }\sqrt{ b}$ is equal to:

Solution

Here, $a$ and $b$ are both negative

$
\begin{aligned}
& \sqrt{-a} \sqrt{-b}=\sqrt{(-1) a} \sqrt{(-1) b} \\
& \Rightarrow \sqrt{-1} \sqrt{a} \sqrt{-1} \sqrt{b}=i^2 \sqrt{a} \sqrt{b} \\
& \Rightarrow-1 \sqrt{|a||b|}
\end{aligned}
$
so the value is $-\sqrt{(|a||b|)}$
Hence, the answer is $-\sqrt{(|a||b|) }$.

Example 2: $(1+i)^8+(1-i)^9$ equals:

Solution:

$
(1+i)^8=\left((1+i)^2\right)^4=\left(1+i^2+2 i\right)^4=(1-1+2 i)^4=(2 i)^4=2^4 \cdot i^4=16
$
and
$
\begin{aligned}
& (1-i)^9=\left((1-i)^2\right)^4 \cdot(1-i)^1=\left(1+i^2-2 i\right)^4 \cdot(1-i) \\
& =(-2 i)^4(1-i)=(-2)^4 \cdot i^4 \cdot(1-i)=16(1-i)=16-16 i \\
& \therefore(1+i)^8+(1-i)^9=16+16-16 i=32-16 i
\end{aligned}
$

Hence, the answer is $32-16 i$

Example 3: The value of $i^{-99}$ is:

Solution:

$i^{-99}=\frac{1}{i^{99}}=\frac{i}{i^{100}}=i \quad\left(\because i^{100}=\left(i^4\right)^{25}=1\right)$

So the value is $i$

Hence, the answer is $i$.

Example 4: If $\left(\frac{1+i}{1-i}\right)^x=1$, then $x=$ ?

Solution:

Power of i in Complex Numbers -

$i^{4 n}=1, i^{4 n+1}=i, i^{4 n+2}=-1, i^{4 n+3}=-i$, where $n \in$ Integer
Now,
$
\begin{aligned}
& \left(\frac{1+i}{1-i}\right)^x=\left[\frac{(1+i)(1+i)}{(1-i)(1+i)}\right]^x=\left[\frac{(1+i)(1+i)}{2}\right]^x \\
& =\left[\frac{(1+i)^2}{2}\right]^x=\left[\frac{1-1+2 i}{2}\right]^x=(i)^x
\end{aligned}
$

And given that
$
\begin{aligned}
& \left(\frac{1+i}{1-i}\right)^x=1 \\
& i^x=(i)^{4 n} \\
& x=4 n
\end{aligned}
$

Hence, the answer is $4n$


Frequently Asked Questions (FAQs)

1. What is iota?

An iota is a complex number represented by i or $\sqrt{-1}$

2. What are the powers of the iota?

Powers of the iota means multiplying the iota many times.

3. Write the first four powers of iota in terms of i.

The first four powers of Iota are-

(i) $i=\sqrt{-1}$
(ii) $i^2=-1$
(iii) $i^3=-\sqrt{-1}=-i$
(iv) $i^4=1$

4. Write some applications of iota.

 Iota is generally used in complex numbers, AC analysis of electrical circuits, and solving quantum mechanics problems.

5. How can we simplify i raised to large powers?
To simplify large powers of i, divide the exponent by 4 and consider the remainder. The result will match the corresponding power in the basic cycle (i, -1, -i, 1).
6. How do we simplify expressions like (1+i)ⁿ?
To simplify (1+i)ⁿ, we can use the binomial theorem and then simplify using the properties of i. Alternatively, we can express 1+i in polar form and use De Moivre's theorem.
7. What is iota (i) in complex numbers?
Iota (i) is defined as the square root of -1. It's an imaginary number that forms the basis of complex numbers. When squared, i² = -1.
8. Why can't we simply calculate the square root of -1 using real numbers?
In the real number system, negative numbers don't have square roots because any number multiplied by itself gives a positive result. Iota (i) was introduced to extend our number system and allow for solutions to equations like x² = -1.
9. How does i relate to the complex plane?
In the complex plane, i represents a 90-degree rotation counterclockwise from the real axis. It forms the basis of the imaginary axis.
10. What's the difference between i and -i?
i and -i are opposite imaginary units. In the complex plane, i represents a 90-degree counterclockwise rotation, while -i represents a 90-degree clockwise rotation.
11. Why is i important in electrical engineering?
In electrical engineering, i is used to represent the 90-degree phase shift between voltage and current in alternating current (AC) circuits. It simplifies calculations involving impedance and power.
12. How do the powers of iota (i) cycle?
The powers of i follow a repeating pattern of four: i¹ = i, i² = -1, i³ = -i, i⁴ = 1. This cycle continues indefinitely for higher powers.
13. Why does i⁴ equal 1?
Since i² = -1, when we multiply i² by itself, we get (i²)² = (-1)(-1) = 1. Therefore, i⁴ = 1.
14. What is the value of i raised to a negative power?
When i is raised to a negative power, we can convert it to a positive power using the rule: i⁻ⁿ = 1/iⁿ. Then simplify using the cyclic pattern of i's powers.
15. Can iota have a square root?
Yes, iota has square roots. They are (1+i)/√2 and (-1-i)/√2. These are complex numbers that, when squared, equal i.
16. How do we multiply complex numbers involving i?
When multiplying complex numbers with i, treat i as a variable and simplify using i² = -1. For example, (2+3i)(4-i) = 8-2i+12i-3i² = 8+10i-3(-1) = 11+10i.
17. What is Euler's formula and how does it relate to i?
Euler's formula, e^(ix) = cos(x) + i*sin(x), relates exponential and trigonometric functions using i. It shows a deep connection between complex numbers and periodic functions.
18. How does i help in solving cubic equations?
Complex numbers, including i, are crucial in solving cubic equations, especially when using Cardano's formula. They allow for solutions even when all roots are real, passing through the "complex plane" in intermediate steps.
19. What happens when we add or subtract powers of i?
When adding or subtracting powers of i, we can't combine like terms unless the powers are the same. For example, i + i² = i - 1, but these terms can't be further simplified.
20. How do we rationalize denominators involving i?
To rationalize denominators with i, multiply both numerator and denominator by the conjugate of the denominator. For example, 1/(2+i) * (2-i)/(2-i) = (2-i)/(4+1) = (2-i)/5.
21. How do we interpret negative powers of i geometrically?
Negative powers of i represent clockwise rotations in the complex plane. For example, i⁻¹ = -i, which is a 90-degree clockwise rotation from the positive real axis.
22. What's the role of i in Fourier transforms?
i plays a crucial role in Fourier transforms, which decompose functions into sums of periodic components. The complex exponential e^(ix), which involves i, is fundamental to these transformations.
23. How do we interpret division by i?
Division by i is equivalent to multiplication by -i. This is because i * -i = 1, so -i is the reciprocal of i. Geometrically, it represents a 90-degree clockwise rotation.
24. How does i relate to hyperbolic functions?
i is related to hyperbolic functions through the identities: cos(ix) = cosh(x) and sin(ix) = i*sinh(x). These relationships connect trigonometric and hyperbolic functions in complex analysis.
25. How do we interpret complex powers of i, like i^i?
Complex powers of i, such as i^i, are multi-valued functions. They can be evaluated using Euler's formula and the properties of complex exponents, resulting in real-valued answers.
26. How do we interpret the absolute value of a complex number involving i?
The absolute value of a complex number a + bi is given by √(a² + b²). This represents the distance from the origin to the point (a, b) in the complex plane, regardless of the presence of i.
27. How do we interpret complex logarithms involving i?
Complex logarithms, like ln(i), are multi-valued functions. The principal value of ln(i) is iπ/2, but there are infinitely many other values differing by multiples of 2πi.
28. How do we interpret the complex exponential function e^(ix) geometrically?
e^(ix) represents a point on the unit circle in the complex plane. As x varies, this point moves around the circle, completing one full revolution when x increases by 2π.
29. What's the role of i in the theory of residues in complex analysis?
i is fundamental in the theory of residues, which is used to evaluate complex integrals. Residues are calculated using the coefficients of Laurent series expansions, which often involve powers of i.
30. What's the significance of i in the context of fractals and complex dynamics?
i plays a key role in the study of fractals and complex dynamics, such as in the Mandelbrot set. These structures arise from iterating complex functions, where i is essential in defining the complex plane.
31. What's the geometric interpretation of multiplying by i?
Multiplying a complex number by i is equivalent to rotating it 90 degrees counterclockwise in the complex plane. This is because i represents a quarter turn in this geometric interpretation.
32. How do we define complex conjugates using i?
For a complex number a + bi, its conjugate is a - bi. The i term changes sign. Conjugates are useful in various calculations, including division of complex numbers.
33. What's the relationship between i and the roots of unity?
The nth roots of unity are complex numbers that, when raised to the nth power, equal 1. i is related to the fourth roots of unity, as i⁴ = 1. Other roots of unity can be expressed using powers of i.
34. How does i relate to trigonometric functions?
i is closely related to trigonometric functions through Euler's formula: e^(ix) = cos(x) + i*sin(x). This relationship allows for powerful connections between complex analysis and trigonometry.
35. What's the significance of i in quantum mechanics?
In quantum mechanics, i appears in the Schrödinger equation and is crucial for describing the wave function of quantum systems. It's essential for representing the probabilistic nature of quantum states.
36. What's the difference between i and j in electrical engineering?
In electrical engineering, j is often used instead of i to represent the imaginary unit. This is to avoid confusion with i used for current. Mathematically, i and j are equivalent.
37. What's the connection between i and the Argand plane?
The Argand plane, also known as the complex plane, uses i as the basis for its vertical axis. Every point on this plane represents a complex number of the form a + bi.
38. How does i relate to the concept of orthogonality?
In the complex plane, i represents a direction perpendicular (orthogonal) to the real axis. This property is useful in various applications, including signal processing and vector analysis.
39. What's the significance of i in control theory?
In control theory, i is used in transfer functions and Bode plots to represent the frequency domain. It allows engineers to analyze system behavior in response to sinusoidal inputs.
40. What's the connection between i and the Riemann sphere?
The Riemann sphere is a way to visualize the extended complex plane, including infinity. i corresponds to a point on the equator of this sphere, representing the pure imaginary unit.
41. What's the role of i in solving differential equations?
i is crucial in solving many differential equations, especially those describing oscillatory systems. It allows for compact representation of solutions using complex exponentials.
42. How does i relate to the concept of phasors in AC circuit analysis?
In AC circuit analysis, phasors use i to represent the phase difference between voltage and current. This allows for simplified calculations using complex numbers instead of trigonometric functions.
43. What's the significance of i in the context of Euler's identity (e^(iπ) + 1 = 0)?
Euler's identity, often called the most beautiful equation in mathematics, connects five fundamental constants (e, i, π, 1, and 0) in a single equation. i plays a crucial role in this profound relationship.
44. What's the relationship between i and the roots of negative numbers?
i allows us to express the roots of negative numbers. For example, √(-a) = i√a for positive a. This extends the concept of roots to all real numbers.
45. How does i relate to the concept of analytic continuation?
i is crucial in analytic continuation, which extends the domain of a function. For example, the complex exponential function e^z (where z can involve i) is an analytic continuation of the real exponential function.
46. What's the significance of i in signal processing?
In signal processing, i is used to represent the imaginary part of complex-valued signals. This is particularly useful in analyzing frequency content and phase relationships.
47. What's the role of i in conformal mapping?
i is essential in conformal mapping, a technique in complex analysis. These mappings preserve angles and local shapes, and often involve functions of complex variables (using i).
48. How does i relate to the concept of impedance in electrical circuits?
Impedance in AC circuits is often expressed as a complex number Z = R + iX, where R is resistance and X is reactance. i allows for a compact representation of both magnitude and phase relationships.
49. What's the significance of i in quantum field theory?
In quantum field theory, i appears in the exponential of the action in Feynman path integrals. It's crucial for describing the quantum behavior of fields and particles.
50. How do we interpret the complex plane when dealing with functions of a complex variable?
When dealing with functions of a complex variable, we often need to consider two complex planes: one for the input (z-plane) and one for the output (w-plane). i is fundamental to both planes.
51. What's the role of i in the theory of special relativity?
In special relativity, i is used in the definition of four-vectors and the Minkowski metric. It allows for a compact representation of spacetime coordinates and transformations.
52. How does i relate to the concept of holomorphic functions?
Holomorphic functions are complex-differentiable functions. The Cauchy-Riemann equations, which involve i, are necessary conditions for a function to be holomorphic.
53. What's the significance of i in the context of Möbius transformations?
Möbius transformations, important in complex analysis and geometry, often involve i. These transformations preserve angles and map circles to circles (or lines) in the complex plane.
54. How does i relate to the concept of analytic signals in signal processing?
Analytic signals are complex-valued signals where the imaginary part is the Hilbert transform of the real part. i is crucial in representing these signals and in analyzing their properties.

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