Power(n)	Expression	Value
-3	1/i3 = 1/(-1 × i) = 1/-i = i	i
-2	1/i2 = 1/-1 = -1	-1
-1	1/i = -i	-i
0	i0 = 1	1
1	i	i
2	i2 = -1	-1
3	i3 = -i	-i
4	i4 = i2 × i2 = -1 × -1 = 1	1
5	i5 = i2 × i2 × i = -1 × -1 × i = i	i
6	i6= i2 × i2 × i2 = -1 × -1 × -1 = -1	-1

General Results

From the above table, we can generalize that i is a cyclic expression that repeats its value after four intervals which can be given by

• i⁴ⁿ = 1
• i⁴ⁿ⁺¹ = i
• i⁴ⁿ⁺² = -1
• i⁴ⁿ⁺³ = -i

Properties of i

Let us take a look at some properties of i

• The square of i i.e. i² is equal to -1 therefore, the value of i is √-1.
• i and -i can be described as the square roots of the number 1.
• The conjugate of i is -i and both have the same absolute value of 1.
• The powers of i form a cyclic pattern of four values namely i, -i, 1, and -1.
• i is one of the complex fourth roots of unity others being -1, 1, and -i.
• When plotted on a complex plane, i can be plotted at point (1, 0) on the x-axis.

Related Articles:

• Real Numbers
• Cube of an Imaginary Number
• Additive Identity and Inverse of Complex Numbers
• Modulus of Complex Numbers
• Polar and Exponential form of complex numbers

Solved Examples of the Value of i

Example 1: Evaluate the value of expression √-6

Solution:

We have √-6

√-6 =√-1 · √6

Since √-1 = i

∴ √-6 = i√6

Example 2: Find the value of √49 + √-8

Solution:

√-8 = √-1 .√8

since √-1 = i and √8 = 2√2

∴ √-8 = 2i√2

Also we know √49 = 7

Hence, √49 + √-8 = 7+ 2i√2

Example 3: Find the value of x+y if x= √49 + √-8 and y=-2i√2.

Solution:

√-8 = √-1 .√8

since √-1 = i and √8 = 2√2

∴ √-8 = 2i√2

∴ √49 + √-8 = 7+ 2i√2 ( As √49 = 7)

∴x = 7+ 2i√2

Given y= -2i√2

∴ x + y = 7+ 2i√2 -2i√2

∴ x + y = 7

Example 4: Find the quadrant where x = √36 + √-4 lies on a complex plane.

Solution:

x=√36 + √-4

∴x = 6 + 2i (As √36 = 6 and √-4 = 2i

∴ real part = 6 and imaginary part = 2

Since both the real and imaginary parts are positive, x lies in the 1^st quadrant.

Example 5: Find the value of x × y if x = √49 + √-8 and y =√49 – √-8.

Solution:

√-8 =√-1 .√8

since √-1 = i

∴√-8 = 2i√2

∴√49 + √-8 = 7+ 2i√2

∴ x = 7+ 2i√2

Similarly, y = 7- 2i√2

x × y = (7+ 2i√2) × ( 7- 2i√2)

∴ x × y = 49 – ( 2i√2)²

∴x × y = 49 – (-8)(As (2√2)² = 8 and i² = -1)

∴x × y = 49 + 8 = 57

Conclusion

The value of i is a fundamental concept in mathematics that extends our understanding of numbers beyond the real number line. Defined as [Tex]\sqrt{-1}[/Tex]​, the imaginary unit i is essential in representing complex numbers and exhibits a cyclic pattern in its powers. By exploring its properties, geometrical interpretation, and practical applications, we see how i plays a crucial role in various fields such as electrical engineering, quantum mechanics, and complex number theory. This guide helps demystify i, providing a solid foundation for further mathematical studies.

FAQs on Value of i

What is the magnitude of a complex number a+ib?

The magnitude of any complex number a+ib is given by √a²+b²

What will be the value of i⁵⁸?

From the general formula of powers i⁵⁸=i^4.14+2=i² which is equal to -1 therefore, i⁵⁸=-1

In which quadrant will the number 6-8i lie in the complex plane?

The value 6-8i lies in the 4^th quadrant.

In which quadrant will the number 5-6-8i lie in the complex plane?

The value 5-6-8i lies is equal to -1-8i therefore, it lies in the third quadrant.

How can we represent real numbers using i?

Any number can be represented by z=x+iy since the imaginary part for real numbers will be 0 therefore, y=0. We can write real numbers as z=x+i.0

What are the absolute values of-2+i and 2+i?

If we calculate the magnitude -2+i =√2²+1²=√5 and the magnitude 2+i =√2²+1²=√5 therefore, the absolute value of -2+i and 2+i is same i.e. √5