Value of i: The imaginary unit number is used to express the complex numbers, where i is defined as imaginary or unit imaginary. we will explain here imaginary numbers rules and chart, which are used in Mathematical calculations. The basic arithmetic operations on complex numbers can be done by calculators. The imaginary number i is also expressed as j sometimes. It is a solution to the quadratic equation or expression, x2+1 = 0, such as;
x2 = 0 – 1
x2 = -1
x = √-1
x = i
Therefore, an imaginary number is the complex number which we can write like a real number multiplied by the imaginary unit i, where i2 = -1. The imaginary number, when multiplied by itself, gives a negative value. For example, consider an imaginary number 3i, if multiplied by itself or if we take the square of 3i, gives 9i2 or we can write it as -9. Also, 0 is considered as both the real number and an imaginary number.
The general form of the complex number can be expressed as;
X + i Y
where X and Y are real numbers and i is an imaginary number.
Squaring of imaginary number i gives a negative value, but what if we square other numbers, like;
- Square of 1 = 12 = 1
- Square of -2 = (-2)2 = 4
- Square of 0.2 = (0.2)2 = 0.04
The above results do not give any negative number. Because as we know, square of any real number is a positive number. But, the square of the imaginary unit is always negative and represents a non-real or complex number. Now we multiply the imaginary unit ‘i’ multiple times, let us see what result we get;
We know, i2 = -1, let us calculate the value of ‘i’ raised to the power other numbers.
|i3||i2 * i||-i|
|i4||i2 * i2 = -1 * -1||1|
|i5||i2 * i2 * i||i|
|i6||i2 * i2* i2||−1|
|i0||i1-1 = i1.i-1 = i1/i = i/i =1||1|
|i−1||1/-i = -i/-i2 = -i/1||−i|
Now let us proceed to find the value of ‘i’.
What is the Value of i?
We have learned till now the imaginary unit ‘i’ is used to represent the complex numbers which are non-real. But if we have to represent the complex numbers into a complete real form, we have to know the value of imaginary number ‘i’.
Basically, first, it was considered as there are no such facts as the imaginary number and real numbers. But when the complex number came into the picture, real numbers and imaginary numbers were differentiated. Some of the examples of complex numbers are;
And there are a lot of examples.
But to solve such complex numbers, we have to know the value of i, which we can conclude from the expression, i2 = -1
If we take the square root on both sides, we get,
√i2 = √-1
Or i = √-1
Therefore, the value if i is √-1, which we cannot define, therefore, it is called an imaginary number.
Example: Solve the complex number √-4?
Solution: √-4 = √(-1 * 4) = 2√-1 = 2i
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