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# Imaginary Number

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 Sub Topics In mathematics, there are many Numbers one of them is imaginary number. Imaginary number definition says that it is a type of number which is always less than or equal to zero whenever it is squared. Let iy, is an imaginary number when it is added to a real number, which is x, then their addition results into a Complex Number of the form x + iy, where x and y are known as the real part and the imaginary part of that complex number. Let us show you an example of imaginary number. If we have a negative number in the Square root then we cannot solve it until we remove the negative sign and we can do that but when we remove the negative sign we have to put a imaginary unit before the square root. So in imaginary numbers definition, we have an example: √ -36 which is an imaginary number and it’s square root is -36 but imaginary number can be written as a real number by putting an imaginary unit before the square root. In imaginary number, we put an imaginary unit because of the property i2 = -1. Imaginary number definition has one more important Point to share and that is a complex number can be equal to an imaginary number when the real part of the complex number is zero. Imaginary numbers definition says that imaginary number has many useful applications in complex number field. Imaginary number has many applications and they are: Imaginary number is useful in construction of non-real complex numbers, which plays a very important role in applications of science and technology and it has many other applications in control theory, fluid dynamics, quantum mechanics, vibration analysis, electromagnetism, signal processing and cartography. So, finally we have discussed all the points of imaginary numbers definition.

## Imaginary numbers rules

In simple mathematics, square root of a positive number is defined as a number that gives a specify number when multiplied by itself. A number can be positive or negative. If we want to find the Square root of a negative number then it can be calculated with the help of Complex Number.

Complex number is a Combination of two Numbers or parts, one is real number and the other is Imaginary Number. Here the concept of imaginary number is introduced. We represent the imaginary number with ‘i’ or sometimes with ‘j’. ‘I’ is another representation of √-1. With help of imaginary number rules you can easily solve several problems related to imaginary numbers or in order to operate imaginary numbers we should follow imaginary numbers rules:
Rule 1: Imaginary number is represented by the ‘I’ which is √-1.

Rule 2: We should have the knowledge of the power on imaginary number. Let us take an example to understand it better.
i0 = 1; i1 = i; i2 = -1; i3 =i.

Rule 3: Property of conjugation: If there is a number as x + iy then its conjugate will be x – iy.
For instance, if a number is 6 + 8i so its conjugate number will be 6 – 8i. -8 stands for the y.

Rule 4: Multiplication of two imaginary numbers: We can elaborate multiplication better with an example so let’s take an example.
(2-3i)*(4=5i)
· We have to multiply these two numbers so first we multiply the real part of first number with the real part of another number.
· Now multiply imaginary part with other number.
· Add the real parts and the imaginary part separately.
· Add these real and imaginary parts together.

Rule 5: Power of imaginary numbers:
We solve the power as we solve in Algebra. For any number of power (x+iy)n
(a+b)2 = a2 +b2+ 2ab

Rule 6: Division by x + iy
Let us understand it with an example, say we have to divide 4 – 3i by 3 + 2i
· First multiply the numerator and the denominator with the conjugate of denominator which is 3 -2i.
· When we multiply the denominator with its conjugate we get a real value (3 +2i)(3-2i)=9 +4=13.
· In numerator, we ignore the imaginary part after the multiplication.
· After multiplication we get the real values (12+6) and we divide it in order to get the divided value 18/13 which is the result.
These are some imaginary number rules.