There are a lot of varieties of numbers in our number system, which helps us in day to day calculations. However, in certain calculations, you would require a number that can be both real and imaginary. That is precisely why you would be required to use complex numbers or the Value of i in Complex Numbers to be more specific. But first, you need to understand what complex numbers are and how to employ those in your calculations.
Concept Of Complex Numbers
In the first century, by the mathematical account of a Greek mathematician, the concept of complex numbers was first found. He was trying to find out the root value of a negative number by simply changing the numeric root value, but he entirely converted the negative value into a positive.
Later, by an Italian mathematician, the real value of complex numbers was found. However, mathematicians are still finding the negative root of the quadratic or cubic expression.
There are numerous applications of this concept. Scientists use complex numbers in their research, vibration analysis and quantum physics problems. Let’s look at what complex numbers are, how they are represented, their properties and different operations of the complex number.
Definition Of Complex Numbers
The real number and imaginary number, when combined together, give us the complex number. It is the sum of real and imaginary numbers.
Complex numbers are usually represented in the form of z = x + iy. Here, z denotes the sum of the value of ‘ x ’, a real part, and the value of ‘ y ’, an imaginary part.
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The ‘ i ‘ in the equation is an imaginary number called iota. All the non-real values are denoted by an iota, i.e., ‘ i ‘. You will come to know more about iota further in the article.
Some Examples Of Complex Numbers
- 12+i5, here, 12 is a real part, and 5 is an imaginary part. Together, i5 is the imaginary number.
- 1- i
- -3+7i, here, -3 is the real number, and 7i is an imaginary number.
- 2+⅓ i
- – ¼ – ⅔i, in this example, -¼ is a real number, and -⅔ is an imaginary number.
Value Of i
Iota is often used to find the value of imaginary numbers, referred to as i. To find the square root of a negative number, iota ( i) is used. To find roots, we must know the exact values of i.
The value of i2 is always -1, which means the value of i2 always gives us the negative number. Let me explain this to you with an example.
What would you do if you were asked to find the value of the square root of -8? See, it’s very simple, and this is how it is done.
√-8 = √i2 8 = 4i
Moreover, 0 is the real number, and we know that the complex number includes all the real numbers. So, 0 is also a complex number. But you must be thinking about how 0 could be represented in a complex number when both the real and imaginary part is zero?
Therefore, you should know that the complex number 0 is represented in the same way as other complex numbers in form or a + ib, just the value of the expression is zero.
z=a + ib
z=0 + i0
Some Other Powers Of i
i0 = 1
i = √-1
i2 = -1
i3 = -1 x i = -i or -1 x √-1 = -1
i-1= -i or -√-1
Tricks To Solve The Advanced Value Of Iota
i4 = (i2)2 = (-1)2 = 1
i4n = 1
i4n+1 = i = √-1
i-1= -i or -√-1
You can easily find the higher values of i in any expression by using the above formulas. To find the exact value, just solve the values by putting n at 0,1,2 and 3.
- i4n+5 = ?
Let n = 0
I4x0+5 = i5
i= i2 xi 3 = -1 x -1= 1
I4x1+5 = i9
i9= i4x i5 = 1 x 1 = 1
The fundamental aspect of mathematics is a conceptually based study and is understood through various applications.
Therefore, you should not memorise the values but try to understand the concept and how the values are derived. You must go through some more examples of complex numbers to understand the concept in a better way.
Question 1: find the value of 2 + √-5.
Solution: 2 + √-5, in this 2 is a real part, and √-5 is an imaginary part.
By substituting the value of i = √-1 in the expression, we will get
=2 + √-1 x 5 = 2+ i5
This is how we solve the complex number into the simplified equation.
Question 2: find the value of i-39?
Solution: i-39 = i-4×9-3
= (i-4)9 x i-3
= (1)-9x i-3
= 1/i3 = 1/-i
= -1/i x i/i = -i/i2 = -i/-1 = i
Various Properties Of Complex Numbers
Properties of complex numbers will help you understand the concept better; also, you will come to know how to perform different operations on complex numbers.
When a sum and product of two complex numbers come to be a real number, they are known as complex conjugate numbers.
Forming a conjugate of complex numbers is a simple task. Just take a real number and add its additive inverse multiplied with ‘i’ to the expression.
For example – z = x + iy
z = x – iy
If you need to divide the complex number with another number, you must find its reciprocal first.
Some other properties are
- When x + iy = 0, where x and y are the real numbers, then x = 0 and y = 0
- Complex numbers follow the rules of associative and commutative law of addition and multiplication. It also follows the distributive law.
- When a + ib = c + id, where a,b,c and d are real numbers, then a = c and b= d.
You should never think of complex numbers as either real or imaginary, as most students often get confused between the two. A complex number combines both the real and imaginary values and represents both real and imaginary numbers.
So, if you are ever in a situation where you need to represent both the values, then only you should use complex numbers. And to represent the value of imaginary numbers, always use iota with the imaginary part of the complex number.