The Value of i
Complex numbers are some of the most interesting and fun parts of mathematics. It is essential to know the basics of the number system in order to understand complex numbers properly. This article has been written in order to teach you about the value of i. There is a common notion that complex numbers are tough to understand and very confusing. This is far from the truth because we can guarantee that complex numbers are very fun and easy to understand.
The real number is denoted by a real number and the imaginary part of the complex number is denoted with the alphabet i. This article is about the same i. It tells you what the exact value of i is and if it has an exact measurable value or not. The addition of the imaginary part to the number makes a number complex. It is a very simple topic that requires some real good focus from the very start, so sit back, relax and enjoy this fun filled article from Vedantu on the value of i.
The value i or the concept of i is used in explaining and expressing complex numbers. Complex numbers are numbers with a real and imaginary part. The imaginary part is defined with the help of i. Basically, “i” is the imaginary part which is also called iota.
Value of i is √-1 A negative value inside a square root signifies an imaginary value. All the basic arithmetic operators are applicable to imaginary numbers. On squaring an imaginary number, we obtain a negative value.
In the study of complex numbers, “i” holds significant importance. We will try to understand the complexities revolving around the value of i with the help of some examples. We will also discuss higher degrees of “i” in detail. With a better understanding of these concepts, students can solve complex number problems with more ease.
What is the Value of i?
A statement was proposed at the start of this article. The statement was about the exact value or measurement of what i stands for. This part of the article deals with it and explains if it has a specific value or not.
The imaginary part of a complex number is defined as ‘iota’. To calculate the value of an imaginary number, we use the notation iota or “i”. The square root of a negative number gives us an imaginary number.
Value of i = √-1
We use this value of i in understanding the concepts of complex numbers.
For a quadratic equation, x2 + 1 = 0
x2 = -1
x = √-1
Here √-1 is the imaginary part.
i = √-1
i2 = -1
As discussed earlier, the squaring of an imaginary number gives us a negative value.
i = √-1
Squaring on both sides
i2 = -1
Any complex number is the combination of a real number and an imaginary number. All non-real values are represented by iota or “i”.
Examples of real numbers are 10, -20, √3 etc.
Examples of imaginary numbers are 2i, √-5, -i etc.
Complex numbers are of the form a + ib, where i signifies the imaginary part. Zero is also a complex number. In a complex number, only the real part can be added or subtracted to the real part and only the imaginary part can be added or subtracted with the imaginary part.
The Value of i
Now we will try to understand the value of i in a complex number. For any complex number,
a + ib, a and b are real numbers, where i signifies the imaginary part. Value of i squared
i2 = -1, gives us a negative number.
On multiplying a negative integer to this value we get,
-i2 = 1
If we multiply i to i2,we get
i x i2 = i(-1)= -i
On further multiplication, we will get
-i x i = -i2 = -(-1)=1
Given below is a table with the commonly used values of i. Students can refer to these values while solving complex numbers problems.
Values of i
Memorising all these values can be confusing and tiring! There’s a trick to solving higher degrees of i. The values of i follow a circular pattern. Let’s look at that pattern:
i4n = 1
i4n+1 = i
i4n+2 = -1
i4n+3 = -i
Using this circle of formulas, students can calculate the values of i.
For example, at n = 0
i0 = 1
i1 = i
i2 = -1
i3 = -i
At n = 1,
i5 = 1
i6 = i
i7 = -1
i8 = -i
At n = 2,
i9 = 1
i10 = i
i11 = -1
i12 = -i
The value of higher degrees of i follows this circular formula. Mathematics is a conceptual and application-based study. It doesn’t promote the learning of values. Getting in tune with the concepts will help you understand mathematical applications better.
Complex numbers is a fundamental topic of study. The concept of imaginary numbers is a significant part of it. Students can clear their concepts further by going through these solved examples.
Find the value of 1 + √-3.
Ans: 1 + √-3 is a complex number with a real and imaginary part.
We know that,
√-1 = i, Substituting this value, we get
1 + 3i
It is the simplified form of the equation.
Conclusion
We hope that this article was able to explain the details of what the value of i is and made the topic of complex numbers easy for you. The goal of the article was to offer a different approach to the topic of complex numbers. An approach that can give you a very distinct and unique take on the topic. Vedantu strives to bring nothing but the best for its students and the article on the value of i was another attempt of ours to do the same.
FAQs on Value of i
1. Define the Value of i Power i.
“i” is an imaginary number, but an imaginary number raised to the power of an imaginary number turns out to be a real number.The value of i is √-1.The value of ii is a real number with its value 0.207 approximately. The value of i to the power i is ii ≃ 0.20788.Let’s calculate this value mathematically. To calculate the value of i, we will need to understand Euler’s formula first.According to Euler’s formula,eix = cosx + isinxAt x = π/2x = π/2cis x = cos π/2 + i sin π/2Here, cis x is just a notation.Cis x = iSo,ii = (cis π/2)iii = ei2 π/2ii = e-π/2ii=0.20788 approximately
2. Solve the following quadratic equation x² + 25 = 0.
To solve the following quadratic equation, we will have to use the concepts revolving around complex numbers. Here, we will use the treatment of imaginary values to obtain the solution of the equation. x² + 25 = 0x2= -25x = √-25x = i√25 As the value of i = √-1. So, x=i(5) Therefore, x = 5i will satisfy the equation. Let’s put this value of x in the equation x² + 25 = 0(5i)2 + 25 = 025i² + 25 = 0 The value of i² = -1. On substituting this value in the equation, we get25(-1) + 25 = 0-25 + 25 = 00 = 0 L.H.S = R.H.S, hence proved.
3. How do I find the values of higher degrees of i?
One thing that is clear right from the start is that you cannot expect yourself to memorize all the values of higher degrees of i. Therefore, it is essential to make sure that there is a way for you to find out these values. After reading the article thoroughly you can understand that the values of i follow a circular pattern. Go through the examples in the article to fully understand how to find the values of higher degrees of i. The chapter of Complex numbers is not the most difficult one that you would encounter. It is definitely one of the most essential ones with respect to the exam and we highly suggest you solve as many problems as possible so that you get a good hold of the concept. There is no doubt that the chapter plays a major role in the chapters coming in the future as well and that is why complex numbers is not a chapter that you must ignore but should make the most out of!
