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Complex Numbers - Introduction

Let us try and solve the equation x2+1 =0. We  can simplify it and write it as x2=-1 or x = ± √−1.

But what is the root of -1? 

It is not on the number line. 

So does it simply not exist? 

Well in mathematics, sticking to reality has never been a priority! 

  • This solution does not lie on the number line, that means it must be off it then.
  • We can write -a = (-1)×(a), where ‘a’ is a real number. 
  • The square root of a negative number can be written as Basics of Complex Numbers | Algebra - Mathematics
  • Let us use a symbol for √−1. Let’s denote it by ‘i’ ( from iota, Greek for extremely small). 
  • Then i = √−1 is a number that doesn’t fall anywhere on the number line! 
  • Such numbers that are not on the real number line, are the imaginary numbers. They are also known as the complex numbers.
    Basics of Complex Numbers | Algebra - Mathematics

“Now wait for a second, we only have ‘i’, why are you calling it numbers? Where are the others?” 

We can get every real number from other real numbers by certain algebraic operations. 

  • Like if you keep on adding the number one to itself, you will get the set of natural numbers and so on.
  • Similarly, all the numbers that have ‘i’ in them are the imaginary numbers. 
  • The number i, is the imaginary unit. 3i, 4i, -i, √−9 etc. are all imaginary numbers.

 

Question for Basics of Complex Numbers
Try yourself:Are complex numbers on number line?
View Solution

Let us have a general definition of the imaginary or complex numbers.

Why So Complex!

Let ‘a’ , ‘b’ be two real numbers. Let i = √−1, then any number of the form a + ib is a complex number.

The reason to define a complex number in this way is to make a connection between the real numbers and the complex ones. For example, we can write, 2 = 2 + 0.i.  Therefore, every real number can be written in the form of a + ib; where b = 0.

Basics of Complex Numbers | Algebra - Mathematics

Also if a complex number is such that a = 0, we call it a purely imaginary number. In general, a is known as the “real” part and b is known as the “imaginary” or the complex part of the imaginary number. 

“Why am I doing this again?” Well, hold on. 

Let’s recall the equation that we couldn’t quite figure out the solution for i.e. x2+1=0.

  • Hence, x = ± √−1 or x = ± i. Substituting for x, in the equation we have (i)2+1=0
    i2 =±[√−1]2=-1, using the above equation we have, (-1)+1=0 satisfies the equation. 
  • Substitution of -i also gives the same results. So there you go, now you can solve equations that you would have rather just left alone. 

To sum up, all the numbers of the form a +ib, where a and b are real and i =√−1 , are called imaginary numbers. We usually denote an imaginary number by ‘z’.

Question for Basics of Complex Numbers
Try yourself:What is i called in complex numbers?
View Solution

Equality of Complex Numbers

Let us practice the concepts we have read this far.
Example 1: There are two numbers z1 = x + iy and z2 = 3 – i7. Find the value of x and y for z1 = z2. What is the sum of Re (z1, z2)?
Solution: We have z1 = x + iy and z2 = 3 – i7.

  • First of all, real part of any complex number (a+ib) is represented as Re(a + ib) = a and imaginary part of (a +ib) is represented as Im(a+ib) = b.
  • Also, when two complex numbers are equal, their corresponding real parts and imaginary parts must be equal. Therefore, if a + ib = c + id, then Re(a+ib) = Re(c+id) and Im(a+ib) = Im(c+id)
  • Conversely if two complex numbers say z1 and z2, are such that Re(z1) = Re(z2) and Im(z1) = Im(z2), then z1=z2. In other words, we must have a = c and b = d. We can’t have a = d because we can’t relate the real and the imaginary parts together. Let us come back to our problem.
    z= z2 or x + iy = 3 – i7
    Re(x + iy) = x              Re(3 – i7) = 3
    Im(x + iy) = y              Im(3 – i7) = -7
  • This means x = 3 and y = -7. Therefore, Re(z1,z2) = 3,3 and hence, the sum of real parts of z1 and z2 = 3 + 3 = 6.
  • Integer Powers of  ‘i’ - As we saw already, i2 = -1. Also, i3 = i2(i) = -i. Similarly, you can find any given power of i, by reducing it to the above two forms.

More Solved Examples For You

Example 2: If 2i2+6i3+3i16−6i19+4i25=x+iy, then (x,y)= ?

A) (1 , 4)

B) (4 , 1)

C) (-1 , 4)

D) (-1 , -4)

Solution: (A) The above expression can be simplified as:

2 (-1) + 6 (-i) + 3 ( i )2×2×2×2 – 6 (i18 . i) + 4 (i24.i) = x + iy

Therefore, -2 – 6i + 3 (1) – 6 (-1)9.i +  4i = x + iy Or 1 + 4i = x + iy

Therefore, x = 1 and y = 4 or (x,y) = (1,4)

Example 3: Perform the indicated operation and write your answer in standard form. 

(4−5i)(12+11i)

Solution: We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is “foil” the two complex numbers to get, 

                                  (4−5i)(12+11i) = 48+44i−60i−55i= 48−16i−55i2

All we need to do to finish the problem is to recall that i2=−1. Upon using this fact we can finish the problem.

                                  (4−5i)(12+11i)=48−16i−55(−1)=103−16i

The document Basics of Complex Numbers | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Basics of Complex Numbers - Algebra - Mathematics

1. What are complex numbers?
Ans. Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (i.e., the square root of -1). They consist of a real part (a) and an imaginary part (bi).
2. How are complex numbers represented graphically?
Ans. Complex numbers can be represented graphically on a coordinate plane known as the complex plane. The real part of the complex number represents the x-coordinate, and the imaginary part represents the y-coordinate. The complex number a + bi is plotted as the point (a, b) in the complex plane.
3. What does it mean for two complex numbers to be equal?
Ans. Two complex numbers a + bi and c + di are equal if and only if their real parts (a and c) are equal and their imaginary parts (b and d) are equal. In other words, a + bi = c + di if and only if a = c and b = d.
4. How can complex numbers be added or subtracted?
Ans. To add or subtract complex numbers, we simply add or subtract their real parts separately and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.
5. How can complex numbers be multiplied or divided?
Ans. To multiply complex numbers, we use the distributive property and the fact that i^2 = -1. For example, (a + bi)(c + di) = (ac - bd) + (ad + bc)i. To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator, which involves changing the sign of the imaginary part.
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