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Complex Numbers Complex numbers is an important topic of IIT JEE Mathematics as it fetches several questions in the exam. Locus based approach of complex numbers is one of the several mathematical tools which can be usefully employed in solving problems of coordinate geometry. This chapter discusses the operations of complex numbers and the application of properties such as multiplication of two complex numbers. The concept of rotation has been dealt with adequate number of tricks of different levels. This has been expressed in detail with ample problems and by illustrations. The application of properties of modulus and conjugate in solving problems has been dealt with in a unique way and discussed by illustrations in depth. What exactly do we mean by complex numbers? A complex number, generally represented by the number z is the number of the form z = a+ib, where a and b are real numbers and ‘i’ is the imaginary number iota satisfying the condition i^{2} = -1. Example: 3-4i is a complex number where 3 is called the real part of the complex number while 4 is the imaginary part. Equality of two complex numbers Two complex numbers say z_{1} and z_{2} are said to be equal if their corresponding real and imaginary parts are equal. For example: If z_{1} = a+ib and z_{2} = c+id then z_{1} = z_{2} implies that a = c and b = d. Conjugate of a complex number If we have a complex number z = a+ib, then its conjugate is denoted by or z* and is equal to a-ib. In fact, for any complex number z, its conjugate is given by z* = Re(z) – Im(z). View the video on complex numbers Geometrically, the reflection of a complex number may be thought of as a reflection about the real axis. As a result, if we multiply the conjugate with itself i.e. conjugate of a conjugate is the number itself. Hence Absolute value or Modulus of a Complex Number: The absolute value of a complex number z = a+ib is defined by |z| = √x^{2}+y^{2}. The absolute basically signifies the magnitude of the complex number. Argument of a complex number Argument of a complex number is a quite tricky but an important concept of complex numbers. The argument of the complex number is the magnitude of the angle which it makes with the positive direction of x-axis. Given the complex number z = x+iy, the below list shows the value of argument according to the positive or negative real and imaginary parts: Operations on Complex Numbers: Addition of complex numbers: In order to add two complex numbers, the corresponding real and imaginary parts get added to each other. To add two complex numbers we add each element separately: (a+bi) + (c+di) = (a+c) + (b+d)i Example: (3 + 2i) + (1 + 7i) = (4 + 9i) Multiplication of complex numbers: The multiplication of two complex numbers is defined by the formula: (a+ib)(c+id) = (ac-bd)+ i(ad + bc), using the value of i^{2} = -1. Division of complex numbers: The division of two complex numbers is performed by first multiplying the term by the conjugate of denominator and then simplifying it further. For example: if we wish to perform the following division (a+ib)/(c+id), then we first multiply both the numerator and denominator by the (c-id) i.e. the conjugate of the denominator and then simplify. Let us have a look at the types of questions asked in the exam from this topic: Illustration: Let a and b be roots of the equation x^{2} + x + 1 =0. Then find the equation whose roots are a^{19 }and b^{7}. Solution: Given that x^{2} + x + 1 =0. This means that either x = ω or x = ω^{2}. Hence, a^{19} = ω^{19} and b^{7} = ω^{14} = ω^{2} Hence, the equation becomes x^{2} – (ω+ω^{2})x + ω ω^{2} = 0. Hence the required equation is x^{2} + x + 1 =0. Illustration: If ω≠ 1 is a cube root of unity and (1+ ω)^{7} = A + Bω then find the values of A and B. Solution: (1+ ω)^{7} = (1+ ω)(1+ ω)^{6} = (1+ ω) (-ω^{2})^{6} = (1+ ω) Hence, A + Bω = (1+ ω) So, A =1 and B = 1. Complex Numbers is important from the perspective of scoring high in IIT JEE as there are few fixed pattern on which a number Multiple Choice Questions are framed on this topic. Ask your questions and get answers form our IIT experts. Related resources:
Complex numbers is an important topic of IIT JEE Mathematics as it fetches several questions in the exam. Locus based approach of complex numbers is one of the several mathematical tools which can be usefully employed in solving problems of coordinate geometry. This chapter discusses the operations of complex numbers and the application of properties such as multiplication of two complex numbers. The concept of rotation has been dealt with adequate number of tricks of different levels. This has been expressed in detail with ample problems and by illustrations. The application of properties of modulus and conjugate in solving problems has been dealt with in a unique way and discussed by illustrations in depth. What exactly do we mean by complex numbers?
A complex number, generally represented by the number z is the number of the form z = a+ib, where a and b are real numbers and ‘i’ is the imaginary number iota satisfying the condition i^{2} = -1.
Example: 3-4i is a complex number where 3 is called the real part of the complex number while 4 is the imaginary part. Equality of two complex numbers
Two complex numbers say z_{1} and z_{2} are said to be equal if their corresponding real and imaginary parts are equal. For example:
If z_{1} = a+ib and z_{2} = c+id then z_{1} = z_{2} implies that a = c and b = d.
Conjugate of a complex number
If we have a complex number z = a+ib, then its conjugate is denoted by or z* and is equal to a-ib. In fact, for any complex number z, its conjugate is given by
z* = Re(z) – Im(z). View the video on complex numbers
Geometrically, the reflection of a complex number may be thought of as a reflection about the real axis. As a result, if we multiply the conjugate with itself i.e. conjugate of a conjugate is the number itself. Hence Absolute value or Modulus of a Complex Number:
The absolute value of a complex number z = a+ib is defined by |z| = √x^{2}+y^{2}. The absolute basically signifies the magnitude of the complex number.
Argument of a complex number
Argument of a complex number is a quite tricky but an important concept of complex numbers. The argument of the complex number is the magnitude of the angle which it makes with the positive direction of x-axis. Given the complex number z = x+iy, the below list shows the value of argument according to the positive or negative real and imaginary parts:
Operations on Complex Numbers:
Addition of complex numbers:
In order to add two complex numbers, the corresponding real and imaginary parts get added to each other.
To add two complex numbers we add each element separately:
(a+bi) + (c+di) = (a+c) + (b+d)i
Example: (3 + 2i) + (1 + 7i) = (4 + 9i)
Multiplication of complex numbers:
The multiplication of two complex numbers is defined by the formula:
(a+ib)(c+id) = (ac-bd)+ i(ad + bc), using the value of i^{2} = -1.
Division of complex numbers:
The division of two complex numbers is performed by first multiplying the term by the conjugate of denominator and then simplifying it further. For example: if we wish to perform the following division (a+ib)/(c+id), then we first multiply both the numerator and denominator by the (c-id) i.e. the conjugate of the denominator and then simplify.
Let us have a look at the types of questions asked in the exam from this topic:
Illustration: Let a and b be roots of the equation
x^{2} + x + 1 =0.
Then find the equation whose roots are a^{19 }and b^{7}.
Solution: Given that x^{2} + x + 1 =0.
This means that either x = ω or x = ω^{2}.
Hence, a^{19} = ω^{19} and b^{7} = ω^{14} = ω^{2}
Hence, the equation becomes x^{2} – (ω+ω^{2})x + ω ω^{2} = 0.
Hence the required equation is x^{2} + x + 1 =0.
Illustration: If ω≠ 1 is a cube root of unity and (1+ ω)^{7} = A + Bω then find the values of A and B.
Solution: (1+ ω)^{7} = (1+ ω)(1+ ω)^{6}
= (1+ ω) (-ω^{2})^{6}
= (1+ ω)
Hence, A + Bω = (1+ ω)
So, A =1 and B = 1.
Complex Numbers is important from the perspective of scoring high in IIT JEE as there are few fixed pattern on which a number Multiple Choice Questions are framed on this topic. Ask your questions and get answers form our IIT experts.
Related resources:
Click here for the Detailed Syllabus of IIT JEE Mathematics.
Look into the Previous Year Papers with Solutions to get a hint of the kinds of questions asked in the exam.
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