4.1 Complex Numbers โ Introduction
The equation xยฒ + 1 = 0 has no real solution because xยฒ = โ1 is impossible for any real number (squares are always non-negative). To solve this, we extend the real number system by introducing the imaginary unit i.
i = โ(โ1) so that iยฒ = โ1
A complex number is any number of the form z = a + ib, where a and b are real numbers.
Re(z) = a (real part) | Im(z) = b (imaginary part)
Two complex numbers zโ = a + ib and zโ = c + id are equal if and only if a = c and b = d.
[ Image: Number system hierarchy: โ โ โค โ โ โ โ โ โ โ add here ]
Powers of i โ they cycle with period 4:
General rule: For any integer k โ iโดแต = 1, iโดแตโบยน = i, iโดแตโบยฒ = โ1, iโดแตโบยณ = โi
Square roots of negative numbers: โ(โa) = โa ยท i for positive real a. Note: โa ร โb โ โ(ab) when both a and b are negative.
๐ฏ Practice MCQs
iโปยณโต = 1/iยณโต. Now 35 = 4ร8 + 3, so iยณโต = iยณ = โi.
Therefore iโปยณโต = 1/(โi) = 1/(โi) ร (i/i) = i/(โiยฒ) = i/(โ(โ1)) = i.
Equating real parts: 4x = 3 โ x = 3/4.
Equating imaginary parts: 3x โ y = โ6 โ 3(3/4) โ y = โ6 โ 9/4 โ y = โ6 โ y = 9/4 + 6 = 33/4.
iโน = i^(4ร2+1) = iยน = i.
iยนโน = i^(4ร4+3) = iยณ = โi.
iโน + iยนโน = i + (โi) = 0.
4.2 Algebra of Complex Numbers
Let zโ = a + ib and zโ = c + id. The four arithmetic operations are defined as:
zโ + zโ = (a + c) + i(b + d)
zโ โ zโ = (a โ c) + i(b โ d)
Example: (2 + i3) + (โ6 + i5) = โ4 + i8
zโzโ = (ac โ bd) + i(ad + bc)
Tip: Just expand like (a + ib)(c + id) using FOIL and substitute iยฒ = โ1.
Example: (3 + i5)(2 + i6) = (6โ30) + i(18+10) = โ24 + i28
To compute zโ/zโ, multiply numerator and denominator by the conjugate of zโ:
zโ/zโ = (zโ ร zฬโ) / |zโ|ยฒ
Example: (6+3i)/(2โi) = (6+3i)(2+i)/[(2โi)(2+i)] = (12โ3+i(6+6))/5 = 9/5 + 12i/5
[ Image: Step-by-step division of complex numbers with conjugate method โ add here ]
Properties of Addition and Multiplication of Complex Numbers:
Closure, Commutative (zโ+zโ = zโ+zโ), Associative โ addition behaves like real numbers.
Closure, Commutative, Associative, Distributive. Multiplicative identity = 1 + 0i. Multiplicative inverse of z = zฬ/|z|ยฒ
Identities hold: (zโ+zโ)ยฒ = zโยฒ + 2zโzโ + zโยฒ, (zโโzโ)ยฒ = zโยฒ โ 2zโzโ + zโยฒ, zโยฒ โ zโยฒ = (zโ+zโ)(zโโzโ)
๐ฏ Practice MCQs
(3+i5)(2+i6) = (3ร2 โ 5ร6) + i(3ร6 + 5ร2)
= (6 โ 30) + i(18 + 10)
= โ24 + i28.
For z = 2โ3i: zฬ = 2+3i, |z|ยฒ = 4+9 = 13.
zโปยน = zฬ/|z|ยฒ = (2+3i)/13 = 2/13 + 3i/13.
(5โ3i)ยณ = 5ยณ โ 3ร5ยฒร(3i) + 3ร5ร(3i)ยฒ โ (3i)ยณ
= 125 โ 225i โ 135 + 27i
= (125โ135) + i(โ225+27) = โ10 โ 198i.
So a = โ10, b = โ198, a + b = โ208.
4.3 Modulus and Conjugate of a Complex Number
|z| = โ(aยฒ + bยฒ)
The modulus is always a non-negative real number. It represents the distance from the origin to the point (a, b) in the Argand plane.
Examples: |3 + i| = โ(9+1) = โ10 | |2 โ 5i| = โ(4+25) = โ29
zฬ = a โ ib (flip the sign of the imaginary part)
Examples: If z = 3 + i โ zฬ = 3 โ i | If z = 2 โ 5i โ zฬ = 2 + 5i
Key result: z ยท zฬ = |z|ยฒ
Multiplicative inverse: zโปยน = zฬ/|z|ยฒ
[ Image: z and zฬ as mirror images on the real axis in Argand plane โ add here ]
Properties for any two complex numbers zโ and zโ:
๐ฏ Practice MCQs
|z|ยฒ = aยฒ + bยฒ = 2ยฒ + (โ3)ยฒ = 4 + 9 = 13.
Note: |z| = โ13, but |z|ยฒ = 13.
If z = 63/25 โ 16i/25 (the computed value from the NCERT example), then zฬ = conjugate = flip sign of imaginary part = 63/25 + 16i/25... wait, actually the expression equals 63/25 โ 16i/25, so its conjugate is 63/25 + 16i/25. The conjugate asked for is of the expression value 63/25 โ 16i/25, giving zฬ = 63/25 + 16i/25. (Both B and the question relate to the conjugate operation โ conjugate flips the imaginary sign.)
Multiply by conjugate of denominator (1+โ2 i):
(5+โ2 i)(1+โ2 i) / [(1โโ2 i)(1+โ2 i)]
Denominator: 1 + 2 = 3
Numerator: (5โ2) + i(5โ2 + โ2) = 3 + 6โ2 i
z = (3 + 6โ2 i)/3 = 1 + 2โ2 i.
4.4 Argand Plane and Geometric Representation
Every complex number z = x + iy corresponds to a unique point P(x, y) in the XY-plane.
This plane (where each point represents a complex number) is called the Argand plane or complex plane.
x-axis = Real axis (points of the form a + 0ยทi)
y-axis = Imaginary axis (points of the form 0 + bi)
[ Image: Argand plane with points A(2,4)=2+4i, B(โ2,3), C(0,1), D(2,0), E(โ5,โ2), F(1,โ2) plotted โ add here ]
|z| = |x + iy| = โ(xยฒ + yยฒ) = distance of point P(x, y) from the origin O(0, 0).
The conjugate zฬ = x โ iy is represented by Q(x, โy), which is the mirror image of P(x, y) in the real axis (x-axis).
Key points to remember for the Argand plane:
Pure real numbers lie on the x-axis (imaginary part = 0)
Pure imaginary numbers lie on the y-axis (real part = 0)
z and zฬ are reflections of each other across the x-axis
|z| represents the distance from origin โ it is always โฅ 0
๐ฏ Practice MCQs
z = โ3 + 4i โ point P(โ3, 4). x = โ3 (negative), y = 4 (positive). A point with negative x and positive y lies in the Second Quadrant.
|(1+i)/(1โi)| = |1+i|/|1โi| = โ(1+1)/โ(1+1) = โ2/โ2 = 1.
This makes sense as both numerator and denominator have the same modulus โ2.
z = โ5 โ 3i โ P(โ5, โ3) in the Argand plane.
zฬ = โ5 + 3i โ Q(โ5, 3) โ which is the mirror image of P across the real axis. So zฬ is represented by Q(โ5, 3).
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