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CBSE Class 11 · Mathematics

Complex Numbers and Quadratic Equations

Official NCERT chapter from Mathematics (book code kemh1). ExamMaster notes are original teaching at CBSE Class 11 depth.

This lesson follows the official chapter “Complex Numbers and Quadratic Equations” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 11
  • Medium level
  • 8 concepts

1Complex Numbers

A complex number is a + bi with a, b real and i² = −1. 3 + 2i is one; 3 is 3 + 0i. Complex is this pair, not “imaginary” as a mood. Two complexes are equal when real parts match and imaginary parts match.

Writing √(−4) as −2 without i is a miss if the lesson used 2i.

Figure. A real number sits on the real axis. A complex number 3+2i is the same real part with a height 2 on the imaginary axis. The dashed drop is the real part, not the number itself.

How it works

  1. Write a + biThe form.
  2. Name a as real part, b as imaginary coefficientThe pair.
  3. Keep i² = −1The engine.

2Algebra of Complex Numbers

Algebra: add real with real, imaginary with imaginary; multiply by expanding and replacing i² with −1. (3+2i)+(1−5i)=4−3i. (3+2i)(1−i)=3−3i+2i−2i²=5−i. Algebra is the same + and × laws, plus that replace.

Adding 3+2i as 5i is a merge-steal.

Figure. Addition is the parallelogram rule on the Argand plane: z_1=3+i and z_2=1+2i meet at z_1+z_2=4+3i. Real parts add; imaginary parts add. The dashed sides complete the parallelogram.

How it works

  1. Add parts separately, or expand the productThe algebra.
  2. Replace i² by −1The reduce.
  3. Write a + bi againThe result.

(3+2i)(1−i)

Multiply and write in a+bi form.

  • Expand3 − 3i + 2i − 2i²
  • i²−2(−1)=+2
  • Sum5 − i

Pro tip. Expand, replace i², collect.

3The Modulus and the Conjugate of a Complex Number

The modulus |z| is the length √(a²+b²); the conjugate z̄ is a − bi. |3+4i|=5; conjugate 3−4i. z z̄ = |z|². Modulus is a length, not the imaginary part.

Calling 4 the modulus of 3+4i is the part-steal.

Figure. The conjugate \bar z is the reflection of z across the real axis. The modulus |z| is the length of the segment from O to z, not a circle. Here z=4+3i and \bar z=4-3i share that length 5.

How it works

  1. Write a and bThe parts.
  2. |z|=√(a²+b²), z̄=a−biThe pair of jobs.
  3. Check z z̄ = |z|²The link.

4Argand Plane and Polar Representation

The Argand plane plots z as (a, b). Polar form is r(cos θ + i sin θ) with r = |z| and θ an argument as taught. 3+0i sits on the real axis. Polar is a length-plus-turn, not a second kind of number.

A sketch with no scale is not r.

Figure. On the Argand plane P is x+iy. Polar form uses the same point: r=|z| is the ray OP, and \theta is the angle that ray makes with the positive real axis. The dashed drops are the Cartesian parts x and y. No arc is drawn; \theta sits in the wedge.

How it works

  1. Plot (a, b)The point.
  2. Read r and a taught θPolar.
  3. Keep r(cos θ + i sin θ) as the same zThe representation.

5A definition is a test you can run

Complex is a test: a+bi, or a point, or r and θ. If you only say “imaginary”, you have a heading.

A dream-poster is not the test.

Figure. The defining test for i is i^2=-1. On the Argand plane that is two true quarter-turns: 1 along the real axis, then i straight up (a square marks the right angle), then -1 on the negative real axis. i is not an unknown to solve for.

How it works

  1. Write a+bi or (a,b) or r,θThe object.
  2. Give the algebra or the lengthThe test.
  3. Then the word has contentThe definition ran.

6Name the given before the unknown

The given is a, b (or r, θ). The unknown is the product, |z|, or z̄. Copy the sign of b before you conjugate.

Using 2i as if i² were still written is a silent leftover.

Figure. Name the given first: real part a=3 and imaginary part b=2 are the two legs. Only then is |z|=\sqrt{a^2+b^2} the unknown hypotenuse. Do not start by guessing a modulus.

How it works

  1. Copy a and bThe given.
  2. Name product, modulus, or conjugateThe unknown.
  3. Then expand or rootGiven first.

7One worked case is enough at this class

One product or one |z| is enough. A page of the same 5-clone does not add a new Argand point.

Ten twins of 3+4i are still one idea.

Figure. Worked point z=4+3i. The legs are 4 and 3; the hypotenuse is |z|=5. Check: 4^2+3^2=16+9=25. The same triangle gives \bar z=4-3i by flipping the 3 down.

How it works

  1. Run one multiply or one modulusThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

8A miss: swapping the school name for the picture

A map-pin photo is a setting. The maths is (a, b) or a+bi. A city is not i.

A compass rose is not arg z.

Figure. The miss is swapping the school name i for every picture of a point off the real axis. i is the unit on the imaginary axis. The point shown is 3+2i — three real units and two imaginary units, not i itself.

How it works

  1. Keep the pin as a storyA setting.
  2. Write a+bi and plot or multiplyThe maths.
  3. Then |z| or z̄Do not swap.
|3+4i| is
  1. 5
  2. 4
  3. 7

√(9+16)=5, not 3+4.

Notes

  • The official chapter title is “Complex Numbers and Quadratic Equations”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • i²=−1
  • |z|=√(a²+b²)
  • z z̄ = |z|²

Recap

Hold these pegs from the official chapter “Complex Numbers and Quadratic Equations”. The wording is ExamMaster’s teaching, not a textbook recap.

Complex Numbers
A complex number is a + bi with a, b real and i² = −1.
Algebra of Complex Numbers
Algebra: add real with real, imaginary with imaginary; multiply by expanding and replacing i² with −1.
The Modulus and the Conjugate of a Complex Number
The modulus |z| is the length √(a²+b²); the conjugate z̄ is a − bi.
Argand Plane and Polar Representation
The Argand plane plots z as (a, b). Polar form is r(cos θ + i sin θ) with r = |z| and θ an argument as taught. 3+0i sits on the real axis. Polar is a length-plus-turn, not a second kind of number.
A definition is a test you can run
Complex is a test: a+bi, or a point, or r and θ.
Name the given before the unknown
The given is a, b (or r, θ). The unknown is the product, |z|, or z̄. Copy the sign of b before you conjugate.

Practise Complex Numbers and Quadratic Equations

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