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Mathematics · Class 11 Science

Ch 4Complex Numbers and Quadratic Equations — Class 11 Mathematics, concept-first.

Consider the simple-looking equation , i.e. . Within the real numbers this equation has no solution, because the square of every real number is never negative: a positive real squares to a positive real, a negative real squares to a positive real (since a negative times a negative is positive), and .

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Key concepts

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Complex Numbers and Algebraic Properties

A complex number is an expression , where are real numbers and is the imaginary unit defined by . The number is its real part and is its imaginary part; two complex numbers are equal exactly when both parts match separat…

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Need for Complex Numbers

Consider the simple-looking equation , i.e. . Within the real numbers this equation has no solution, because the square of every real number is never negative: a positive real squares to a positive re…

2

Algebraic Properties of Complex Numbers

Definition. A complex number is an expression , where and satisfies . The real number is called the real part of , written , and is called the imaginary part, written .

2.1

Addition and Subtraction of Complex Numbers

For and (), addition is defined componentwise: That is, real parts add with real parts and imaginary parts add with imaginary parts — exactly as if were an ordinary algebraic symbol being collected.

2.2

Multiplication of Complex Numbers

For and , multiplication is carried out exactly like multiplying two binomials, and then is replaced by : so, collecting real and imaginary parts, Because and are real numbers whenever are, the produc…

2.3

Conjugate of a Complex Number

For , the conjugate of , written , is obtained by reversing the sign of the imaginary part only: Geometrically (once the Argand plane is introduced in §3) is the mirror image of in the real axis.

2.4

Reciprocal (Multiplicative Inverse) of a Complex Number

For a non-zero complex number (so ), the multiplicative inverse is the complex number satisfying . It is found using exactly the identity from §2.3: multiply and divide by the conjugate , which turns…

3

The Argand Plane

Every complex number can be represented as a unique point in a plane, by plotting (the real part) along a horizontal axis and (the imaginary part) along a perpendicular vertical axis.

4

Modulus and Argument of a Complex Number

Modulus. For , the modulus of , written (also called ), is defined as Geometrically, is the length of the segment from the origin to the point representing in the Argand plane: dropping perpendiculars…

5

Polar Representation of a Complex Number

From the right triangle used to define the modulus and argument (§4), the real and imaginary parts of can themselves be written in terms of and : Substituting these into gives the polar form: Here is…

6

Quadratic Equations in the Complex Number System

For the general quadratic equation with and , completing the square gives the familiar quadratic formula, where is the discriminant.

Summary

- Need for complex numbers. A quadratic with negative discriminant has no real root, since no real number squares to a negative number.

Worked Examples

Solved examples, worked out step by step.

+Show 6 questions6 questions
  1. Example 1Simplify $\sqrt{-9}+\sqrt{-16}$ and express the result in the form $a+ib$.Free
  2. Example 2Find the product $(3+2i)(1-4i)$ and express the result in the form $a+ib$.Free
  3. Example 3Find the conjugate and the multiplicative inverse of $z=4-3i$.Preview
  4. Example 4Find the modulus and the argument of $z=-1+i$.Preview
  5. Example 5Express $z=1+i\sqrt3$ in polar form.Preview
  6. Example 6Solve the quadratic equation $x^2+2x+5=0$ in the complex number system.Preview

More questions

22 Q
+Show 2 questions2 questions
  1. Q27If $z=3+4i$, find $|z|$ and verify that $z\bar z=|z|^2$.Free
  2. Q28Express $\dfrac{1+2i}{3-i}+\dfrac{1-2i}{3+i}$ in the form $a+ib$ and show that it is a real number.Preview
+Show 6 questions6 questions
  1. Q7Express $(5-3i)-(2+4i)$ in the form $a+ib$.Free
  2. Q8Find $(2-i)(3+4i)$ and express the result in the form $a+ib$.Free
  3. Q9Find the multiplicative inverse of $z=2+3i$.Preview
  4. Q10Simplify $\dfrac{3+4i}{4-3i}$ and express the result in the form $a+ib$.Preview
  5. Q11Evaluate $i^{37}+\dfrac{1}{i^{67}}$.Preview
  6. Q12If $z_1=3+2i$ and $z_2=1-i$, verify that $\overline{z_1z_2}=\bar z_1\,\bar z_2$.Preview
+Show 5 questions5 questions
  1. Q13Find the modulus of $z=3-4i$.Free
  2. Q14Find the modulus and the argument of $z=-\sqrt3-i$.Free
  3. Q15Represent $z=2+3i$ and its conjugate $\bar z$ on the Argand plane, and state the geometric relationship between the two points.Preview
  4. Q16Find the modulus and the argument of $z=4i$.Preview
  5. Q17Find the argument of $z=1-i$, giving your answer in the range $(-\pi,\pi]$.Preview
+Show 4 questions4 questions
  1. Q18Express $z=1-i$ in polar form.Free
  2. Q19Express $z=-4$ in polar form.Free
  3. Q20Convert $r(\cos\theta+i\sin\theta)$ back to the form $a+ib$ for $r=2,\ \theta=\dfrac{\pi}{3}$.Preview
  4. Q21Express $z=-2+2\sqrt3\,i$ in polar form and state its modulus and argument.Preview
+Show 5 questions5 questions
  1. Q22Solve $x^2+x+1=0$.Free
  2. Q23Solve $2x^2-x+1=0$.Free
  3. Q24Solve $x^2+3x+5=0$.Preview
  4. Q25Solve $-x^2+x-2=0$.Preview
  5. Q26For the quadratic equation $x^2-4x+13=0$, find the roots and verify that the sum and product of the roots equal $-b/a$ and $c/a$ respectivel…Preview