- Multiplying i is a rotation by 90 degrees counter-clockwise
- Multiplying by -i is a rotation of 90 degrees clockwise

z=1

If we multiply it by i, it becomes

z=i so that it has rotated by the angle 90 degrees

- Real numbers such as natural number,rational number , irrational number are invented in the history as and when we encounter various mathematical needs. Same happen with the complex numbers.

We had no solution for the problem

x^{2}=-1

Eular was the first mathematicain to introduce solution to this problem,he introduced the symbol i

$ i=\sqrt{(-1)}$

So i^{2}=-1

So solution to the problem becomes

x=i or -i - He called the symbol i as imaginary unit.
- Just like all the other number ,this number was added to our Number vocabulary. This like other numbers is useful in explaining where physical explanation.
- It is very useful in the field Electrical and electronics.

i

i

i

i

i

i

This can be generalized as

For any integer k, i

Find the value $(\frac {i}{2})(\frac {-2i}{3}) (\frac {i^3}{4})$

$(\frac {i}{2})(\frac {-2i}{3}) (\frac {i^3}{4})$

$=\frac {-i^5}{12} = \frac {-i}{5}$

For all complex numbers z

- (z
_{1}+ z_{2})^{2}= z_{1}^{2}+ z_{2}^{2}+ 2z_{1}z_{2} - (z
_{1}- z_{2})^{2}= z_{1}^{2}+ z_{2}^{2}- 2z_{1}z_{2} - (z
_{1}+ z_{2})^{3}= z_{1}^{3}+ z_{2}^{3}+ 3z_{1}z_{2}^{2}+ 3z_{1}^{2}z_{2} - (z
_{1}- z_{2})^{3}= z_{1}^{3}- z_{2}^{3}+ 3z_{1}z_{2}^{2}-3z_{1}^{2}z_{2} - z
_{1}^{2}- z_{2}^{2}= (z_{1}+ z_{2})(z_{1}- z_{2}) **Notes**- What is complex numbers
- Algebra Of complex Numbers
- Conjugate of Complex Numbers
- Modulus of complex numbers
- Argand Plane
- Polar Representation of the complex number
- Rotation of Complex Number
- Identities for Complex Numbers
- Eulers formula and De moivre's theorem
- Cube Root of unity
- Complex roots of Quadratic equations

**NCERT Solutions & Worksheets**

Find the non-zero integral Solution for this equation

$|1-i|^x =2^x$

$|1-i|= \sqrt {2} =2^{1/2}$

So

$2^{x/2}=2^x$

2

2

or x=0

So there is no non-zero integral solution for this equation

Class 11 Maths Class 11 Physics Class 11 Chemistry Class 11 Biology

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