**Complex numbers :**

Complex numbers are one form of numbers. It is a combination of real and imaginary numbers. It can be generally expressed as a+ib.

Where "a" is the real number and 'i' is the imaginary number.

**Complex number system :**

A complex number is of the form a + ib where ‘a’ and ‘b’ are real numbers and i is called the imaginary unit, having the property that i^{2} = − 1.

If z = a + ib then a is called the real part of z, denoted by Re(z) and b is called the imaginary part of z and is denoted by Im(z)

Some examples of complex numbers are 3 − 2i, 2 + 3i

Let z_{1 } = 3 - 2i and z_{2 } = 2 + 3i

z Re (z Imz (z |
z Re (z Imz (z |

**Negative of a complex number :**

If z = a + ib is a complex number then the negative of z is denoted by − z and it is defined as − z = − a + i(− b)

**Basic Algebraic operations :**

**Addition :**

(a + ib) + (c + id) = (a + c) + i (b + d)

To add two or more complex numbers, we have to combine the real parts and imaginary parts respectively.

**Subtraction :**

(a + ib) − (c + id) = (a − c)+ i(b − d)

To subtract two or more complex numbers, we have to combine the real parts and imaginary parts respectively.

To perform the operations with complex numbers we can proceed as in the algebra of real numbers replacing i^{2} by − 1 whenever it occurs.

**Multiplication :**

(a + ib) (c + id) = ac + iad + ibc + i2bd

= (ac − bd) + i (ad + bc)

**Division :**

(a + ib) / (c + id) = ((a + ib) / (c + id)) ((c - id) / (c - id))

To divide two complex numbers, we have to multiply the given fraction by the conjugate of denominator.

**Conjugate of a complex number :**

If z = a + ib, then the conjugate of z is denoted by

- Properties of complex numbers
- Add and subtract complex numbers
- How to find the modulus and argument of a complex number
- How to find complex roots of a 4th degree polynomial
- How to find the square root of a complex number
- Product of two complex numbers in polar form
- Product of two complex numbers in exponential form

After having gone through the stuff given above, we hope that the students would have understood "Complex numbers".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**