Number System : Imaginary and Complex Number
Imaginary Number: Since no real number satisfies the equation , a number of the form , where b is a real number, b 0 and is called an imaginary number. Since real numbers does not contain the square root of -ve number, an imaginary number was introduced to overcome this.
Imaginary numbers when squared gives a negative number.
Complex Numbers: A number that can be expressed in the form of where a and b are real numbers and is a solution of the equation
In the complex number, a is called the real part, b is called the imaginary part and .
For e.g. :
Either a or b or both can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers.
If a = 0 and b 0, then the number become a imaginary number. Hence, all Imaginary numbers are complex numbers.
If b = 0 , then the number becomes a real number. Hence, all real numbers are complex numbers.
Conjugate of a Complex Number : If we change the sign of the imaginary part, we get the conjugate of a complex number.
So, if (a-bi) is a complex number, its conjugate will be (a+bi)
and, if (a+bi) is a complex number, its conjugate will be (a-bi)
For e.g. : 3 – 5i is the conjugate of 3 + 5i.
5+3i is the conjugate of 5-3i.
Addition of a Complex Numbers : To add two complex numbers,
~ Add the real parts to get the final real part.
~ Add the imaginary parts to get the final imaginary part.
For e.g. Add 3+5i & 4+7i
Multiplication of complex numbers : To multiply two complex numbers,
~ multiply each part of first complex number with each part of second complex number.
For e.g. Multiply 3+5i and 4+7i
Division of complex numbers : To divide one complex number by another, we actually use conjugate of the number.
~ Multiply the numerator and denominator with the conjugate of the denominator
For e.g. Divide 3+4i by 2+3i
Multiplication of Conjugates : In the previous example, what we did was multiplication of conjugate.
Lets see what multiplication of conjugates results in.