A number which is of the form a+ib, where a and b are real numbers and i2 =-1, is called a complex number and is usually denoted by z.
i.e. z=a+ib.
Here ‘a’ is called the real part of z , i.e. Re(z) = a
And ‘b’ is called the imaginary part of z, i.e. Im(z)=b.
Sum of complex numbers:
If z1 = a+ib and z2 = c+id then the sum of z1 and z2 is denoted by z1+z2 and is defined as
Z1+z2 = (a+c)+i(b+d).
Example:
Find the sum of z1 = 3+5i and z2 = 4-2i.
Solution:
By the definition of sum of complex numbers,
Z1+z2 = (3+4) + i(5-2) = 7 + 3i.
Difference of complex numbers:
If z1 = a+ib and z2 = c+id then the difference of z1 and z2 is denoted by z1-z2 and is defined as
Z1-z2 = (a-c)+i(b-d).
Example:
If z1 = 3+5i and z2 = 4-2i find z1-z2.
Solution:
By the definition of difference of complex numbers,
z1-z2 = (3-4) + i(5+2) = -1 + 7i.
Product of complex numbers:
If z1 = a+ib and z2 = c+id then the product of z1 and z2 is denoted by z1.z2 and is defined as,
z1.z2 = (ac-bd) + i(bc + ad).
Example:
Find the product of z1 = 3+5i and z2 = 4-2i.
Solution:
By the definition of product of complex numbers,
z1.z2 = (12+10)+i(20-6) = 22 + 14i
Division of complex numbers:
If z1 = a+ib and z2 = c+id then the quotient of z1 and z2 is denoted by z1/z2 and is defined as,
Example:
If z1 = 3+5i and z2 = 4-2i find z1/z2.
Solution:
By the definition of division of complex numbers,
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