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21 Questions around this concept.
The equation represents a part of a circle having radius equal to :
Let the complex number z = x + iy be such tha t is purely imaginary. If then is equal to :
For two non-zero complex numbers if , then which of the following are possible?
Choose the correct answer from the options given below:
1. Addition of Two Complex Numbers
z_{1} = a + ib and z_{2} = c + id be any two complex numbers. Then, the sum z_{1} + z_{2} is defined as
z_{1} + z_{2} = (a + ib) + (c + id) = (a + c) + i(b + d)
For example, z_{1}= (3 - 4i) and z_{2} = (2 + 5i), then z_{1} + z_{2} is
(3 − 4i) + (2 + 5i) = (3 + 2) + (−4 + 5)i = 5 + i
2. Difference of Two Complex Numbers
z_{1} = a + ib and z_{2} = c + id be any two complex numbers. Then, the difference z_{1} - z_{2} is defined as
z_{1} - z_{2} = (a + ib) - (c + id) = (a - c) + i(b - d)
For example, z_{1}= (-5 + 7i) and z_{2} = (-11 + 2i), then z_{1} - z_{2} is
(−5 + 7i) − (−11 + 2i) = −5 + 7i + 11 − 2i
= −5 + 11 + 7i − 2i
= (−5 + 11) + (7 − 2)i
= 6 + 5i
3. Multiplication of Two Complex Numbers
z_{1} = a + ib and z_{2} = c + id be any two complex numbers. Then, the multiplication z_{1}・z_{2} is defined as
z_{1}・z_{2} = (a + ib)・ (c + id)
= ac + iad + ibc + i^{2}bd
= ac + i(ad + bc) - bd
= (ac - bd) + i(ad + bc)
For example, z_{1}= (4 + 3i) and z_{2} = (2 - 5i), then z_{1}・ z_{2 }is
(4 + 3i)(2 − 5i) = 4(2) − 4(5i) + 3i(2) − (3i)(5i)
= 8 − 20i + 6i − 15(i^{2})
= (8 + 15) + (−20 + 6)i
= 23 - 14i
4. Division of Two Complex Numbers
z_{1} = a + ib and z_{2} = c + id (and z_{2} is non-zero) be any two complex numbers. Then, the division is defined as
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