How to Distribute Complex Numbers?
Distributing means dividing. Complex numbers satisfy the distributive property. The property states that if we multiply a complex number by the sum of two complex numbers it will be the same as multiplying the complex number with each complex number separately and then adding the result. Let (a + ib), (c + id) , (x + iy) be three complex numbers. Therefore the distributive Property is (a + ib) × {(c+id) + (x + iy)} = {(a + ib)×(c + id)} + {(a + ib)×(x + iy)}. Let us illustrate it with the help of an example
Let there be three complex numbers : 2i, 3 + 7i , 7 + 3i
= (2i) × {(3 + 7i) +( 7 + 3i)}
= 2i × (10 + 10 i)
= 20i -20 —–(1)
Now let us distribute the 2i individually with the given operands
= (2i) × {(3 + 7i) +( 7 + 3i)}
= 2i×(3 + 7i) + (2i)×(7 + 3i)
= 6i -14 + 14i – 6
= 20i -20 (i2 = -1) ——(2)
from eq (1) and (2)
Thus, distributive property is satisfied.
Sample Problems
Problem 1: Compute (2i + 6i )×(4i + 8i -2i)
Solution:
2i×(4i + 8i -2i) + 6i×(4i + 8i -2i)
= 2i×(10i) + 6i×(10i)
= -20 – 60
= -80
Problem 2: Find the value of 2i×(9 + i/9 + 8i)
Solution:
2i×( 9 + i/9 + 8i)
= 2i×9 + 2i ×( i/9 + 8i)
= 18i + 2i ×( 73i/9)
= -146/9 + 18i
Problem 3: Find (4 + 5i)×(9 + 3i)
Solution:
(4 + 5i)×(9 + 3i)
= 4×(9 + 3i) + 5i×(9 + 3i)
=36 + 12i + 45i -15
= 21 +57i
Problem 4: Solve (4 + 5i)×(4 – 5i)
Solution:
(4 + 5i)×(4 – 5i)
= 4×(4 – 5i) + 5i×(4 – 5i)
=16 -20i + 20i +25
=41
Problem 5: Solve (4 + 5i)×(9 – 5i)
Solution:
(4 + 5i)×(9 – 5i)
= 4×(9 – 5i) + 5i×(9 – 5i)
= 36 – 20i + 45i +25
= 41 + 25i
How to distribute complex numbers?
Complex numbers are used to find the square root of negative numbers. It comprises the real and imaginary parts. The complex number is of the form a + ib where a is the real part and b is the imaginary part. The real part of the complex number is any number and is represented by Re(z) where ‘z’ is any complex number. The imaginary part of the complex number comprises any number multiplied with ‘i’, The ‘i’ stands for iota. The complex numbers cannot be represented on the number line. They are represented on a plane called Argand Plane. For example: Let z = 3 + 5i. Here Re(z) = 3 and Im(z) = 5.
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