Which property of addition is shown below?
a + bi + c + di = a + c + bi + di

Solution:

This is the commutative property of addition as the arrangement of the terms on both sides of the equation is different. 

According to this property if we add two or more numbers we arrive at the same sum irrespective of their arrangement in any order.

Here we have the complex numbers with the real and the imaginary parts. The commutative property of addition holds true for complex numbers.

For two complex numbers z\(_1 \) and z\(_2 \),

where z\(_1 \) = a + ib and z\(_2 \) = c + id
z\(_1 \)+ z\(_2 \) = (a + ib) +(c +id)

= (a + c) + i (b + d) [By grouping the real and the imaginary parts]

= a + c + bi + di

= a + bi + c + di [By rearrangement property of addition]

Thus z\(_1 \)+ z\(_2 \) = a + bi + c + di

z\(_2 \)+ z\(_1 \) = (c + id) +(a + ib)

= (c + a) + i(d + b)[By grouping the real and the imaginary parts]

=c + a +  di + bi

= a + c + bi + di [By rearrangement property of addition]

Thus z\(_2 \)+ z\(_1 \) = a + c + bi + di

a + bi + c + di = a + c + bi + di

Using the rearrangement property of addition we can merely rearrange the numbers without affecting the final sum.

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Which property of addition is shown below?
a + bi + c + di = a + c + bi + di

Summary:

The property depicted above is the rearrangement property of the addition of complex numbers.