"The conjugate of the sum of two complex numbers is equal to the sum of the conjugates of the two complex numbers."
This is true right?
Yes, that statement is indeed true. The conjugate of a complex number is obtained by changing the sign of its imaginary part. Let's say we have two complex numbers, a + bi and c + di.
The sum of these two complex numbers is (a + bi) + (c + di), which simplifies to (a + c) + (b + d)i. Now, let's find the conjugate of this sum.
The conjugate of (a + c) + (b + d)i would be (a + c) - (b + d)i. This is obtained by changing the sign of the imaginary part, (b + d)i.
On the other hand, let's find the conjugates of the two complex numbers separately. The conjugate of a + bi would be a - bi, and the conjugate of c + di would be c - di.
Now, if we add these two conjugates, (a - bi) + (c - di), we can simplify it to (a + c) - (b + d)i.
Comparing the two expressions we obtained, (a + c) - (b + d)i and (a + c) - (b + d)i, we can see that they are equal.
Therefore, the conjugate of the sum of two complex numbers is indeed equal to the sum of the conjugates of the two complex numbers.
Yes, that statement is true. To understand why, let's first define what the conjugate of a complex number is.
The conjugate of a complex number a + bi, where 'a' and 'b' are real numbers, is denoted as a - bi. In other words, the conjugate of a complex number is obtained by changing the sign of the imaginary part.
Now, let's consider two complex numbers z₁ = a₁ + b₁i and z₂ = a₂ + b₂i, where a₁, b₁, a₂, and b₂ are real numbers.
The sum of these two complex numbers is given by z = z₁ + z₂ = (a₁ + a₂) + (b₁ + b₂)i.
To find the conjugate of the sum, we replace 'i' with '-i' in the imaginary part:
Conjugate of z = (a₁ + a₂) - (b₁ + b₂)i.
Now, let's find the sum of the conjugates of the two complex numbers:
Conjugate of z₁ = a₁ - b₁i
Conjugate of z₂ = a₂ - b₂i
Sum of the conjugates = (a₁ - b₁i) + (a₂ - b₂i) = (a₁ + a₂) + (-b₁ - b₂)i.
As you can see, the conjugate of the sum of the two complex numbers, z, is equal to the sum of the conjugates of the two complex numbers, z₁ and z₂. Hence, the statement is true.
This property is often used in complex number arithmetic and is helpful in various applications involving complex numbers.