Is the sum of two complex numbers always a real number?

Question

nick1337 · Accepted Answer

Step 1 Not at all. In some rare cases, e.g. 3+5i and 8-5i their sum is indeed a real number (with no imaginary component). But thats

star233 · Accepted Answer

Step 1
Nop, a complex number has always two parts, an imaginary one and a real one!
Like:
a+bi
Where a is the real part and b is the imaginary part
Supose we have 2 numbers

3+5i
and
5+3i
the sum of them would be
8+8i
which is a complex number!
I think you mistook 2 complexes by the sum of a complex with it’s conjugated part
where if a complex is a+bi it’s conjugated part is a-bi and if you sum those two numbers you always have $2 \times (real part of the number)$ in this case we have 2a!

alenahelenash · Accepted Answer

Step 1 No. Heavens no. This is like asking “is the sum of two real numbers always 0”? If&amp;nbsp;z=a+bi&amp;nbsp;and&amp;nbsp;w=c+di&amp;nbsp;then the sum&amp;nbsp;z+w=a+c+(b+d)i&amp;nbsp;This would not be real unless&amp;nbsp;b+d=0

Vasquez · Accepted Answer

Let&#039;s consider two complex numbers, z1 and z2, which can be expressed in the form z1=a1+b1i and z2=a2+b2i, where a1,b1,a2,b2∈ℝ and i represents the imaginary unit.The sum of these two complex numbers can be calculated as:z1+z2=(a1+b1i)+(a2+b2i)To simplify the expression, we can combine the real parts and the imaginary parts separately:z1+z2=(a1+a2)+(b1+b2)iNow, we can observe that the sum consists of two terms: (a1+a2) and (b1+b2)i. The first term is the sum of two real numbers, which will always be a real number. However, the second term involves the imaginary unit i, which implies that it is purely imaginary.Hence, we can conclude that the sum of two complex numbers is not always a real number. It contains a real part and an imaginary part.The sum of two complex numbers, z1 and z2, is given by:z1+z2=(a1+b1i)+(a2+b2i)Simplifying the expression, we have:z1+z2=(a1+a2)+(b1+b2)iWe can observe that the sum consists of a real part, (a1+a2), and an imaginary part, (b1+b2)i. Therefore, the sum of two complex numbers is not always a real number.

RizerMix · Accepted Answer

No, the sum of two complex numbers is not always a real number.
The sum of two complex numbers

z_{1}

and

z_{2}

is given by

z_{1} + z_{2}

. However, this sum is only a real number if the imaginary parts of

z_{1}

and

z_{2}

cancel out, i.e., if the sum of their imaginary parts is zero. If the imaginary parts do not cancel out, then the sum will have a non-zero imaginary part and therefore be a complex number.

Is the sum of two complex numbers always a real

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