The problem asks us to add two complex numbers: $(3 + i\sqrt{2})$ and $(-4 + i\sqrt{2})$.

AlgebraComplex NumbersAddition
2025/4/9

1. Problem Description

The problem asks us to add two complex numbers: (3+i2)(3 + i\sqrt{2}) and (4+i2)(-4 + i\sqrt{2}).

2. Solution Steps

To add two complex numbers, we add their real parts and their imaginary parts separately.
Let z1=a+biz_1 = a + bi and z2=c+diz_2 = c + di be two complex numbers, where a,b,c,da, b, c, d are real numbers and ii is the imaginary unit (i2=1i^2 = -1).
Then, z1+z2=(a+c)+(b+d)iz_1 + z_2 = (a+c) + (b+d)i.
In our case, z1=3+i2z_1 = 3 + i\sqrt{2} and z2=4+i2z_2 = -4 + i\sqrt{2}. So, a=3a = 3, b=2b = \sqrt{2}, c=4c = -4, and d=2d = \sqrt{2}.
Therefore,
z1+z2=(3+(4))+(2+2)i=(34)+(22)i=1+22iz_1 + z_2 = (3 + (-4)) + (\sqrt{2} + \sqrt{2})i = (3 - 4) + (2\sqrt{2})i = -1 + 2\sqrt{2}i.

3. Final Answer

The sum of the two complex numbers is 1+2i2-1 + 2i\sqrt{2}.

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