A complex number is a combination of a real and imaginary number. To subtract complex numbers, subtract each element separately.

In equation \(\displaystyle a\), \(\displaystyle 3\) is the real component and \(\displaystyle 1\) is the imaginary component (designated by \(\displaystyle i\)). In equation \(\displaystyle b\), \(\displaystyle 4\) is the real component and \(\displaystyle -2\) is the imaginary component. Solving for \(\displaystyle b - a\),

\(\displaystyle b-a = \left ( 4-2i\right ) - \left ( 3 + i\right ) = \left ( 4-3\right ) + i\left ( -2-1\right ) = 1 - 3i\)

When you have an exponent on the outside of parentheses while another is on the inside of the parentheses, such as in \(\displaystyle (3^{6})^{2}\), multiply the exponents together to get the answer: \(\displaystyle 3^{12}\).

This is different than when you have two numbers with the same base multiplied together, such as in \(\displaystyle x^{2} \cdot x^{3} = x^{5}\). In that case, you add the exponents together.

Solving this equation is very similar to solving a linear binomial like \(\displaystyle (a+bx)\). To solve, just combine like terms, being careful to watch for double negatives.

\(\displaystyle (2-2i) - (4-i)\)

\(\displaystyle (2-4) + (-2i-(-i))\)

\(\displaystyle -2 -(-2i+i)\)

\(\displaystyle -2 -i\)

A problem like this can be solved similarly to a linear binomial like \(\displaystyle (a +bx)\)/

\(\displaystyle 7i - (3-i) = 3-8i\)

\(\displaystyle 7i + (-3 + i) = 3-8i\)

\(\displaystyle -3 +8i \neq 3+8i\)

This equation can be solved very similarly to a binomial like \(\displaystyle a+bx\).

\(\displaystyle (8-5i) + (-4 +7i)\)

\(\displaystyle (8-4) + (-5i + 7i)\)

\(\displaystyle 4 + 2i\)

Substituting for \(\displaystyle a\) and \(\displaystyle b\), we have

\(\displaystyle 3(-2)^{3}+\frac{1}{2}(2(4)^{2})\)

 This simplifies to

\(\displaystyle -24+16\)

which equals \(\displaystyle -8\)

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

First, distribute:

\(\displaystyle 3\left ( 8 + 5i\right ) + \frac{1}{2}\left (4 + 2i\right )\)

\(\displaystyle \left ( 24 + 15i\right ) + \left (2 + i\right )\)

Then, group the real and imaginary components:

\(\displaystyle \left ( 24 + 2\right ) + i\left (15 + 1\right )\)

Solve to get:

\(\displaystyle 26 + 16i\)

A complex number is a combination of a real and imaginary number. To add complex numbers, add each element separately.

In equation \(\displaystyle a\), \(\displaystyle 5\) is the real component and \(\displaystyle 3\) is the imaginary component (designated by \(\displaystyle i\)).

In equation \(\displaystyle b\), \(\displaystyle 2\) is the real component and \(\displaystyle 1\) is the imaginary component.

When added, 

\(\displaystyle a + b = \left ( 5+3i\right ) + \left ( 2 + i\right ) = \left ( 5 + 2\right ) + i\left ( 3 + 1\right ) = 7 + 4i\)

When adding or subtracting complex numbers, the real terms are additive/subtractive, and so are the nonreal terms.

\(\displaystyle (5 +7i) + (17-4i) = (5+17) + (7i - 4i) = 22 + 3i\)

Complex numbers take the form a + bi, where a is the real term in the complex number and bi is the nonreal (imaginary) term in the complex number. Taking this, we can see that for the real number 8, we can rewrite the number as \(\displaystyle 8 + 0i\), where \(\displaystyle 0i\) represents the (zero-sum) non-real portion of the complex number.

Thus, any real number can be added to any complex number simply by considering the nonreal portion of the number to be \(\displaystyle 0i\).