Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part, and $i$ is the imaginary unit, defined by $i^2 = -1$
Addition and Subtraction
Addition
To add two complex numbers, simply add their real parts and their imaginary parts separately. For example, if you have $(3 + 2i)$ and $(1 + 4i)$, the addition would be:
$(3 + 2i) + (1 + 4i) = (3 + 1) + (2i + 4i) = 4 + 6i$
Subtraction
Subtraction works similarly. Subtract the real parts and the imaginary parts separately. For instance, for $(5 + 3i)$ and $(2 + 1i)$, the subtraction would be:
$(5 + 3i) – (2 + 1i) = (5 – 2) + (3i – 1i) = 3 + 2i$
Multiplication
To multiply two complex numbers, use the distributive property (FOIL method). For example, for $(2 + 3i)$ and $(1 + 4i)$, the multiplication would be:
$(2 + 3i)(1 + 4i) = 2(1) + 2(4i) + 3i(1) + 3i(4i) = 2 + 8i + 3i + 12i^2$
Since $i^2 = -1$, this simplifies to:
$2 + 11i + 12(-1) = 2 + 11i – 12 = -10 + 11i$
Division
To divide two complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a – bi$. For example, to divide $(3 + 2i)$ by $(1 – i)$, first find the conjugate of the denominator, which is $(1 + i)$. Then multiply both the numerator and the denominator by this conjugate:
$frac{3 + 2i}{1 – i} times frac{1 + i}{1 + i} = frac{(3 + 2i)(1 + i)}{(1 – i)(1 + i)}$
The denominator simplifies to $1^2 – (-i^2) = 1 + 1 = 2$. The numerator simplifies to:
$(3 + 2i)(1 + i) = 3(1) + 3(i) + 2i(1) + 2i(i) = 3 + 3i + 2i + 2i^2 = 3 + 5i + 2(-1) = 3 + 5i – 2 = 1 + 5i$
So, the division is:
$frac{1 + 5i}{2} = frac{1}{2} + frac{5}{2}i = 0.5 + 2.5i$
Conclusion
Understanding how to perform arithmetic with complex numbers is essential for many fields, including engineering and physics. By mastering addition, subtraction, multiplication, and division, you can tackle a wide range of problems involving complex numbers.