Imaginary and Complex Numbers Note Sheet
Imaginary and Complex Numbers
Imaginary Unit (\(i\))
Evaluating \(\sqrt{25}\) vs. \(\sqrt{-25}\)
\(\sqrt{25} = 5\)
Because \(5^2 = 25\) (Real number)
\(\sqrt{-25} = \sqrt{25 \cdot (-1)} = 5i\)
No real number squared produces \(-25\)
Taking the square root of a negative value yields an imaginary number.
Definition of \(i\)
\(i = \sqrt{-1}\)
\(i^2 = -1\)
For any real \(b > 0\):
\(\sqrt{-b} = i\sqrt{b}\)
Powers of \(i\) Evaluated:
\(i^1 = i\) (Base definition)
\(i^2 = -1\) \((\sqrt{-1})^2 = -1\)
\(i^3 = -i\) \(i^2 \cdot i = (-1) \cdot i\)
\(i^4 = 1\) \(i^2 \cdot i^2 = (-1)(-1)\)
Complex Number: \(a + bi\)
A complex number is a combination of a real number and an imaginary number written in the standard form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i = \sqrt{-1}\).
a
Real Part
The standalone real quantity.
In \(7 - 4i\), the real part is \(7\).
b
Imaginary Part
The real coefficient multiplying \(i\).
In \(7 - 4i\), the imaginary part is \(-4\).
Adding & Subtracting Complex Numbers
Example 1: Addition
Simplify: \((5 - 6i) + (-2 + 11i)\)
Step 1:
Group real terms and imaginary terms:
\(= [5 + (-2)] + [-6i + 11i]\)
Step 2:
Combine real parts and imaginary parts:
\(= 3 + 5i\)
Example 2: Subtraction
Simplify: \((8 + 3i) - (4 - 7i)\)
Step 1:
Distribute the negative sign to the second set:
\(= 8 + 3i - 4 + 7i\)
Step 2:
Combine real parts and imaginary parts:
\(= (8 - 4) + (3i + 7i) = 4 + 10i\)
Multiplying Complex Numbers
Multiply complex numbers using the distributive property just like algebraic expressions. Whenever \(i^2\) appears, substitute \(-1\), then combine like terms to write the final result in standard form \(a + bi\).
Example 1: Distributing a Single Term
Multiply: \(4i(3 - 5i)\)
- Distribute \(4i\) to both terms:
\(= (4i \cdot 3) - (4i \cdot 5i) = 12i - 20i^2\)
- Substitute \(i^2 = \mathbf{-1}\) and simplify:
\(= 12i - 20(\mathbf{-1}) = 12i + 20\)
Standard Form: \(= 20 + 12i\)
Example 2: Distributing Two Complex Numbers
Multiply: \((2 + 3i)(4 - i)\)
- Distribute each term in the first binomial:
\(= 8 - 2i + 12i - 3i^2\)
- Combine middle terms & substitute \(i^2 = \mathbf{-1}\):
\(= 8 + 10i - 3(\mathbf{-1}) = 8 + 10i + 3\)
Standard Form: \(= 11 + 10i\)
Solving Variables in Complex Numbers
Two complex numbers are equal when their real parts are equal and their imaginary parts are equal. Equate the real parts to solve for one variable, and equate the imaginary coefficients to solve for the other variable.
Example 1: Find \(x\) and \(y\)
Solve: \((2x - 3) + (5y + 1)i = 11 - 14i\)
Real Parts:
\(2x - 3 = 11\)
\(2x = 14\)
\(x = 7\)
Imaginary Parts:
\(5y + 1 = -14\)
\(5y = -15\)
\(y = -3\)
Example 2: Find \(m\) and \(n\)
Solve: \(3m + 16i - 8 = 7 + (4n - 4)i\)
Real Parts:
\(3m - 8 = 7\)
\(3m = 15\)
\(m = 5\)
Imaginary Parts:
\(16 = 4n - 4\)
\(20 = 4n\)
\(n = 5\)
