An imaginary number is a number that, when squared, results in a negative number.
The most basic imaginary number is \$\sqrt{-1}\$, which is denoted by the symbol \$i\$.
The symbol \$i\$ is not a variable. It is a constant representing the square root of \$-1\$.
A complex number is a number of the form \$a + bi\$, where \$a\$ and \$b\$ are real numbers, and \$i\$ is the imaginary unit.
- The real part of the complex number \$a + bi\$ is \$a\$.
- The imaginary part is \$b\$.
Real numbers are a subset of complex numbers where the imaginary part is zero (i.e., \$b = 0\$).
Simplifying Imaginary Numbers
The square root of any negative number can be expressed using \$i\$.
Always simplify \$i^2\$ to \$-1\$ when multiplying complex numbers.
Complex Solutions to Quadratic Equations
When solving quadratic equations, if the discriminant is negative, the solutions are complex.
Real numbers lie on the real axis, and imaginary numbers lie on the imaginary axis.
Absolute Value of a Complex Number
The absolute value of a complex number \$a + bi\$ is the distance from the origin \$(0, 0)\$ to the point \$(a, b)\$ on the complex plane.
The absolute value of a complex number is always a non-negative real number.
1. Simplify \$\sqrt{-50}\$. 2. Calculate \$i^{23}\$. 3. Add \$(4 + 3i) + (2 - 5i)\$. 4. Multiply \$(1 + 2i)(3 - i)\$. 5. Solve \$x^2 + 2x + 5 = 0\$ using the quadratic formula. 6. Plot \$-3 + 4i\$ on the complex plane. 7. Find the absolute value of \$-6 + 8i\$.
How do imaginary and complex numbers expand our understanding of mathematics? Are they "real" in the same sense as real numbers?
