Vocabulary
Imaginary unit i- i = √(-1) , so i² = -1. Complex Number- A number a + bi where a and b are real numbers and I is the imaginary unit. Imaginary Number- A complex number a + bi where b≠0. Complex Conjugates- Two complex numbers of the form a + bi and a - bi. Complex Plane- A coordinate plane in which each point (a,b) represents a complex number a + bi. The horizontal axis is the real axis and the vertical axis is the imaginary axis. Absolute value of a complex number- If z=a + bi, then the absolute value of z, denoted ∣z∣, is a nonnegative real number defined as ∣z∣ = √(a²+b²).
Essential Information
Not all quadratic equations have real-number solutions. For example, x² = -1 has no real-number solutions because the square of any real number x is never a negative number.
Mathematicians created an expanded system of numbers using the imaginary unit I, defined as i = √(-1). i² = -1. The imaginary unit i can be used to write the square root of any negative number.
The Square Root of a Negative Number Properties 1. If r is a positive real number then, √(-r) = i√(r). 2. (i√(r))² = -r
Mathematicians created an expanded system of numbers using the imaginary unit I, defined as i = √(-1). i² = -1. The imaginary unit i can be used to write the square root of any negative number.
The Square Root of a Negative Number Properties 1. If r is a positive real number then, √(-r) = i√(r). 2. (i√(r))² = -r
Complex Numbers
A complex number written in standard form is a number a + bi where a and b are real numbers. The number a is the real part of the complex number, and the part bi is the imaginary part. If b≠0, then a + bi is an imaginary number. If a=0 and b≠0, then a + bi is a pure imaginary number. Two complex numbers a + bi and c + di are equal only if a=c and b=d. |
Sums and Differences of Complex Numbers
To add or subtract two complex numbers, add or subtract their real parts and their imaginary parts separately
To add or subtract two complex numbers, add or subtract their real parts and their imaginary parts separately
- Sums: (a+bi) + (c+di) = (a+c) + (b+d)i
- Differences: (a+bi - (c+di) = (a-c) + (b-d)i
Multiplying Complex Numbers
To multiply two complex numbers, use the distributive property (FOIL). Just like when multiplying real numbers or algebraic expressions.
To multiply two complex numbers, use the distributive property (FOIL). Just like when multiplying real numbers or algebraic expressions.
Complex Conjugates
Two complex numbers of the form a + bi and a - bi are called complex conjugates. The product of complex conjugates is always a real number.
(2+4i)(2-4i) = 4 - 8i + 8i + 16 = 20.
This can be used to write the quotient of two complex numbers in standard form.
Two complex numbers of the form a + bi and a - bi are called complex conjugates. The product of complex conjugates is always a real number.
(2+4i)(2-4i) = 4 - 8i + 8i + 16 = 20.
This can be used to write the quotient of two complex numbers in standard form.
Complex Plane
Just as every real number corresponds on the real number line, every complex number corresponds to a point in the complex plane. The complex plane has a horizontal axis called the real axis and a vertical axis called the imaginary axis.
Just as every real number corresponds on the real number line, every complex number corresponds to a point in the complex plane. The complex plane has a horizontal axis called the real axis and a vertical axis called the imaginary axis.