1.1 The imaginary unit i and imaginary numbers

Complex numbers expand our mathematical toolkit beyond real numbers. The imaginary unit 'i' is defined as the square root of -1, allowing us to solve equations that were previously unsolvable. This concept opens up new possibilities in algebra and geometry.

Imaginary numbers are represented on the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis. This visualization helps us understand their properties and perform arithmetic operations, laying the foundation for further study of complex analysis.

The imaginary unit 'i'

Definition and properties of the imaginary unit 'i'

• The imaginary unit 'i' is defined as a number that satisfies the equation $i^2 = -1$
• 'i' is not a real number and cannot be represented on the real number line
• The square of 'i' equals -1, which is a real number
• When 'i' is multiplied by itself an odd number of times, the result is either 'i' or '-i'
  • $i^1 = i$
  • $i^3 = i^2 \cdot i = -1 \cdot i = -i$
• When 'i' is multiplied by itself an even number of times, the result is either 1 or -1
  • $i^2 = -1$
  • $i^4 = (i^2)^2 = (-1)^2 = 1$

Imaginary numbers in the complex plane

Representation of imaginary numbers in the complex plane

• An imaginary number is a number that can be written as a real number multiplied by 'i'
  • $3i$, $-2i$, and $\sqrt{2}i$ are examples of imaginary numbers
• The complex plane is a two-dimensional representation of complex numbers
  • The real part is plotted on the horizontal axis
  • The imaginary part is plotted on the vertical axis
• Pure imaginary numbers are complex numbers with a real part equal to zero
  • Pure imaginary numbers are represented on the vertical axis of the complex plane
  • Examples of pure imaginary numbers: $2i$, $-5i$, and $-\frac{1}{2}i$
• The distance of an imaginary number from the origin in the complex plane represents its magnitude or absolute value
• The direction of an imaginary number in the complex plane is always along the vertical axis
  • Positive imaginary numbers are directed upward
  • Negative imaginary numbers are directed downward

Arithmetic with imaginary numbers

Addition and subtraction of imaginary numbers

• Addition and subtraction of imaginary numbers are performed by adding or subtracting their coefficients while keeping 'i'
  • $(2i) + (3i) = 5i$
  • $(-4i) - (7i) = -11i$
• The real parts and imaginary parts are added or subtracted separately
  • $(3 + 2i) + (4 - 5i) = (3 + 4) + (2 - 5)i = 7 - 3i$

Multiplication and division of imaginary numbers

• Multiplication of imaginary numbers is performed by multiplying their coefficients and applying the property $i^2 = -1$ when necessary
  • $(2i) \cdot (3i) = 2 \cdot 3 \cdot i^2 = 6 \cdot (-1) = -6$
  • $(4i) \cdot (-2i) = 4 \cdot (-2) \cdot i^2 = -8 \cdot (-1) = 8$
• Division of imaginary numbers is performed by multiplying the numerator and denominator by the complex conjugate of the denominator
  • The complex conjugate of $a + bi$ is $a - bi$, obtained by changing the sign of the imaginary part
  • $\frac{2i}{3i} = \frac{2i}{3i} \cdot \frac{3i}{3i} = \frac{2 \cdot 3 \cdot i^2}{3^2 \cdot i^2} = \frac{-6}{9} = -\frac{2}{3}$
• The absolute value (modulus) of an imaginary number $a + bi$ is given by the formula $\sqrt{a^2 + b^2}$
  • The absolute value represents the distance from the origin in the complex plane
  • $|3i| = \sqrt{0^2 + 3^2} = \sqrt{9} = 3$

Solving equations with imaginary numbers

Solving techniques for equations involving imaginary numbers

• Equations involving imaginary numbers can be solved by applying the same algebraic techniques used for solving equations with real numbers
• When solving equations with imaginary numbers, keep the real and imaginary parts separate and equate the corresponding parts on both sides of the equation
  • $2x + 3i = 5 + 7i$ becomes $2x = 5$ (real part) and $3i = 7i$ (imaginary part)
• Equations involving higher powers of 'i' can be simplified by applying the properties of 'i'
  • $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$ can be used to reduce the powers of 'i'
  • $x^2 + 2ix - 3 = 0$ can be simplified to $x^2 - 3 + 2ix = 0$

Solving quadratic equations with imaginary solutions

• Quadratic equations with negative discriminants ($b^2 - 4ac < 0$) have complex solutions involving imaginary numbers
• The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ can be used to find the complex solutions of quadratic equations
  • When the discriminant ($b^2 - 4ac$) is negative, the solutions will involve imaginary numbers
  • Example: $x^2 + 4x + 5 = 0$ has solutions $x = -2 \pm i$, as the discriminant is $4^2 - 4 \cdot 1 \cdot 5 = -4$