Section 1.3: Complex Numbers Study Guide

Definition and Classification of Complex Numbers

  • The Imaginary Unit ii:

    • The real number system does not contain any numbers whose square is negative. Consequently, equations such as x2=−1x^2 = -1 have no real solutions.

    • To expand the number system to solve such equations, the imaginary unit ii is defined as:         i=−1i = \sqrt{-1}

    • Squaring both sides yields the foundational identity:         i2=−1i^2 = -1

  • Standard Form of Complex Numbers:

    • A complex number is any number that can be expressed in standard form:         a+bia + bi

    • In this expression, aa and bb are real numbers:

      • aa is designated as the real part.

      • bb is designated as the imaginary part.

  • Classifications of Complex Numbers:

    • Real Numbers: Form a subset of the complex numbers where the imaginary coefficient b=0b = 0 (expressed as a+0i=aa + 0i = a).

    • Nonreal Complex Numbers: Complex numbers where b≠0b \neq 0.

    • Pure Imaginary Numbers: Complex numbers where the real part a=0a = 0 and the imaginary part b≠0b \neq 0 (expressed as 0+bi=bi0 + bi = bi).

Equality and Standard Form

  • Criteria for Equality:

    • Two complex numbers a+bia + bi and c+dic + di are equal if and only if both their real parts are identical and their imaginary parts are identical.

    • Equivalence relation:         a+bi=c+di  ⟺  a=c and b=da + bi = c + di \iff a = c \text{ and } b = d

Simplifying Square Roots of Negative Numbers

  • Definition of −a\sqrt{-a}:

    • For any positive real number a > 0, the radical expression −a\sqrt{-a} is defined and rewritten as:         −a=ia\sqrt{-a} = i\sqrt{a}

  • Examples of Radical Simplification:

    • Simplification of −16\sqrt{-16}:         −16=i16=4i\sqrt{-16} = i\sqrt{16} = 4i

    • Simplification of −70\sqrt{-70}:         −70=i70\sqrt{-70} = i\sqrt{70}

    • Simplification of −48\sqrt{-48}:         −48=i48=i16×3=4i3\sqrt{-48} = i\sqrt{48} = i\sqrt{16 \times 3} = 4i\sqrt{3}

  • Crucial Caveat for Radical Rules:

    • The standard product rule for radicals, cd=cd\sqrt{c}\sqrt{d} = \sqrt{cd}, holds true only when at least one of cc or dd is non-negative.

    • It is invalid when both c < 0 and d < 0.

    • Rule: Expressions involving negative radicands must always be rewritten in terms of ii (i.e., as iai\sqrt{a}) before performing multiplication, division, or applying any radical rules.

Operations: Addition and Subtraction

  • Methodology:

    • Complex numbers are added or subtracted by combining their real parts and imaginary parts separately.

    • This follows the standard algebraic procedure for combining like terms.

  • General Formula:     (a+bi)±(c+di)=(a±c)+(b±d)i(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i

  • Procedural Steps:

    • Group the real parts: a±ca \pm c.

    • Group the imaginary parts: (b±d)i(b \pm d)i.

    • Simplify and write the final output in standard form a+bia + bi.

Operations: Multiplication and Complex Conjugates

  • Algebraic Multiplication:

    • Multiplication of complex numbers proceeds similarly to multiplying algebraic binomials using the FOIL method (First, Outside, Inside, Last).

    • Every instance of i2i^2 that arises during multiplication must be replaced with −1-1.

  • General Multiplication Formula:     (a+bi)(c+di)=ac+adi+bci+bdi2(a + bi)(c + di) = ac + adi + bci + bdi^2     Since i2=−1i^2 = -1, this simplifies to:     (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a + bi)(c + di) = (ac - bd) + (ad + bc)i

  • Complex Conjugates:

    • For any complex number a+bia + bi, its complex conjugate is a−bia - bi.

    • Conjugates differ only in the sign of their imaginary part.

  • The Conjugate Product Property:

    • Multiplying a complex number by its complex conjugate always yields a non-negative real number.

    • Derivation:         (a+bi)(a−bi)=a2−abi+abi−b2i2=a2−b2(−1)=a2+b2(a + bi)(a - bi) = a^2 - abi + abi - b^2i^2 = a^2 - b^2(-1) = a^2 + b^2

    • Identity:         (a+bi)(a−bi)=a2+b2(a + bi)(a - bi) = a^2 + b^2

Operations: Division

  • Procedure for Division:

    • To divide complex numbers, eliminate the imaginary unit from the denominator by multiplying both the numerator and the denominator by the complex conjugate of the denominator.

    • This applies both to general complex denominators (c+dic + di) and purely imaginary denominators (didi).

  • General Division Formula:     a+bic+di=(a+bi)(c−di)(c+di)(c−di)=(ac+bd)+(bc−ad)ic2+d2\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}     a+bic+di=ac+bdc2+d2+bc−adc2+d2i\frac{a + bi}{c + di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i

  • Simplification Requirement:

    • After expanding and simplifying, express the final quotient explicitly in standard form a + bi$.\n\n# Powers of i and Periodic Patterns\n\n* **The Four-Term Cyclic Pattern:**\n * Powers of i repeat infinitely in a periodic cycle of four distinct values:\n        i^1 = i\n        i^2 = -1\n        i^3 = i^2 \cdot i = (-1)i = -i\n        i^4 = (i^2)^2 = (-1)^2 = 1\n * Subsequent powers repeat this pattern: i^5 = i,,i^6 = -1,,i^7 = -i,,i^8 = 1, and so forth.\n\n* **Rule for Exponent Multiples of 4:**\n * Because i^4 = 1,anyintegerpowerof, any integer power ofiwheretheexponentisamultipleofwhere the exponent is a multiple of4equalsequals1:\n        i^{4k} = 1 \quad \text{for any integer } k\n\n* **Simplification of Positive Exponents:**\n * To simplify i^n,expresstheexponentintermsofmultiplesof, express the exponent in terms of multiples of4:\n        i^n = i^{4k + r} = (i^4)^k \cdot i^r = 1^k \cdot i^r = i^r\n * **Example (i^{15}):**\n        i^{15} = i^{12 + 3} = (i^4)^3 \cdot i^3 = (1)^3 \cdot (-i) = -i\n\n* **Simplification of Negative Exponents:**\n * Negative powers of icanbesimplifiedbymultiplyingtheexpressionbyanappropriatepowerofcan be simplified by multiplying the expression by an appropriate power ofi^4 = 1 to obtain the least positive exponent.\n * **Example (i^{-3}):**\n        i^{-3} = i^{-3} \cdot 1 = i^{-3} \cdot i^4 = i^{-3 + 4} = i^1 = i$$