Algebra 2 Conjugates Quiz

Algebra 2 Conjugates

• The conjugate of a binomial expression changes the sign of the second term.
• Example of a conjugate: For √2 + x, its conjugate is √2 - x.
• For expressions involving square roots:
  • The conjugate of √3 + y is √3 - y.
  • For 5 - √a, the conjugate is 5 + √a.

Complex Conjugates

• Complex conjugates consist of a real part that remains the same and an imaginary part that is negated.
• The complex conjugate of 5 - 2i is 5 + 2i.
• For 1 + 4i, the complex conjugate is 1 - 4i.
• The conjugate of a purely imaginary number like -5i is simply its positive counterpart, 5i.
• Real numbers are their own conjugates (e.g., the conjugate of 12 remains 12).

Properties of Complex Conjugates

• For any complex number of the form a - bi, its conjugate is a + bi.
• The conjugate of negative imaginary numbers like 9i is -9i, with the sign reversed.
• When complex numbers have the same real parts but opposite imaginary parts, they are considered conjugates.

Operations Involving Complex Numbers

• Division by complex numbers involves multiplying both numerator and denominator by the conjugate of the denominator to simplify the expression.
• Example operations:
  • (4-5i) + (7+9i) simplifies to 11 + 4i.
  • (6+12i) + (7-5i) - (3-10i) simplifies to 10 + 7i.
  • The product of 6i and (2-3i) gives 18 + 12i.

Basic Complex Number Properties

• i represents the imaginary unit, defined as √-1.
• Powers of i cycle every four terms:
  • i^0 = 1
  • i^1 = i
  • i^2 = -1
  • i^3 = -i
  • i^4 = 1 (and repeats from there).
• To simplify higher powers of i, reduce the exponent modulo 4.

Simplifications

• Simplifying i^59 results in -i since 59 mod 4 = 3.
• The expression 3i^8 + 5i^15 simplifies to 3 - 5i, with each power calculated using the properties of i.
• (i^5)^7 simplifies to -i, using the cycle of powers.

Key Reminders

• Understanding the conjugate and its properties is essential in manipulating and solving complex equations.
• Always apply conjugates when simplifying or dividing expressions involving complex numbers.