Number and Quantity
Number and Quantity
Algebra II extends the real number system to the complex numbers and treats radicals as rational exponents. Know that i = sqrt(-1), i^2 = -1, and powers of i cycle with period 4, so reduce the exponent modulo 4. Be able to add, subtract, multiply, and divide complex numbers; division uses the conjugate of the denominator, because (a+bi)(a-bi) = a^2+b^2 is real. A complex number a+bi has modulus sqrt(a^2+b^2). Complex numbers are closed under addition, subtraction, and multiplication. The discriminant b^2-4ac tells whether a quadratic has two real, one real, or two complex conjugate solutions. For rational exponents, sqrtn = x^(m/n) when defined; the denominator is the root, the numerator is the power. When simplifying radicals, combine like radicals and rationalize denominators. Watch domains: even roots of negative real numbers are not real, while odd roots are. Traps include mishandling the four-step cycle of i (i^35 = i^3 = -i), forgetting to distribute the conjugate to the whole numerator, dropping the real part during division, and applying exponent rules such as (x^a)^b = x^(ab) outside their domain. Also check extraneous solutions after raising both sides of a radical equation to an even power.
Worked example: Simplify (3+2i)/(1-4i). Multiply numerator and denominator by 1+4i: numerator (3+2i)(1+4i) = 3+12i+2i+8i^2 = 3+14i-8 = -5+14i; denominator 1^2+4^2 = 17. Result -5/17 + (14/17)i.
Rule of thumb: for i^n, divide n by 4 and use the remainder; a remainder of 0 gives 1, 1 gives i, 2 gives -1, 3 gives -i.