Concept guide
Complex numbers: computing (2+3i)(4−5i) as an ordered pair and deciding equality
Let z=2+3i and w=4−5i. Find z+w, z−w and z·w as ordered pairs (real part, imaginary coefficient). For the product, first derive the general formula (a+bi)(c+di)=(ac−bd)+(ad+bc)i from distributivity and i²=−1, then substitute the numbers and cross-check by direct expansion; finally decide whether z·w equals the real number 23.