ACT Mathematics — complex numbers, matrices, vectors and radicals

ACT's description of the mathematics section lists, under Number and Quantity, real and complex number systems, integer and rational exponents, and vectors and matrices; the Algebra subcategory adds systems represented by simple matrices. Number and Quantity is the smallest of the five subcategories, about a tenth of the section, but its topics are often missing from a student's preparation altogether rather than merely weak, so a few hours of targeted practice can turn guesses into sure points. This material covers them in short, direct questions of the kind the ACT asks.

The complex-number questions use the four-step cycle of the powers of i, multiply two complex numbers (the step where i² must become −1), divide by a complex number with the conjugate, and find a modulus as a distance in the complex plane. The matrix questions ask for 2A − B entry by entry, multiply a 2 × 2 matrix by a column, compute a 2 × 2 determinant as ad − bc, and decide the dimensions of a product from the inner and outer sizes. Two vector questions find the magnitude of a vector given by its endpoints and a linear combination 2u + w in component form. The material ends with a rational exponent, 16 to the power three fourths, and a sum of two radicals that has to be simplified before the terms can be combined.

The material offers two formats. The quiz has twelve multiple-choice questions with four answer choices each, as on the enhanced ACT, and every explanation names the slip behind each wrong choice, so a miss tells you what to fix. The flashcards collect the rules and formulas worth knowing by heart.

The content is based on ACT's published description of the mathematics section (reporting categories and the topics listed for each). It is independent practice and is not produced or endorsed by ACT.

• Simplify powers of i and add, multiply and divide complex numbers
• Find the modulus of a complex number
• Add, scale and multiply matrices and compute a 2 × 2 determinant
• Find vector components, magnitudes and linear combinations
• Evaluate rational exponents and simplify sums of radicals

Practice material written by Zestly, based on ACT's published description of the ACT mathematics section (act.org, Mathematics Test Description: Preparing for Higher Math — Number & Quantity and Algebra). Formulas beyond the basics are given in the questions, as ACT states that recall of complex formulas is not required.