Math: Numbers, Ratios and Percents

This is Unit Seven of ACT Preparation, and the start of the mathematics: numbers, ratios and percents. The English units closed on order and flow; the mathematics opens on the rules every computation obeys. One discipline runs through this unit: follow the rule for the form in front of you, and change forms only on purpose. By the end of this part you will be able to classify real numbers and compute in the right order. To simplify with the exponent rules and work in scientific notation. To simplify radicals and read rational exponents. To compute with complex numbers. And to work with vectors, matrices and unit rates.

Four entries on the kinds of numbers and the order of operations.

The kinds of numbers.

ACT 0601 The Real Number System

The natural numbers are one, two, three and so on; adding zero gives the whole numbers; adding the negatives gives the integers. Every integer is rational, and the rationals and irrationals together make the real numbers. Negative four is an integer and rational, but not whole. The sets nest.

Rational and irrational.

ACT 0602 Rational and Irrational Numbers

A rational number is a fraction of two integers with a nonzero denominator, and its decimal terminates or repeats: three eighths is 0.375, and two elevenths repeats. An irrational number's decimal never ends and never repeats, like the square root of two or pi. But the square root of forty-nine is seven, an integer.

The order of operations.

ACT 0603 Order of Operations

Evaluate grouping symbols first, then exponents and roots, then multiplication and division from left to right, then addition and subtraction from left to right. Six plus two times three squared is six plus eighteen, twenty-four. Multiplication and division share a level: twelve divided by three, times two, is eight.

A factor that multiplies every term.

ACT 0604 The Distributive Property

The distributive property says a factor outside parentheses multiplies every term inside, and it runs both ways, so a sum can be factored back. Three times the quantity x minus four is 3x minus twelve. The negative sign distributes too: negative two times negative five is positive ten.

Five entries on exponents and on scientific notation.

Three rules for the same base.

ACT 0605 Product, Quotient and Power Rules

For the same base, multiplying adds exponents, dividing subtracts them, and a power of a power multiplies them: x cubed times x to the fourth is x to the seventh, and x cubed, to the fourth, is x to the twelfth. The rules need the same base, and a power of a product applies to each factor: 2x, cubed, is 8x cubed.

Zero and negative exponents.

ACT 0606 Zero and Negative Exponents

Any nonzero number to the zero power is one. A negative exponent means a reciprocal: two to the negative three is one eighth. A negative exponent never makes the number negative; two to the negative three is a positive eighth.

Every rule at once, step by step.

ACT 0607 Simplifying With Exponent Rules

Apply any power of a product to each factor, combine powers of the same base, rewrite negative exponents as reciprocals, then simplify the numbers. In the worked example, squaring gives four x to the sixth, y to the negative fourth; dividing by four x squared y leaves x to the fourth, y to the negative fifth; and y moves below the bar, giving x to the fourth over y to the fifth. Move only the factor carrying the negative exponent: three x to the negative two is three over x squared, not one over 3x squared.

Writing very large and very small numbers.

ACT 0608 Scientific Notation

Scientific notation writes a number as a leading number at least one and less than ten, times a power of ten; small numbers take a negative exponent. Forty-five million is 4.5 times ten to the seventh. Forty-five times ten to the sixth is equal, but not in scientific notation.

Computing in scientific notation, step by step.

ACT 0609 Computing in Scientific Notation

Multiply or divide the leading numbers, combine the powers of ten with the exponent rules, and restore the form if the leading number left the range. Six times ten to the fifth, times four times ten cubed, gives twenty-four times ten to the eighth, which becomes 2.4 times ten to the ninth. Three times ten to the fourth, divided by six times ten to the negative two, gives five times ten to the fifth. Making the leading number ten times smaller makes the power of ten one larger.

Eight entries on radicals, rational exponents and complex numbers.

Square roots, simplified.

ACT 0610 Square Roots and Simplifying Radicals

The principal square root is the non-negative root, and the root of a product is the product of the roots, so a radical simplifies by factoring out perfect squares: the square root of seventy-two is six root two. The square root of twenty-five is five, not plus or minus five, and a root does not split over a sum: the square root of nine plus sixteen is five, not seven.

Which radicals combine.

ACT 0611 Like Radicals

Radicals add or subtract only with the same radicand and index; then the coefficients combine, as with like terms. Three root five plus two root five is five root five. Simplify first: root twelve and root three look unlike until root twelve becomes two root three, and the sum is three root three.

Clearing a radical from a denominator.

ACT 0612 Rationalizing a Denominator

To clear a radical from a denominator, multiply top and bottom by the radical, or for a two-term denominator by its conjugate, the same terms with the sign changed. Three over root two becomes three root two over two. Multiply numerator and denominator by the same factor, so the value is unchanged.

Exponents that are fractions.

ACT 0613 Rational Exponents

A rational exponent is a root: a to the one over n is the nth root of a, and a to the m over n is that root raised to the m. Eight to the two thirds is the cube root of eight, squared, which is four. Take the root first: twenty-seven to the two thirds is nine, without ever computing twenty-seven squared.

Simplifying a radical expression, step by step.

ACT 0614 Simplifying a Radical Expression

Rewrite rational exponents or radicals in whichever form is easier, pull out perfect squares, rationalize any denominator, then combine like radicals. Root fifty plus root eighteen minus four over root two becomes five root two plus three root two minus two root two, which is six root two. Twelve to the one half plus root twenty-seven becomes two root three plus three root three, five root three. Do not combine before simplifying: three terms that looked unlike were all multiples of root two.

The square root of a negative number.

ACT 0615 The Imaginary Unit and Its Powers

The imaginary unit i satisfies i squared equals negative one, so the square root of negative thirty-six is 6i. Powers of i cycle every four: i, negative one, negative i, one, so i to the tenth is negative one. Convert to i before multiplying: root negative four times root negative nine is negative six, not root thirty-six.

Adding and multiplying complex numbers.

ACT 0616 Adding and Multiplying Complex Numbers

A complex number is a plus b i. Add by combining real parts and imaginary parts; multiply by distributing, then replace i squared with negative one. Two plus i, times three minus four i, gives six minus five i plus four, which is ten minus five i. The i squared term turns real.

Dividing complex numbers, step by step.

ACT 0617 Dividing Complex Numbers

Multiply the numerator and the denominator by the denominator's conjugate, which makes the denominator the real number c squared plus d squared; multiply out the numerator, replace i squared, and divide each part. Five plus five i over one plus two i: the denominator becomes five, the numerator fifteen minus five i, and the quotient three minus i. Check by multiplying back: three minus i, times one plus two i, is five plus five i.

Four entries on vectors, matrices, and rates.

Magnitude and direction.

ACT 0618 Vectors

A vector has magnitude and direction and can be written by its components. Vectors add component by component, a scalar multiplies each component, and the magnitude is the square root of the sum of the squared components: the vector three, four has magnitude five. Magnitude is a length, never negative.

Rows and columns.

ACT 0619 Matrices

An m by n matrix has m rows and n columns. Same-size matrices add entry by entry, and A times B exists only when A's column count equals B's row count: a two by three times a three by four gives a two by four. Order matters: AB can exist while BA does not.

Ratios and rates.

ACT 0620 Ratio, Rate and Unit Rate

A ratio compares two quantities by division, in lowest terms: twelve boys to eighteen girls is two to three. A rate compares quantities with different units, and a unit rate gives the amount per one unit: 150 miles in three hours is fifty miles per hour. Order matters in a ratio.

Comparing unit rates, step by step.

ACT 0621 Finding and Comparing Unit Rates

Write each rate with the same quantity on top, divide to get unit rates, and decide first whether higher or lower is better. A twelve-ounce box at $3.60 costs thirty cents an ounce; a twenty-ounce box at $5.40 costs twenty-seven cents an ounce, so the larger box is the better buy, by three cents an ounce. Divide in one direction: mixing price per ounce with ounces per dollar reverses the answer.

Look at what you can now do. You can place a number in its nested sets and compute in the right order, distributing signs as well as numbers. You can apply the exponent rules to the same base, read negative exponents as reciprocals, and keep scientific notation in form. You can simplify radicals before combining them, rationalize denominators, and take the root first in a rational exponent. You can compute with i, and divide by the conjugate. And you can add vectors, check matrix dimensions, and compare unit rates in one direction. Part two turns to proportions and percents.