OAR

Numbers are grouped into nested sets. Each set contains the one before it, so every natural number is also a whole number, an integer, a rational number, and a real number.

Counting numbers beginning with 1 Natural numbers

Whole numbers Natural numbers plus zero

Integers Whole numbers and their negatives

Any number expressible as a/b with integers a and b, b ≠ 0; decimals terminate or

Cannot be written as a ratio of integers; decimals never terminate or repeat

All rational and irrational numbers; every point on the number line

The square root of a whole number is rational only when the number is a perfect square: √49 = 7 is rational, but

A prime number is a natural number greater than 1 with exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. The number 2 is the only even prime. A composite number has more than two factors. The number 1 is neither prime nor composite.

To test whether a number is prime, divide it by each prime up to its square root. For 91, √91 ≈ 9.5, so test 2, 3, 5, and 7: 91 = 7 × 13, so 91 is composite. For 97, √97 ≈ 9.8, and none of 2, 3, 5, or 7 divides 97, so 97 is prime.

Digit sum is divisible by 3

Last two digits form a multiple of 4

Last digit is 0 or 5

Divisible by both 2 and 3

Digit sum is divisible by 9

Even ± even = even

Even × any integer = even

Consecutive integers

n, n + 1, n + 2; consecutive even or odd integers differ by 2.

Zero is an even integer; it is neither positive nor negative.

In base 10, each place is a power of 10. In 4,527.36 the 4 is in the thousands place (4 × 1,000), the 3 is in the tenths place (3/10), and the 6 is in the hundredths place (6/100). Rounding looks only at the digit immediately to the right of

WHAT TO REMEMBER F OR THE EXAM

Every integer is rational; π and √ of non-perfect squares are irrational.

Test primality only with primes up to the square root. 1 is not prime; 2 is the only even prime.

Evaluate expressions in this order: Parentheses and other grouping symbols, Exponents and roots, Multiplication and Division from left to right, then Addition and Subtraction from left to right (PEMDAS). Multiplication does not automatically come before division; whichever appears first from the left is done first.

Subtraction and division are neither commutative nor associative. Division by zero is undefined.

aᵐ × aⁿ = aᵐ⁺ⁿ Product aᵐ ÷ aⁿ = aᵐ⁻ⁿ Quotient (aᵐ)ⁿ = aᵐⁿ Power of a power (ab)ⁿ = aⁿbⁿ Power of a product

A square root of x is a value that multiplied by itself gives x. Memorize perfect squares through 15² = 225 and cubes through 5³ = 125. Simplify roots by factoring out perfect squares: √72 = √(36 × 2) = 6√2. Only like radicals combine: 3√2 + 5√2 = 8√2, but √2 + √3 cannot be simplified. √a × √b = √(ab).

A number in scientific notation is written a × 10ⁿ with 1 ≤ a < 10. To multiply, multiply the coefficients and add the

WHAT TO REMEMBER F OR THE EXAM

Multiply and divide left to right; add and subtract left to right. Add exponents when multiplying like bases; multiply exponents for a power of a power. Simplify radicals by pulling out the largest perfect-square factor.

Signed Numbers and Absolute Value

On a number line, values increase to the right, so −2 > −7. The absolute value |x| is the distance from zero and is never

Add, same signs Add absolute values; keep the sign

Subtract absolute values; take the sign of the larger

Multiply or divide, same signs Result is positive

Multiply or divide, different signs Result is negative

With several factors, count the negatives: an even count gives a positive product, an odd count a negative product. Watch exponents: (−3)² = 9, but −3² = −9 because the exponent applies before the negative sign.

Subtracting a negative is adding a positive. (−a)² is positive; −a² is negative. An odd power of a negative number is negative.

Every composite number has exactly one prime factorization. Divide repeatedly by the smallest prime: 60 ÷ 2 = 30, 30

The GCF is the largest number that divides each number evenly. Multiply the shared primes at their lowest powers. For 48 = 2⁴ × 3 and 72 = 2³ × 3², GCF = 2³ × 3 = 24. The GCF answers questions about splitting items into the largest

The LCM is the smallest positive number that each number divides. Multiply every prime at its highest power. For 12 = 2² × 3 and 18 = 2 × 3², LCM = 2² × 3² = 36. The LCM answers questions about events that repeat on different cycles

Example: Two signal lights flash every 12 and 18 seconds and flash together at time 0. They next flash together after

WHAT TO REMEMBER F OR THE EXAM

A proper fraction is less than 1; an improper fraction is 1 or greater; a mixed number combines a whole number and a fraction (11/4 = 2 3/4). Simplify by dividing numerator and denominator by their GCF: 24/36 = 2/3.

Add or subtract Convert to the least common denominator

Multiply across; cross-cancel first Multiply

Divide Multiply by the reciprocal

Every percent problem fits part = percent × whole. What is 35% of 80? 0.35 × 80 = 28. Fifteen is what percent of 60?

A percent change is always measured from the original value. A percentage-point difference is the simple difference between two percentages (a rate rising from 20% to 25% is a 5-point rise but a 25% increase).

A ratio compares quantities: if officers and enlisted personnel are in the ratio 2:7, then officers make up 2/9 of the total. A proportion states that two ratios are equal; solve by cross-multiplying: 3/5 = x/20 gives 5x = 60, x = 12.

In direct variation y = kx: doubling x doubles y. In inverse variation y = k/x: doubling x halves y. Crew size and time to finish a job vary inversely.

Percent change uses the original value as the base. Successive percent changes multiply; they do not add.

Ratio a:b means the parts are a/(a + b) and b/(a + b) of the whole.

Most Math Skills items are word problems. The method is always the same: define the unknown, translate each sentence into an equation, solve, and check the answer against the question actually asked.

more than, increased by, sum + less than, decreased by, difference −

“miles per gallon”: miles ÷ gallons the difference. Average speed is total distance ÷ total time, not the average of the speeds.

Example: A ship steams 60 nautical miles at 20 knots and returns at 30 knots. Time is 3 h + 2 h = 5 h for 120 nautical miles, so the average speed is 24 knots, not 25.

If one crew finishes a job in a hours and another in b hours, together they complete 1/a + 1/b of the job per hour. Two pumps that empty a tank in 6 h and 3 h work at 1/6 + 1/3 = 1/2 tank per hour, so together they need 2 hours.

Average = sum ÷ count, so sum = average × count. If four scores average 85, a fifth score of 95 raises the average to

Simple interest

P = favorable outcomes ÷ total outcomes. For independent events, multiply: two heads in two tosses = 1/2 × 1/2 = 1/4.

Without replacement

Unit conversion

Average speed = total distance ÷ total time. Work rates add; times do not.

Check that the answer matches what was asked (the other number, the total, the percent).

For inequalities, the same steps apply, except that multiplying or dividing by a negative number reverses the inequality

Each linear equation in two variables is a line; the solution of a system is the intersection point. Parallel lines (same slope, different intercepts) give no solution; identical lines give infinitely many.

Substitution

When a question asks for x + y or another combination, look for a way to add or subtract the equations directly instead of solving for each variable.

WHAT TO REMEMBER F OR THE EXAM

Reverse the inequality sign when multiplying or dividing by a negative.

Same slope, different intercept → no solution. Always substitute back to check both equations.

A polynomial is a sum of terms with non-negative integer exponents, such as 3x² + 2x − 5. The degree is the highest exponent. Add or subtract by combining like terms; subtract by distributing the negative sign to every term: (3x² + 4x −

IDENTITIES TO MEMORIZE Pattern Expansion

Distribute the negative across every term when subtracting. (a + b)² ≠ a² + b²; the middle term 2ab is required. Always check for a common factor before factoring further.

A quadratic equation has the standard form ax² + bx + c = 0 with a ≠ 0. Choose the fastest method for the equation in

Factoring The trinomial factors over the integers

No x term, or a perfect square equals a constant

The Discriminant and Root Relationships

Sum and product

For ax² + bx + c = 0, the roots sum to −b/a and multiply to c/a.

The product of two consecutive positive integers is 72: x(x + 1) = 72 → x² + x − 72 = 0 → (x + 9)(x − 8) = 0. Reject the negative root; the integers are 8 and 9. Area problems work the same way: a rectangle 3 m longer than it is wide with

Set the equation equal to zero before factoring.

Reject roots that make no sense in context (negative lengths, counts).

Roots sum to −b/a and multiply to c/a.

An acute angle is less than 90°, a right angle is 90°, an obtuse angle is between 90° and 180°, and a straight angle is 180°. Complementary angles sum to 90°, supplementary angles to 180°, and vertical angles are equal.

PARALLEL LINES CUT BY A TRANSVERSAL

Angles 1 and 3 are corresponding angles and are equal; angles 2 and 3 are alternate interior angles and are equal; same-side interior angles sum to 180°. Schematic for study purposes.

The interior angles of a triangle sum to 180°, and an exterior angle equals the sum of the two remote interior angles. The longest side is opposite the largest angle, and any side must be shorter than the sum of the other two. Area = ½ × base

Right triangle with legs a and b and hypotenuse c; memorize common Pythagorean triples and their multiples (6-8-10, 9-12-15, 10-24-26).

Sides in the ratio 1 : √3 : 2; the short leg is opposite 30°.

Similar triangles

Equal angles and proportional sides; areas scale by the square of the side ratio.

Isosceles triangle

Two equal sides and two equal base angles.

AREA AND PERIMETER FORMULAS Figure Area

The interior angles of an n-sided polygon sum to (n − 2) × 180°; each angle of a regular hexagon is 720° ÷ 6 = 120°. An arc that subtends a central angle θ has length (θ/360) × 2πr, and the sector has area (θ/360) × πr².

Scaling: if every length is multiplied by k, area is multiplied by k² and volume by k³. Doubling the radius of a sphere multiplies its volume by 8.

COORDINATE FORMULAS Quantity Formula

Slope-intercept form y = mx + b

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1 (negative reciprocals). A horizontal line has slope 0; a vertical line has an undefined slope.

WHAT TO REMEMBER F OR THE EXAM

Triangle angles sum to 180°; quadrilateral angles sum to 360°.

Area scales with the square of length; volume with the cube. Perpendicular slopes are negative reciprocals.