Difficulty: Foundational | Prerequisites: None, this is the pre-requisite material for everything else in Calc II.
This material is the bedrock everything in Calculus II rests on. Before you touch integration techniques, sequences, or series, you need fluency with the number system, algebraic manipulation, and the basic geometry of lines. If you are rusty on exponents, factoring, or interval notation, this is where to start. Every later topic in the course assumes you can do this work quickly and without thinking about it.
Real numbers break into naturals, wholes, integers, rationals, and irrationals. Algebra's core rules (commutative, associative, distributive, identity, inverse) let you rearrange and simplify expressions. Exponent laws, radical rules, absolute value properties, and the standard forms of line equations round out the toolkit you will use constantly in Calc II.
Natural numbers
The counting numbers: {1, 2, 3, ...}. Think of these as the numbers you learned first as a child.
Whole numbers
Natural numbers plus zero: {0, 1, 2, 3, ...}. In simple terms, you just add zero to the naturals.
Integers
All positive and negative whole numbers, including zero: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Think of it as the whole number line with no gaps but no fractions.
Rational numbers
Numbers expressible as a/b where a and b are integers and b is not equal to 0. In simple terms, any number you can write as a fraction, including terminating and repeating decimals.
Irrational numbers
Real numbers that are not rational. They cannot be written as a fraction of two integers. Their decimal expansions neither terminate nor repeat. Classic examples: pi and the square root of 2.
Interval notation
A shorthand for describing sets of real numbers between endpoints. Parentheses ( ) mean the endpoint is excluded (open); square brackets [ ] mean it is included (closed). For example, (a, b) = {x | a < x < b} and [a, b] = {x | a <= x <= b}.
Set-builder notation
A way of describing a set by stating the property its members must satisfy: {x | condition on x}. Think of it as a filter applied to all real numbers.
Absolute value
The distance of a number from zero on the number line, always non-negative. Formally: |x| = x if x >= 0, and |x| = -x if x < 0.
Slope
The ratio of vertical change to horizontal change between two points on a line: m = (y2 - y1) / (x2 - x1). Think of it as "rise over run."
Rational exponents
Exponents written as fractions. a^(1/n) = the nth root of a, and a^(m/n) = (nth root of a)^m. In simple terms, the denominator of the exponent tells you which root, and the numerator tells you which power.
Commutative: a + b = b + a; ab = ba
Associative: (a + b) + c = a + (b + c); (ab)c = a(bc)
Distributive: a(b + c) = ab + ac; a(b - c) = ab - ac
Identity: a + 0 = a; a * 1 = a
Inverse: a + (-a) = 0; a * (1/a) = 1 (a not equal to 0)
Multiplication properties: (-1)a = -a; (-1)(-a) = a; a * 0 = 0; (-a)(b) = (a)(-b) = -ab; (-a)(-b) = ab
b^m * b^n = b^(m+n)
b^m / b^n = b^(m-n)
(b^m)^n = b^(mn)
(ab)^n = a^n * b^n
(a/b)^n = a^n / b^n
a^(1/n) = nth root of a
a^(m/n) = (nth root of a)^m or nth root of a^m
a^(-m/n) = 1 / a^(m/n)
If n is even, the nth root of a^n = |a|
If n is odd, the nth root of a^n = a
Product rule: (nth root of a) * (nth root of b) = nth root of ab
Quotient rule: (nth root of a) / (nth root of b) = nth root of (a/b)
|x| = c implies x = c or x = -c (c > 0)
|x| < c implies -c < x < c (c > 0)
|x| > c implies x < -c or x > c (c > 0)
Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept
General form: Ax + By + C = 0
Point-slope form: y - y1 = m(x - x1), where (x1, y1) is a known point
Horizontal line (parallel to x-axis): y = b
Vertical line (parallel to y-axis): x = a
Difference of two squares: A^2 - B^2 = (A + B)(A - B)
Perfect square trinomials: A^2 + 2AB + B^2 = (A + B)^2 and A^2 - 2AB + B^2 = (A - B)^2
Sum of two cubes: A^3 + B^3 = (A + B)(A^2 - AB + B^2)
Difference of two cubes: A^3 - B^3 = (A - B)(A^2 + AB + B^2)
m = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2)A^2 - B^2 = (A+B)(A-B)A^3 + B^3 = (A+B)(A^2 - AB + B^2)A^3 - B^3 = (A-B)(A^2 + AB + B^2)a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^mStudents often think (-a)^2 and -(a^2) are the same thing. They are not. (-a)^2 = a^2 (positive), while -(a^2) is always non-positive.
Confusing open and closed intervals: (a, b) excludes both endpoints, [a, b] includes both. Mixing these up in domain questions loses marks.
Students sometimes distribute an exponent over addition: (a + b)^2 does not equal a^2 + b^2. You must expand using the perfect square trinomial formula.
The absolute value |x| = -x when x is negative does not mean the result is negative. Since x is already negative, -x flips it positive.
Exponent and radical manipulation appear in nearly every integration problem. If you fumble these, you will be slow on every exam question.
Factoring (especially difference of squares and sum/difference of cubes) is essential for partial fraction decomposition, a core Calc II integration technique.
Interval notation is used to express domains, ranges, and intervals of convergence for series. Expect to read and write it constantly.
Line equations come back in linear approximation and tangent-line problems.
True or false: every integer is a rational number.
Simplify: x^(3/2) * x^(1/2).
True or false: (a + b)^2 = a^2 + b^2.
Write the interval (-3, 5] in set-builder notation.
What is |-7| ?
Q: Factor completely: 8x^3 - 27.
A: This is a difference of cubes. A = 2x, B = 3. So 8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9).
Q: Solve |2x - 5| < 3.
A: By the absolute value inequality rule: -3 < 2x - 5 < 3. Add 5: 2 < 2x < 8. Divide by 2: 1 < x < 4. In interval notation: (1, 4).
Q: Simplify (x^4 * x^(-2)) / x^3.
A: Combine exponents in the numerator: x^(4 + (-2)) = x^2. Then x^2 / x^3 = x^(2-3) = x^(-1) = 1/x.
Q: Write the equation of a line with slope -2 passing through (1, 5).
A: Point-slope form: y - 5 = -2(x - 1). Simplify: y = -2x + 7.
Q: True or false: the square root of 4 is plus or minus 2.
A: False. The square root symbol denotes the principal (non-negative) root only: the square root of 4 = 2. The equation x^2 = 4 has solutions x = 2 and x = -2, but the radical itself returns only the positive value.
This material connects directly to trigonometry (next set of notes), where you will combine algebraic manipulation with trig identities. Factoring skills feed into partial fractions, and exponent rules are essential for integration by substitution. Interval notation becomes the standard language for expressing convergence of series and power series later in the course.
Real numbers, natural numbers, whole numbers, integers, rational numbers, irrational numbers, interval notation, set-builder notation, open interval, closed interval, half-open interval, commutative property, associative property, distributive property, exponent rules, laws of exponents, rational exponents, fractional exponents, radicals, nth root, absolute value, slope formula, rise over run, slope-intercept form, point-slope form, general form of a line, difference of squares, perfect square trinomial, sum of cubes, difference of cubes, factoring, algebraic identities, Calc II prerequisites, precalculus review