Symbols

42 pieces of notation from the pathway: how to say each one out loud, what it means, and the node where it first matters.

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Sentences

$a = b$
“a equals b”
The two sides name the same number. An equation is a claim that can be true or false; a formula is an equation that is always true for its letters.
Introduction to algebra
$a \neq b$
“a is not equal to b”
The two sides are different numbers.
Equations and inequalities (one step)
$\pi \approx 3.14$
“pi is approximately 3.14”
Close, not equal. Used for rounded decimals of irrational numbers.
Real numbers · Perimeters and areas of plane figures (circles)
$x < 5$
“x is less than 5”
Every number to the left of 5 on the number line. The open end of the symbol faces the larger number.
Equations and inequalities (one step)
$x \le 5$
“x is less than or equal to 5”
Like < but 5 itself is included — a closed dot on the number line.
Equations and inequalities (one step) · Linear equations and inequalities (two variables)
$\therefore$
“therefore”
What follows is a consequence of what came before. Common in geometry reasons.
Geometry and angle theorems · Euclidean geometry: transformations and congruence
$x = 3 \pm 2$
“x equals 3 plus or minus 2”
Two answers at once: 3 + 2 and 3 − 2. The quadratic formula ends this way.
Quadratic equations and functions

Operations

$3 \times 4 = 3 \cdot 4 = 12$
“three times four”
Multiplication. The dot replaces × once letters appear, so × is not confused with x.
Multiplication · Algebraic expressions
$12 \div 4 = \frac{12}{4} = 12/4$
“twelve divided by four”
Three ways to write one division. The fraction bar is a division sign.
Division · Fractions as division
$17 \div 5 = 3 \text{ R } 2$
“seventeen divided by five is three, remainder two”
Three full groups of 5 and 2 left over. What the remainder means depends on the story.
Division
$2^5 = 32$
“two to the fifth, or two to the power five”
Five factors of 2 multiplied. The small number counts factors; it is not a multiplier.
Multiples and factors (prime factorization) · Exponents
$\sqrt{49} = 7$
“the square root of 49”
The non-negative number whose square is 49. √50 is a point on the number line between 7 and 8, not a 'broken' number.
Real numbers
$\sqrt[3]{27} = 3$
“the cube root of 27”
The number whose cube is 27.
Real numbers
$a^{m/n} = \sqrt[n]{a^m}$
“a to the m over n”
A fractional exponent is a root: the denominator is the root, the numerator the power.
Exponents
$\lvert -7 \rvert = 7$
“the absolute value of negative seven”
Distance from zero, so never negative.
Negative numbers and absolute value
$35\% = \frac{35}{100} = 0.35$
“thirty-five percent”
Per hundred. A percent is a ratio with denominator 100.
Percentage · Percent and percent change
$a : b = 2 : 3$
“a to b is two to three”
For every 2 of a there are 3 of b. A ratio compares; it is not a fraction of the whole unless you add the parts.
Ratio · Ratios
$4.56 \times 10^{6}$
“four point five six times ten to the sixth”
4,560,000. The first factor is between 1 and 10; the power of ten moves the decimal point.
Real numbers
$\gcd(a, b) = \gcd(b, a \bmod b)$
“the g-c-d of a and b equals the g-c-d of b and a mod b”
Euclid's algorithm: a mod b is the remainder when a is divided by b. Repeat until the remainder is 0.
Factors and multiples: Euclid's algorithm

Algebra

$3x + 2$
“three x plus two”
An expression. x stands for a number we have not named; 3x means 3 times x. There is no equals sign, so there is nothing to solve.
Algebraic expressions · Introduction to algebra
$2(x + 3) = 2x + 6$
“two times the quantity x plus three”
Do what is inside first; or distribute the 2 over both terms. An identity — true for every x.
Writing and evaluating expressions · Algebraic expressions
$f(x) = 2x + 1$
“f of x equals two x plus one”
f is a rule; f(x) is its output at input x. f(3) = 7. Not 'f times x'.
Relations and functions
$y = mx + b$
“y equals m x plus b”
A line: m is the slope (rise over run), b the y-intercept. Dimensions 7 writes the intercept as c; same form.
Coordinates and linear graphs · Linear equations and inequalities (two variables)
$y - y_1 = m(x - x_1)$
“y minus y-one equals m times x minus x-one”
A line through the known point (x₁, y₁) with slope m. The subscript 1 means 'the given one'.
Linear equations and inequalities (two variables)
$a_n = a_1 + (n - 1)d$
“a sub n equals a sub one plus n minus one times d”
The n-th term of an arithmetic sequence. a₁ is the first term, d the common difference.
Patterns, sequences, and series
$D = b^2 - 4ac$
“D equals b squared minus four a c”
The part under the square root in the quadratic formula. D > 0: two real roots; D = 0: one; D < 0: the parabola misses the x-axis.
Quadratic equations and functions
$f(x) = a \cdot b^{x}$
“f of x equals a times b to the x”
Exponential growth (b > 1) or decay (0 < b < 1). a is the starting value.
Exponential functions

Geometry

$\angle ABC = 40^\circ$
“angle A B C equals forty degrees”
The angle at B between rays BA and BC. The middle letter is the vertex.
Angles · Geometry and angle theorems
$AB \parallel CD, \quad AB \perp EF$
“A B is parallel to C D; A B is perpendicular to E F”
Parallel lines never meet; perpendicular lines meet at 90°.
Lines and shapes · Angles, triangles, and quadrilaterals
$\triangle ABC \cong \triangle DEF, \quad \triangle ABC \sim \triangle DEF$
“triangle A B C is congruent to triangle D E F; is similar to”
Congruent: same shape and size. Similar: same shape, scaled. The letter order tells which vertices match.
Euclidean geometry: transformations and congruence · Euclidean geometry: transformations and congruence
$C = 2\pi r, \quad A = \pi r^2$
“C equals two pi r; A equals pi r squared”
Circumference and area of a circle. π is the ratio of any circle's circumference to its diameter — irrational, so 22/7 and 3.14 are approximations.
Perimeters and areas of plane figures (circles)
$(x, y) = (3, -2)$
“the point three, negative two”
Three right, two down from the origin. Order matters: (3, −2) and (−2, 3) are different points.
Data analysis and coordinate graphs · Coordinates and graphs (four quadrants)
$A \to A'$
“A maps to A prime”
A′ is the image of A after a transformation. The tick mark means 'the new one'.
Euclidean geometry: transformations and congruence
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
“d equals the square root of x-two minus x-one squared plus y-two minus y-one squared”
Distance between two points — the Pythagorean theorem on the coordinate plane.
Coordinate geometry

Statistics

$\bar{x} = \frac{\sum x}{n}$
“x bar equals the sum of the x's over n”
The mean: add everything, divide by how many. The bar means 'average of'; Σ means 'add them all'.
Line graphs, line plots, and the mean · Statistics: center and spread
$\text{IQR} = Q_3 - Q_1$
“the interquartile range equals Q three minus Q one”
The spread of the middle half of the data. A value more than 1.5 × IQR beyond a quartile is an outlier.
Displaying and comparing data · Statistics: center and spread
$\sigma$
“sigma”
Standard deviation: a typical distance from the mean. Lowercase sigma for a population, s for a sample.
Statistics: center and spread
$r = 0.9$
“r equals zero point nine”
Correlation coefficient, between −1 and 1. Near ±1 means the points hug a line; sign gives the direction. It says nothing about cause.
Data analysis: two variables
$P(\text{heads}) = \frac{1}{2}$
“the probability of heads is one half”
Favourable outcomes over equally likely outcomes. Always between 0 and 1.
Probability

Number sets

$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$
“the naturals are inside the integers, inside the rationals, inside the reals”
Counting numbers; then negatives; then fractions; then every point on the line, including √2 and π.
Real numbers
$\mathbb{R} \setminus \mathbb{Q}$
“the reals minus the rationals”
The irrationals: real numbers that are not fractions. Their decimals never terminate or repeat.
Real numbers
$0.\overline{3} = \frac{1}{3}$
“zero point three repeating”
The bar marks the digits that repeat forever. Every repeating decimal is a fraction.
Decimals: terminating and repeating