Ternary relations

Addition, multiplication, and exponentiation are ternary relations: sets of triples. One triple is drawn as a triangle, and the vertices are its three components. No component is distinguished as the output, so equivalent equations denote the same diagram.

Marks

The color of a triangle names the relation, and so does the letter inside it; the settings choose which of the two to show. One tick marks the compound component: the sum, the product, or the power. Two ticks mark the exponent. Every other vertex is an operand and carries no mark: an addend, a factor, or the base. The operands of a sum or a product are interchangeable, and because they are unmarked they look alike. Only a power needs a second mark, since its base and exponent cannot trade places. With corner caps, a filled corner stands for one tick and an empty corner for two.

AdditionMultiplicationExponentiation

Filled corner: the sum, the product, or the power
Empty corner: the exponent
Unmarked vertex: an addend, a factor, or the base
①A circled numeral names an unnamed component

A — addition

The relation is a = b + c. The ticked vertex is the sum, and the two unmarked vertices are the addends. Subtraction is the same relation, solved for an addend.

a = b + c

b = a - c

c = a - b

M — multiplication

The relation is a = bc. The ticked vertex is the product, and the two unmarked vertices are the factors. Division is the same relation, solved for a factor. A factor may be zero. The triple 1 = b · 0 is not in the relation, and 0 = b · 0 holds for every b. The diagram is the relation, so neither case is special.

a = b c

b = \frac{a}{c}

c = \frac{a}{b}

P — power

The relation is a = b^c. The single tick marks the power, the double tick the exponent, and the unmarked vertex is the base. The root solves the relation for the base. The logarithm solves it for the exponent.

a = b^{c}

b = \sqrt[c]{a}

c = \log_{b} a

Composition

A compound expression is the composition of these relations along a shared vertex. That vertex is the component the composition quantifies. With no letter, it is a bound variable. Each occurrence of a numeral is a distinct vertex, so the three squares in a² + b² = c² are three copies of 2.

a^{2} + b^{2} = c^{2}

c = \sqrt{a^{2} + b^{2}}

a = \sqrt{c^{2} - b^{2}}

b = \sqrt{c^{2} - a^{2}}

These expressions denote the same relation. The radical adds no triangle: it solves the power with exponent 2 for the base.

A chain of additions, or of multiplications, is one relation on all its terms, so it is one shape: a corner for each term, the sum or the product ticked, and the terms unmarked. Drawn as triangles instead, the chain needs corners for values the formula never names.

d = a + b + c

a = d - b - c

b = d - a - c

The expression

The field takes one expression, or one equation between two expressions. Direct notation uses the signs + − * / ^, parentheses, sqrt(a), and log(b, a). A name is one word. Digits at the end are a subscript, and a Greek name is that letter, so q1 is q₁ and pi is π. Adjacent letters are one name, and a space multiplies: m c is a product, while mc is one variable.

The same expression may be written in LaTeX, enclosed in $. There, adjacent factors multiply. The commands are \frac, \sqrt, \log, \ln, \cdot, \times, \div, and the Greek commands.

a ^ bpower
sqrt(a)square root
log(b, a)logarithm of a in base b
q1, pi, tauq₁, π, τ
2 pi ra space multiplies
$mc^{2}$ product, then a power
$\frac{a}{b}$ quotient
$\sqrt[c]{a}$ root of index c
$\log_{b}{a}$ logarithm