Speak Maths · Last school years
How a paper’s sentences are built
Papers use a small set of sentence frames: definitions, theorems, proofs, and a handful of abbreviations. Once you know the frames, a page stops looking like a wall.
Draft: written by an AI helper and not yet checked by a teacher.
The alphabet
Read each sign aloud, the way it is written under it.
- Say: is defined to beMeans: Introduces a name. Nothing is claimed or checked: the left side is new, and the right side tells what it stands for.
- Say: such thatMeans: Introduces the property that picks the objects just named: “there is x such that …”.
- Say: if and only ifMeans: Both directions at once: each side follows from the other. A proof owes two arguments.
- Say: without loss of generalityMeans: We prove only one case because the others become the same after renaming. The reader must be able to see why.
- Say: that isMeans: Says the same thing again in other words. What follows is equal to what came before, not an example.
- Say: for exampleMeans: Gives one case among many. It illustrates; it proves nothing about the others.
- Say: equation oneMeans: A number in round brackets beside a line names it, so that later lines can say “by (1)”. Square brackets [1] point to a source instead.
- Say: end of the proofMeans: A small filled square: what had to be shown has been shown. Nothing new follows in this proof.
The story: A tiny theorem
Tap a line to see its words.
- Definition. E is the set of whole numbers n for which some whole number k gives n equal to 2k: the even numbers.
- Theorem. The sum of two even numbers is even.
- Proof. Let n and m be two even numbers.
- Then there are whole numbers a and b such that n equals 2a and m equals 2b.
- Add them: n plus m equals 2 times a plus b. We call this line (1).
- By (1), two divides n plus m; that is, n plus m is even. The proof ends here.
- For example, four plus six is ten, which is even. An example shows the theorem; it does not prove it.
- Remark. n is even if and only if two divides n: each statement follows from the other.
- If we ever need an order, assume without loss of generality that n is at most m: adding gives the same result either way.
In words this story is 730 letters long; in symbols, 126.
Read it
What does it say?
Write it
Which notation says this?
Define S to be the set one, two, three.
There is a real x such that x squared is four.
a is at most b if and only if b minus a is not negative.
Without loss of generality, assume a is at most b.
n equals 2k, that is, two divides n.
The even number m, for example six.
Equation one: x plus y equals ten.
The proof is finished.
For every natural n, n squared is at least n.
Define f of x to be x squared plus one.
A is contained in B if and only if every member of A is a member of B.
Define T to be the sum of i from one to four.
Then there are whole numbers a and b such that n equals 2a and m equals 2b.
Build it
Tap the tiles in order. One tile does not belong.
Define S to be the set of natural numbers below five.
There is an x such that x squared is nine.
a is at most b if and only if b minus a is at least zero.
Equation one: u plus v equals w.
Without loss of generality, a is at most b.
n equals 2k, that is, two divides n; the proof ends.
For example, four plus six is ten.
Challenges
Creative puzzles that use every sign of this chapter. Take your time; there is no clock.
1 Definitions in a table
Three sets were defined on the numbers one to six: E, S and D. Judge each line in this table.
| ✓ | |||
| ✓ | ✓ | ||
| ✓ | ✓ | ||
| ✓ | |||
| ✓ | ✓ |
A tick means: the row number is a member of the column set.
E is exactly the even numbers, so the iff holds. D (multiples of three) and E (even) share only six; and 5 belongs to no set at all.
2 For every epsilon, a delta
W(ε, δ) means: this delta works for this epsilon. Judge each line.
| ✓ | ✓ | ||
| ✓ | ✓ | ||
| ✓ |
A tick means: the column delta works for the row epsilon.
Every row has a tick, so each epsilon gets its own delta. But no column is full: no single delta works for all epsilons. The order of the quantifiers is the whole point of the definition.
3 Who uses which line?
Equations (1) to (3) are used in four proofs, the columns P. Judge each line.
| ✓ | ✓ | ✓ | ||
| ✓ | ||||
| ✓ | ✓ |
A tick U(e, P) means: the proof P cites the equation numbered e.
Every proof cites something, but no equation is cited by all four. The third equation is cited in the third proof, where the second is not.
4 Six little marks
Pair each mark with what it says.
Tap a line, then its meaning.
i.e. and e.g. look alike but differ: the first restates, the second illustrates. The rest are quick signs of the structure of an argument.
5 Five ways to join x and 3
Pair each line with what it says.
Tap a line, then its meaning.
Only := introduces a name. = states a fact to check or use; the arrow is a command in a procedure; ≡ and ≈ say other things entirely.
6 If, only if, iff
Pair each line with what it says.
Tap a line, then its meaning.
The last two lines say the same. “iff” is the word form of the double arrow, and a proof of it needs both directions.
7 Both directions
This should say: a is at most b if and only if b minus a is at least zero. One tile is wrong.
Tap the tile that is wrong.
The arrow covers only one direction, but the sentence promises both: that is exactly iff.
8 Which property?
This should say: there is a real x such that x cubed is eight. One tile is wrong.
Tap the tile that is wrong.
After “there is x” the property that picks x is introduced by “such that”, not by “that is”.
9 An example, not a restatement
The line should give an example: for example, four plus six is ten. One tile is wrong.
Tap the tile that is wrong.
“That is” would claim that all sums are ten. An example is one case, marked by “for example”.
10 A line, not a source
This should say: by equation one, x plus y equals ten. One tile is wrong.
Tap the tile that is wrong.
Round brackets name an equation of this paper; square brackets [1] would point to a reference list instead.
11 How a proof ends
This proof should end with the end-of-proof mark. One tile is wrong.
Tap the tile that is wrong.
The filled square says the proof is complete. The ⊥ would signal a contradiction, which is a different event.
12 Break the iff
This claim is not true for every real x. Which x breaks it?
The “only if” half fails: when x is −2 the square is four but x is not 2. The other values make both sides false, so they keep the iff true.
13 How many cases are left?
After the w.l.o.g. line, how many ordered pairs (a, b) still need checking?
Of the nine pairs, those with a at most b are (1,1), (1,2), (1,3), (2,2), (2,3), (3,3): six.
14 Adding two numbered lines
Add the lines (1) and (2). What is x?
Adding gives 2x = 8, so x = 4 (and y = 3).
15 Name it, then add it
S is defined as the numbers from one to ten that three divides. What is the sum of its members?
S = {3, 6, 9}, and 3 + 6 + 9 = 18.
16 Name and value
Define Q to be the sum of the first four squares, and say its value.
One way:
The := gives the name; the value 30 is a computation (1 + 4 + 9 + 16) that the reader can check.
17 Theorem, then the mark
The product of two odd whole numbers is odd. Write this as a statement with the end-of-proof mark after it (as if the proof were complete).
One way:
The statement is a general rule about all whole numbers a and b; the mark closes an argument that has been given elsewhere.
18 Even if and only if
A whole number is even if and only if its square is even. Write it.
One way:
A real iff: even n gives even n²; and if n² is even then n cannot be odd. Both directions are true.
19 Why the symmetry
Because adding is symmetric, assume without loss of generality that m is at most n. Write that line, with the reason in brackets.
One way:
The bracket is the reader’s ticket: it shows why the other case is the same.
20 A numbered fact
The three angles of a triangle add up to 180 degrees. Write this as equation (1).
One way:
The number in brackets lets later lines say “by (1)” instead of repeating the fact.
Chapter finished. The symbols are yours.
Tell your own
Write a tiny result of your own in paper style: a definition, a statement, a one-line proof, and the end mark.
Kept on this device only.