Speak Maths Β· University
How to read a dense passage
Do not read a dense passage from left to right and hope. Find the definitions, list the symbols, read it aloud, say it in one sentence, then check one case with real numbers.
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: x belongs to AMeans: Tells where a letter lives. Before reading a formula, find this line for every letter: it says what kind of thing the letter is.
- Say: f from A to BMeans: The signature of a function: what goes in (from A) and what comes out (in B). It says nothing yet about the rule.
- Say: x goes to x squaredMeans: The rule itself, written without naming the function: whatever x is, put it in the rule and read off the result.
- Say: for every x in AMeans: Binds the letter x for what follows. Its reach is the rest of the formula or the bracket; outside it, x means nothing.
- Say: a one, a two, and so on up to a nMeans: A pattern left unfinished. Do not guess: write out the first terms with real numbers, then see what the dots stand for.
- Say: if P then QMeans: The main arrow of a passage: the claim in one sentence. Find it first, then ask what P and Q stand for.
- Say: the size of x is at most CMeans: A bound: a limit that something never crosses. Find which letter is fixed (C) and which ones range freely.
The story: Five steps on one function
Tap a line to see its words.
- Step one: find the definitions. A function f takes a real number and gives a real number.
- The rule: every x goes to two x plus one.
- Step two: list the symbols. The letters x and y are any real numbers.
- Step three: read it aloud. The distance between two outputs is twice the distance between the two inputs.
- Step four: try numbers. Three goes to seven, one goes to three.
- The outputs are four apart; twice the inputsβ distance of two is four. The claim holds for this case.
- The dots hide a pattern: the outputs go up by two each time.
- Step five: one sentence. The function never stretches a distance by more than two.
- One consequence: different inputs always give different outputs.
In words this story is 682 letters long; in symbols, 117.
Read it
What does it say?
Write it
Which notation says this?
f takes real numbers to real numbers.
The rule that sends x to x plus three.
Two outputs are at most five times as far apart as the two inputs.
For every real x there is a real y greater than x.
Add the whole numbers from one up to n.
The terms start three, five, seven, and the pattern goes on.
Different inputs give different outputs.
If x is greater than two, its square is greater than four.
f takes every member of A to a member of B.
Each member of A has exactly one partner in B.
Try numbers: for f of x equal to two x plus one, f of three is seven.
The matrix A stretches no vector by more than a factor C.
From some place N on, every term is within epsilon of L.
Build it
Tap the tiles in order. One tile does not belong.
f takes natural numbers to real numbers.
The rule that sends x to its square.
For every x in A.
Two outputs are at most C times as far apart as the two inputs.
If x is greater than two, its square is greater than four.
The terms three, five, seven and so on.
Different inputs give different outputs.
Challenges
Creative puzzles that use every sign of this chapter. Take your time; there is no clock.
1 A function in a table
F(a, b) means: f sends a to b. Judge each line about this table.
| β | |||
| β | |||
| β |
A tick means: f sends the row element to the column element.
Every row has exactly one tick, so f is a function. But two rows share a column (so different inputs can meet), and the third column is never reached.
2 Check a case with numbers
Here f(x) = xΒ² on the numbers 0 to 3, and B(x, y) says: the outputs are at most twice as far apart as the inputs. The table is filled in. Judge each line.
| β | β | β | ||
| β | β | |||
| β | β | |||
| β |
A tick means B(x, y): the outputs f(x) and f(y) are at most 2 times |x β y| apart.
Squaring is gentle near zero and wild further out: between 3 and 1 the outputs differ by 8 but the inputs by only 2, so the factor 2 is broken. Testing one case beats believing a formula.
3 Bigger than, in a small world
G(x, y) means: y is bigger than x. The world has only three numbers. Judge each line.
| β | β | ||
| β | |||
A tick means: the column number is bigger than the row number.
In a world with a top number, βfor every x there is a bigger yβ breaks at the top. Testing a claim on a small world is the quickest way to see how far its scope reaches.
4 Roles in a passage
Pair each piece with the job it does when you read.
Tap a line, then its meaning.
Every dense passage is built of these jobs. Naming the job of each piece is the first half of understanding.
5 What the dots hide
Pair each line with what it says.
Tap a line, then its meaning.
Dots on the baseline with commas make a list; dots in the middle between plus signs make a sum; next to each other they make a product. Check the first terms with real numbers.
6 Five arrows
Pair each line with what it says.
Tap a line, then its meaning.
Arrows of different shapes do different jobs: a long arrow joins statements, an arrow with a foot names a rule, a plain arrow between sets names where a function goes.
7 A signature
This should say: f takes natural numbers to real numbers. One tile is wrong.
Tap the tile that is wrong.
The colon says βf is a function with this signatureβ; the foot arrow is for a rule on a single element.
8 Whose partner?
This should say: for every x in A there is a y in B with y equal to f of x. One tile is wrong.
Tap the tile that is wrong.
Only some y in B need equal f(x); βfor every yβ would force all of B to equal one value. And y must live in B.
9 A bound, not a floor
This should say: two outputs are at most C times as far apart as the two inputs. One tile is wrong.
Tap the tile that is wrong.
βAt mostβ is β€. With β₯ the sentence would give a lower limit instead of a ceiling.
10 A rule, not a consequence
This should name the rule that sends x to its cube. One tile is wrong.
Tap the tile that is wrong.
A double arrow joins two statements (if β¦ then); the rule that sends x to xΒ³ is written with the foot arrow.
11 Plug in, twice
Read the rule, then use it twice. What is f(f(2))?
f(2) = 5, and f(5) = 11. Try with numbers, from the inside out.
12 Name the pattern
The dots hide a rule. Which formula gives a_n?
Test the first terms: 2n + 1 gives 3, 5, 7, 9. The others fail at n = 1 or n = 2.
13 Where does it fail?
Test the claim x by x. Which x breaks it?
For x = 1 take y = 3; for x = 2 take y = 2; for x = 4 no y in {1, 2, 3} works, because y would have to be 0.
14 The smallest ceiling
What is the smallest C for which the bound holds for all x and y in [0, 1]?
|xΒ² β yΒ²| = |x + y|Β·|x β y|, and x + y is at most 2 on [0, 1]. So 2 works; any smaller C fails near x = y = 1.
15 Where does it live?
Whatever natural number n is, in which set does 1/n always lie?
1/n is always a fraction of whole numbers, so it lies in the rationals. It is a natural number only when n = 1.
16 The one sentence, in notation
The function f never stretches a distance by more than three. Write it.
One way:
This is the one-sentence summary of a passage, turned back into notation: the loop of the reading strategy.
17 Different in, different out
Different members of A always go to different members of B. Write it.
One way:
This is injectivity: the arrow goes from βdifferent inputsβ to βdifferent outputsβ, and the quantifier covers every pair.
18 A rule and its first terms
The rule sends n to twice n; the first terms are two, four, six, and so on. Write the rule and the terms.
One way:
The rule explains the dots; the terms let a reader check the rule.
19 Signature and rule
h takes natural numbers to natural numbers and sends n to n squared. Write the signature and the rule in one line.
One way:
Signature first (what goes in and out), rule second (what happens): the two halves of defining a function.
20 One case breaks a claim
The claim says: for every real x, x squared is at least x. Write the case x equal to one half in notation, showing that the claim breaks.
One way:
One number is enough to break a βfor everyβ. Squares of numbers between 0 and 1 are smaller than the numbers.
Chapter finished. The symbols are yours.
Tell your own
Take a two-line passage you do not understand and walk the five steps on it. Write the one-sentence version and one number case.
Kept on this device only.