Speak Maths · Last school years

One sign, many meanings

The same sign means different things on different pages. Learn the traps, and learn to read the field from what stands around the sign.

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.

$∣ x ∣$ Say: the bars round x: size, or the number of membersMeans: Round one thing, bars give its size (a number) or the count of its members (a set). Look at what is inside.
$a ∣ b$ Say: a divides b; or ‘such that’; or ‘given’Means: Between two things the single bar is another sign: divides between numbers, ‘such that’ in a set, ‘given’ in a probability.
$(a, b)$ Say: bracket, a, comma, b: an open interval, a point, or the common divisorMeans: The same two numbers in round brackets can be the numbers between a and b, a point on a grid, or their gcd. The page decides.
$f^{'}$ Say: f prime: the derivative, or a second copyMeans: After f the prime is the derivative; after a point A it is the moved copy A′; after a set it can be the complement.
$\times$ Say: times: multiply, cross product, or all pairsMeans: Between numbers it multiplies; between vectors in space it is the cross product; between sets it forms all pairs.
$\sim$ Say: is similar to; or is distributed asMeans: Between shapes the tilde says same shape, other size. Between a random variable and a law it says which law X follows.
$\nabla f$ Say: nabla f: the gradientMeans: Alone it makes the gradient vector; with a dot it is divergence; with a cross it is curl; squared it is the Laplacian.
$i$ Say: i: a counter, or the square root of minus oneMeans: In a sum or a list, i counts. On a page about complex numbers, i squared is minus one. Decide from the page.

The story: One sign, ten pages

Subtitles:

Tap a line to see its words.

$∣ - 5 ∣ = 5$ On the first page, bars round minus five give its size, without the sign: five.
$∣{a, b, c}∣ = 3$ On the second page, bars round a set count its members: three.
$3 ∣ 12$ On the third page, one bar between two numbers says three divides twelve.
$x \in (1, 3)$ On the fourth page, round brackets with a comma make an open interval: x lies strictly between one and three.
$P = (1, 3)$ On a grid, the same brackets name a point: one across, three up.
$f (x) = x^{2}, f^{'} (2) = 4$ A prime on f means slope. For f of x equal to x squared, the slope at two is four.
$2 \times 3 = 6, e_{1} \times e_{2} = e_{3}$ Between numbers the cross multiplies: six. Between the unit vectors of space it is the cross product and gives the third unit vector.
$X \sim N (0, 1), △ A B C \sim △ D E F$ On a page about chance, the tilde says X follows the standard normal law. On a page about shapes, two triangles are similar.
$f = x^{2} + y^{2}, \nabla f = (2 x, 2 y)$ Nabla f is the gradient: the vector of the partial slopes. Here it is two x, two y.
$i = 1 \sum 3 i = 6, i^{2} = - 1$ In a sum, i is the counter: one plus two plus three is six. On a page of complex numbers, i squared is minus one.

In words this story is 931 letters long; in symbols, 109.

Read it

What does it say?

$∣ - 7 ∣ = 7$

$∣{2, 4, 6, 8}∣ = 4$

$4 ∣ 20$

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$

${x ∣ x > 3}$

$x \in (2, 5)$

$P = (2, 5)$

$(12, 18) = 6$

$f (x) = x^{3}, f^{'} (x) = 3 x^{2}$

$A^{'} = (3, 1)$

$u \times v$

$△ A B C \sim △ D E F$

$Y \sim Bin (10, 0.5)$

$\nabla f = (f_{x}, f_{y})$

$i = 1 \sum 4 i = 10$

$e^{iπ} = - 1$

Write it

Which notation says this?

The size of minus four is four.

The set one, two, three, four, five has five members.

Six divides thirty.

The set of whole numbers n such that n squared is less than ten.

Let R be “it rains” and C “it is cloudy”. The probability of rain, given cloud.

y lies strictly between zero and one.

The point three across and four up is called Q.

The greatest common divisor of eight and twelve is four.

For f of x equal to x squared, the derivative is two x.

The cross product of the vectors u and v.

The gradient of f.

The random variable X follows the normal law with mean zero and variance one.

Triangle ABC is similar to triangle DEF.

The sum of one, two and three, with i as the counter.

Build it

Tap the tiles in order. One tile does not belong.

The set of all x such that x is greater than three.

Four divides twelve.

The size of minus nine is nine.

The probability of A given B.

X follows the standard normal law.

The gradient of f is the vector of its partial slopes.

e to the i pi equals minus one.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 The same bar, twice

Rows are divisors, columns are numbers. Judge each line: true or false in this table? Note the two jobs of the bar in the set-builder lines.

	$6$	$8$	$9$	$12$
$2$	✓	✓		✓
$3$	✓		✓	✓
$4$		✓		✓

A tick means: the row number divides the column number.

$3 ∣ 12$
$∣{b ∣ 2 ∣ b}∣ = 3$
$∣{b ∣ 4 ∣ b}∣ = 3$
$\exists b : 2 ∣ b \land 3 ∣ b \land 4 ∣ b$
$\forall a : a ∣ 6$

2 Brackets or points?

Here every pair of numbers in brackets is an interval, round end open, square end closed. Rows are whole numbers. Judge each line.

	$(1, 3)$	$[1, 3]$	$(2, 4]$
$1$		✓
$2$	✓	✓
$3$		✓	✓
$4$			✓

A tick means: the row number belongs to the column interval.

$2 \in (1, 3)$
$3 \in (1, 3)$
$\forall x : x \in (1, 3) \Rightarrow x \in [1, 3]$
$\exists x : x \in (1, 3) \land x \in (2, 4]$
$\exists x : x \in [1, 3] \land x \in (2, 4]$

3 Triangles of one shape

Four triangles; the tilde means similar. Judge each line about this table.

	$T_{1}$	$T_{2}$	$T_{3}$	$T_{4}$
$T_{1}$	✓	✓		✓
$T_{2}$	✓	✓		✓
$T_{3}$			✓
$T_{4}$	✓	✓		✓

A tick means: the row triangle is similar to the column triangle.

$T_{1} \sim T_{4}$
$T_{2} \sim T_{3}$
$\forall T : T \sim T$
$\forall S, T : S \sim T \Rightarrow T \sim S$
$\exists T : T \sim T_{1} \land T \sim T_{3}$

4 Counting inside the bars

Rows are sets, columns are the numbers that may belong to them. Judge each line.

	$1$	$2$	$3$	$4$
$S_{1}$	✓	✓
$S_{2}$		✓	✓	✓
$S_{3}$				✓

A tick means: the column number is a member of the row set.

$∣ S_{2} ∣ = 3$
$∣ S_{1} ∣ > ∣ S_{3} ∣$
$∣ S_{3} ∣ = 4$
$\exists x : ∣{S ∣ x \in S}∣ = 2$
$\forall x : ∣{S ∣ x \in S}∣ \geq 2$

5 Five jobs of the bar

Pair each line with what it says.

Tap a line, then its meaning.

6 Three primes, three stories

Pair each line with what it says.

Tap a line, then its meaning.

7 Same brackets

Pair each line with what it says.

Tap a line, then its meaning.

8 Four nablas

Pair each line with what it says.

Tap a line, then its meaning.

9 A slope gone wrong

This should say: for f of x equal to x squared, the slope at two is four. One tile is wrong.

Tap the tile that is wrong.

10 A vector from two vectors

This should say: the cross product of two vectors in space is again a vector in space. One tile is wrong.

Tap the tile that is wrong.

11 A number-theory page

This should say: the greatest common divisor of twelve and eighteen is six. One tile is wrong.

Tap the tile that is wrong.

12 Which law?

This should say: X follows the standard normal law. One tile is wrong.

Tap the tile that is wrong.

13 Bars inside bars

How many members does this set have?

$∣{x \in Z ∣ ∣ x ∣ \leq 2}∣$

14 Powers of i

On a page of complex numbers, what is this?

$i^{3} = ?$

15 Counter, not root

Here i is only the counter of the sum. What is the total?

$i = 1 \sum 4 i^{2}$

16 Counting pairs

Here the cross makes all pairs. How many members does the new set have?

$∣{1, 2, 3} \times {a, b}∣$

17 A gradient at a point

What vector is this?

$f (x, y) = x^{2} + y^{2}, \nabla f (1, 2) = ?$

18 One bar, three jobs

Write one line with three statements: the size of minus eight is eight; three divides twelve; the set of all x such that x is greater than zero.

Your version

19 A slope worth a line

The slope of x cubed at one is three. Write it with f and a prime.

Your version

20 Same angles, same shape

If two triangles S and T have two equal angles, they are similar. Write it.

Your version

21 Gradient of a product

For f of x and y equal to x times y, the gradient at the point three, two is the vector two, three. Write it.

Your version

22 The famous line

Euler’s number to the power i times pi, plus one, is zero. Write it.

Your version

Tell your own

Write three lines that use one sign three ways, for example the bar: a size, a divisor, a “given”. Then say which page each line belongs to.

Kept on this device only.

One sign, many meanings

The alphabet

The story: One sign, ten pages

Read it

Write it

Build it

Challenges

1 The same bar, twice

2 Brackets or points?

3 Triangles of one shape

4 Counting inside the bars

5 Five jobs of the bar

6 Three primes, three stories

7 Same brackets

8 Four nablas

9 A slope gone wrong

10 A vector from two vectors

11 A number-theory page

12 Which law?

13 Bars inside bars

14 Powers of i

15 Counter, not root

16 Counting pairs

17 A gradient at a point

18 One bar, three jobs

19 A slope worth a line

20 Same angles, same shape

21 Gradient of a product

22 The famous line

Tell your own

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