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Reading a fraction of sums

Long expressions are short ones nested. Learn where a fraction bar, a bracket, or a big sigma begins and ends, and the longest line becomes a few small steps in the right order.

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.

$\frac{a + b}{c + d}$ Say: the whole of a plus b over the whole of c plus dMeans: The fraction bar works as a pair of brackets: everything above it is one block, everything below is another.
$a - (b - (c - d))$ Say: a minus, in brackets, b minus, in brackets, c minus dMeans: Nested brackets are read from the inside out: do the innermost first. A minus in front of a bracket flips every sign inside it.
$\sum_{i = 1}^{n} a_{i} b_{i}$ Say: the sum, for i from 1 to n, of a i times b iMeans: The body of a sum reaches over products and stops at the next plus or minus: here the whole product a i b i is added.
$\sum_{i = 1}^{n} a_{i} + b$ Say: the sum of a i, and then plus bMeans: After a plus the sum has ended. Here b is added once, outside. To add b for every i, write brackets: the sum of a i plus b.
$lim_{n \to \infty} \sum_{i = 1}^{n} a_{i}$ Say: the limit, as n goes to infinity, of the sum of a i for i up to nMeans: First the sum for a fixed n, then the limit as n grows. The n of the sum is the same n the limit moves.
$\prod_{i = 1}^{n} a_{i}$ Say: the product, for i from 1 to n, of a iMeans: Like the sigma, but the terms are multiplied. A single zero makes the whole product zero.
$\sum_{i = 1}^{m} \sum_{j = 1}^{n} a_{ij}$ Say: the sum over i from 1 to m of the sum over j from 1 to n of a i jMeans: Two loops: i is the outer loop, j the inner one and it runs fastest. Together they add every entry of an m by n table.
$a / b + c$ Say: a divided by b, then plus cMeans: A slash on one line binds tighter than plus: divide first, then add. To divide by a sum, put the sum in brackets.

The story: Nine small steps

Subtitles:

Tap a line to see its words.

$i = 1 \sum 4 i = 1 + 2 + 3 + 4 = 10$ A big sum is a loop in short form. For i from one to four, add i: ten.
$i = 1 \sum 4 i^{2} = 1 + 4 + 9 + 16 = 30$ The body can be longer. Add i squared instead: thirty.
$i = 1 \sum 4 i + 1 = 11, i = 1 \sum 4 (i + 1) = 14$ Where the body stops matters. Without brackets, one is added once after the sum: eleven. With brackets, one is added for each i: fourteen.
$\frac{1 + 2}{3 + 4} = \frac{3}{7}$ The fraction bar is a pair of brackets. All of the top over all of the bottom: three sevenths.
$12/3 + 1 = 5, 12/ (3 + 1) = 3$ On one line, the slash divides first. To divide by the sum, use brackets: five against three.
$2 - (3 - (4 - 1)) = 2$ Nested brackets: inside out. Four minus one is three; three minus three is zero; two minus zero is two.
$i = 1 \sum 2 j = 1 \sum 3 ij = 18$ Two loops. For i equal to one, j gives one, two, three: six. For i equal to two, twice that: twelve. Together, eighteen.
$i = 1 \prod 4 i = 1 \cdot 2 \cdot 3 \cdot 4 = 24$ The same loop with a product sign: twenty-four.
$n \to \infty lim i = 1 \sum n \frac{1}{2 ^{i}} = 1$ Last, a limit of sums: half, a quarter, an eighth, and so on. However long we add, the total creeps up to one.

In words this story is 837 letters long; in symbols, 151.

Read it

What does it say?

$\frac{1 + 2}{3 + 4}$

$1 + \frac{2}{3} + 4$

$i = 1 \sum 3 i^{2} = 14$

$i = 1 \sum 3 i + 5 = 11$

$i = 1 \sum 3 (i + 5) = 21$

$i = 1 \sum 2 j = 1 \sum 3 ij = 18$

$12/3 + 1$

$12/ (3 + 1)$

$2 - (3 - (4 - 1))$

$i = 1 \prod 4 i = 24$

$n \to \infty lim i = 1 \sum n \frac{1}{2 ^{i}} = 1$

$\frac{\sum _{i = 1}^{4} i}{4} = \frac{5}{2}$

$\frac{\frac{1}{2}}{\frac{1}{4}} = 2$

$k = 0 \sum n 2^{k} = 2^{n + 1} - 1$

Write it

Which notation says this?

The whole of one plus two, divided by the whole of three plus four.

Add i squared for i from one to four.

Add i for i from one to four, then add one to the total.

Add i plus one for each i from one to four.

Twelve divided by the sum three plus one.

Two minus the bracket: three minus the bracket four minus one.

The product of the numbers from one to five.

For each i from one to two add i times j over j from one to three; add over i.

The total of one over two to the i, for i from one to n, as n grows without bound.

A half divided by a quarter.

The mean of x one up to x n: their total divided by n.

Add the powers of two from two to the zero up to two to the n.

Six over two, and then one added.

Build it

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

Twelve divided by the sum three plus one.

Add i for i from one to n.

Add i plus one for each i from one to n.

Add i for each i, then add one once, outside the sum.

The product of the numbers from one to n.

The limit, as n grows, of the sum of a i for i up to n.

Two minus the bracket three minus one.

Challenges

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

1 Sums over a table

A table of ticks: a_ij is 1 if the cell in row i and column j is ticked, otherwise 0. Judge each line.

	$1$	$2$	$3$	$4$
$1$	✓		✓	✓
$2$		✓	✓
$3$	✓	✓

A tick means a_ij = 1; no tick means a_ij = 0.

$\sum_{j = 1}^{4} a_{1 j} = 3$
$\sum_{i = 1}^{3} a_{i 2} = 2$
$\sum_{i = 1}^{3} \sum_{j = 1}^{4} a_{ij} = 8$
$\sum_{i = 1}^{3} a_{i 1} + a_{22} = 5$
$\sum_{i = 1}^{3} (a_{i 1} + a_{22}) = 5$

2 The partial sums creep up

Let S_n be the sum of 1/2^i for i from one to n. The rows are n, the columns are thresholds. Judge each line.

	$0.5$	$0.7$	$0.9$
$1$	✓
$2$	✓	✓
$3$	✓	✓
$4$	✓	✓	✓

A tick means: S_n is at least the column number.

$S_{4} \geq 0.9$
$S_{3} \geq 0.9$
$\forall n \leq 4 : S_{n} \geq 0.5$
$\exists n \leq 4 : S_{n} \geq 0.95$
$\forall n : S_{n} < 1$

3 A product as “all of them”

Entries are 0 or 1. A product of entries is 1 only if every entry is 1. Judge each line.

	$1$	$2$	$3$
$1$	✓	✓	✓
$2$	✓		✓
$3$

A tick means a_ij = 1; no tick means a_ij = 0.

$\prod_{j = 1}^{3} a_{1 j} = 1$
$\prod_{j = 1}^{3} a_{2 j} = 1$
$\sum_{i = 1}^{3} \prod_{j = 1}^{3} a_{ij} = 1$
$\prod_{i = 1}^{3} \sum_{j = 1}^{3} a_{ij} = 0$
$\prod_{j = 1}^{3} \sum_{i = 1}^{3} a_{ij} = 6$

4 Five fractions

Pair each line with what it says.

Tap a line, then its meaning.

5 One line, many readings

Pair each line with its value.

Tap a line, then its meaning.

6 How far does the sigma reach?

Pair each line with its value.

Tap a line, then its meaning.

7 Four big operators

Pair each line with what it says.

Tap a line, then its meaning.

8 One added once

This should say: add i for i from one to four, and then add one to the total, once. The value is 11. One tile is wrong.

Tap the tile that is wrong.

9 Adding fractions

This should say: a half plus a third is five sixths. One tile is wrong.

Tap the tile that is wrong.

10 Inside out

This should say: two minus the bracket three minus the bracket four minus one. One tile is wrong.

Tap the tile that is wrong.

11 Two loops, one total

This should say: add i times j for i from one to two and j from one to three. One tile is wrong.

Tap the tile that is wrong.

12 Which letter moves?

This should say: the limit, as n grows, of the sum of 1 over 2 to the i, for i from one to n. One tile is wrong.

Tap the tile that is wrong.

13 Two sums, one minus

Read each sum, then subtract. What is the value?

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

14 A tower of fractions

Work from the inside out. What is the value?

$\frac{1}{1 + \frac{1}{1 + \frac{1}{2}}}$

15 A sum that folds up

The brackets put the whole difference inside the body. What is the value?

$i = 1 \sum 4 (\frac{1}{i} - \frac{1}{i + 1})$

16 Many tiny pieces

Each term shrinks to zero, but there are more of them as n grows. What is the limit?

$n \to \infty lim i = 1 \sum n \frac{1}{n}$

17 A product that folds up

Multiply the four factors. What is the value?

$i = 2 \prod 5 (1 - \frac{1}{i})$

18 The mean

The mean of x one up to x n is their total divided by n. Write it as a formula for x bar.

Your version

19 Every entry

Write the total of all entries of a table a with m rows and n columns.

Your version

20 Brackets under a bar

Ten minus the bracket seven minus three, all divided by two, equals three. Write it.

Your version

21 Halves forever

As n grows, the sum of one over two to the i, for i from one to n, comes to one. Write it with the limit.

Your version

22 A line of code

Write, on one line, twelve divided by the sum three plus one.

Your version

Tell your own

Write a long expression of your own with a fraction, a sum and a bracket. Then say aloud, in order, what you would compute first, second and third.

Kept on this device only.

Reading a fraction of sums

The alphabet

The story: Nine small steps

Read it

Write it

Build it

Challenges

1 Sums over a table

2 The partial sums creep up

3 A product as “all of them”

4 Five fractions

5 One line, many readings

6 How far does the sigma reach?

7 Four big operators

8 One added once

9 Adding fractions

10 Inside out

11 Two loops, one total

12 Which letter moves?

13 Two sums, one minus

14 A tower of fractions

15 A sum that folds up

16 Many tiny pieces

17 A product that folds up

18 The mean

19 Every entry

20 Brackets under a bar

21 Halves forever

22 A line of code

Tell your own

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