Counting and numbers

You already met 0 and 1 in Zero and one. Now look around: how many birds? how many cups? You answer by counting — pointing at one thing at a time until everything has been pointed at. This chapter uses pictures of dots to stand for real things, like marbles or fingers.

What is counting?

To count is to touch (or point at) each object once. Say a word for each touch: “one, two, three…” The last word tells you how many there are.

Try it on your fingers: one finger up for each thing you counted. If every object has a finger, the number of fingers matches the number of objects.

Zero and one (recap)

0 means nothing to count; 1 means exactly one. When you count aloud you still say zero before one, then two, and so on. If you want the full picture of nothing versus one — and how that pairs with false and true — revisit Zero and one.

Numbers 1 to 10

Each number has a digit (the symbol), a word, and a picture of dots — one dot for each thing you counted.

Which group has more dots — the bigger picture wins:

7 more

4 fewer

Patterns

A pattern is when something repeats or grows the same way each time. You can often guess what comes next if you look at the pictures.

One more dot each time

Each step adds exactly one dot:

1. 1
2. 2
3. 3
4. 4
5. 5

Two dots at a time (even numbers)

Even counts split into pairs with nothing left over. We jump two dots at a time:

1. 0 none
2. 2
3. 4
4. 6
5. 8
6. 10

One extra dot (odd numbers)

Odd counts also grow by two dots each step, but there is always one dot left alone — it does not fit in a pair. Look for the single dot in orange:

1. 1
2. 3
3. 5
4. 7
5. 9

Circle, square, circle, square…

Patterns can use shapes, not only dots. Watch the pictures repeat:

1. circle
2. square
3. circle
4. square
5. circle
6. square
7. ? next?

Addition

Addition means putting two groups together. Three dots and four more dots make seven dots in one big group. We write 3 + 4 = 7.

Multiplication

Multiplication is many equal groups. Three bags with two sweets each is the same as three pairs of dots. We write 3 × 2 = 6 — “three groups of two.”

You can lay the dots in a rectangle: 3 rows with 2 dots in each row fills a block of 6 dots.

Division

Division is about sharing equally or making equal groups. If 12 biscuits are shared among 4 friends, each friend gets the same number. We write 12 ÷ 4 = 3 — “twelve split into four equal piles gives three in each pile.”

Division is the partner of multiplication. If 3 × 4 = 12, then 12 ÷ 3 = 4 and 12 ÷ 4 = 3. One picture of dots can answer both kinds of question.

History

Long before writing, people kept track of herds, harvests, and trade by making marks — tallies — one mark for each item. Archaeologists have found tally bones tens of thousands of years old.

Different cultures invented symbols for amounts. The idea of zero as a number took time: Indian mathematicians such as Brahmagupta (7th century CE) wrote rules for arithmetic that included zero. The Maya used a shell sign for zero in their calendar. Zero let people write “none” clearly and build place-value systems later on.

Multiplication grew from repeated addition in trade and land measurement — counting equal rows of goods or fields without adding one row at a time forever.

Division appears in sharing problems as old as fair trade: split grain, land, or money into equal parts. Large-number names (million, billion) expanded as empires, astronomy, and finance needed to record bigger totals than everyday counting.

Past ten: tens and hundreds

You can count higher than ten on your fingers — people use place value. Ten single dots can be thought of as one group of ten. Ten groups of ten make one hundred.

1. 1 one
2. × 10
3. 1 ten
4. × 10
5. 1 hundred

The number 47 means 4 tens and 7 ones — forty-seven dots, without drawing every dot one by one.

Big numbers in real life

The world is huge. Counting every grain of sand on a beach one by one would take forever, so we use big number words to talk about amounts we cannot easily picture dot by dot. They matter because governments, scientists, and shops all work with large totals.

Each big word is one thousand times the word before (ten hundreds make a thousand; one thousand thousands make a million). We bundle in tens and thousands so we can compare and add without losing track.

• Money — prices and taxes use thousands and millions.
• Distance — kilometres between cities add up to thousands.
• Time — minutes in a year are in the hundreds of thousands.
• Nature — insects in a forest, cells in your body — counts go huge fast.

Try it yourself

Use the four demos below to practise what you read: count objects one by one, add groups together, build a multiplication array, and share dots equally. Then explore how big a number is — from one dot up to a billion.

Derivation

Why addition combines groups

Suppose the first group has \(a\) objects and the second has \(b\) objects, with no overlap. If you put them in one pile and count from the start, you say \(a\) number words for the first group, then continue with \(b\) more words. The last word is the total \(a + b\). That is why \(3 + 4 = 7\): three objects, then four more, gives seven in all.

Multiplication as repeated addition

If there are \(r\) equal rows and each row has \(c\) dots, you can add \(c + c + \cdots + c\) (\(r\) times) to get the total. Writing \(r \times c\) saves repeating the same addition. For example, \(4 \times 3 = 3 + 3 + 3 + 3 = 12\).

Division as fair sharing

If \(T\) dots are split into \(g\) equal piles with nothing left over, each pile has \(T \div g\) dots. That is the same as asking: what number times \(g\) gives \(T\)? So \(T \div g = q\) when \(g \times q = T\). Example: \(12 \div 4 = 3\) because \(4 \times 3 = 12\).

Place value and big numbers

In base ten, each column is worth ten times the column to its right. So \(47 = 4 \times 10 + 7\). Keep bundling: ten tens make one hundred, ten hundreds make one thousand, and each “thousand step” is one thousand times the step before — ten hundreds make \(1{,}000\), and one thousand thousands make \(1{,}000{,}000\). That is why we group digits in threes when writing millions and billions.

Checkpoints

Pause and answer in your head or with a partner before moving on.

Real-life applications

• Sharing snacks — \(6\) biscuits shared among \(3\) friends is \(6 \div 3 = 2\) each; check with \(3 \times 2 = 6\).
• Rows of chairs — a room with \(4\) rows of \(5\) chairs has \(4 \times 5 = 20\) seats.
• Egg cartons — two rows of \(5\) cups hold \(2 \times 5 = 10\) eggs.
• Counting steps — climbing \(8\) steps then \(2\) more is \(8 + 2 = 10\) steps.
• Population — countries and cities are counted in millions or billions.
• Packaging — eggs in cartons (\(2 \times 5\)) and sweets split among friends (\(12 \div 4\)).

Question bank

Easy

How many fingers on one hand?

Answer: \(5\).

What digit means “none”?

Answer: \(0\) (zero).

Compute \(2 + 5\).

Answer: \(7\).

Steps: Two objects plus five more gives seven.

Intermediate

Compute \(6 + 4\).

Answer: \(10\).

Compute \(3 \times 4\).

Answer: \(12\).

Steps: \(4 + 4 + 4 = 12\), or four rows of three dots.

Which is larger: \(8\) or \(6\)?

Answer: \(8\) is larger.

Difficult

Compute \(5 + 3 + 2\).

Answer: \(10\).

Steps: \(5 + 3 = 8\), then \(8 + 2 = 10\).

A tray has \(3\) rows of \(3\) muffins. How many muffins?

Answer: \(9\).

Steps: \(3 \times 3 = 9\).

Hardcore

Without counting one by one: if \(7 + 3 = 10\), what is \(3 + 7\)? Why?

Answer: \(10\).

Explanation: Addition of whole numbers can be done in either order (commutative property).

Show \(5 \times 2\) as both repeated addition and as a dot rectangle.

Answer: \(10\).

Steps: \(2 + 2 + 2 + 2 + 2 = 10\); five rows of two dots (or two columns of five).

Share \(12\) dots equally among \(4\) piles. How many per pile?

Answer: \(3\) per pile.

Steps: \(12 \div 4 = 3\) because \(4 \times 3 = 12\).

How many zeros are in one million written in digits?

Answer: Six zeros: 1,000,000.

References

• Ifrah, Georges — The Universal History of Numbers (origins of counting and zero).
• Crossley & Henry — early Indian arithmetic including zero.