B. Number · Number: Number Sets, Powers and Rational Numbers in Action

The Story of Numbers: Zero, Place Value and Number Systems Around the World

B1.1 A2About 70 minutes

Learning goals

I can explain how place value and zero work in different number systems, including base 10, base 20 and base 60.
I can convert numbers between our base-10 system and Maya (base 20), Babylonian (base 60) and binary (base 2) numbers.
I can research a number concept and tell the story of how a culture developed and used it.
I can describe how that number concept still matters today.

Words to know

number system
A set of symbols and rules for writing numbers, such as Roman numerals or our base-10 digits.

base
In a place-value system, the number of a place that makes one of the next place. In base 10, ten ones make one ten.

place value
The value a digit has because of where it sits in a number. In 507, the 5 means 5 hundreds.

placeholder
A symbol that shows an empty place, so 507 and 57 cannot be confused. Our 0 is a placeholder.

Hindu-Arabic numerals
The digits 0 to 9 we use today. They were developed in India and spread to Europe through the work of Arab and Persian scholars.

binary
Base 2. It uses only the digits 0 and 1. Computers store information in binary.

vigesimal
Another word for base 20.

Remember from Grade 8

In Grade 8 you sorted numbers into the real number system, wrote very large and very small numbers in scientific notation, and used place value with powers of ten, like 10³ = 1000 and 10⁻¹ = 0.1. All of that depends on one big idea: place value. This lesson asks where place value and zero came from, and why they are such a powerful invention.

Why does 2026 not need 2026 symbols?

Try writing this year in Roman numerals: MMXXVI. That is M (1000) + M (1000) + X (10) + X (10) + V (5) + I (1) = 2026. Roman numerals add up the values of letters. There is no symbol for zero, and the position of a letter does not multiply its value.

Now look at 2026 in our digits. The 2 on the left means 2 thousands, not 2. The 0 tells you there are no hundreds. The position of each digit tells you what it is worth. That idea is place value, and it is one of the most important inventions in the history of mathematics.

How a place-value system works

Every place-value system has a base. In our base-10 system, each place is worth 10 times the place to its right:

2026 = 2 × 1000 + 0 × 100 + 2 × 10 + 6 × 1
2026 = 2 × 10³ + 0 × 10² + 2 × 10¹ + 6 × 10⁰

In base 20 each place is worth 20 times the place before it: 1, 20, 400, 8000, ... In base 60 the places are 1, 60, 3600, ... In base 2 (binary) they are 1, 2, 4, 8, 16, ...

A place-value system needs a way to show an empty place. Without zero, how would you tell 26 from 206 or 2006? That is why zero was such a big step.

Base	Place values (right to left)	Digits needed
2 (binary)	1, 2, 4, 8, 16, 32, ...	0 and 1
10 (decimal)	1, 10, 100, 1000, ...	0 to 9
20 (vigesimal)	1, 20, 400, 8000, ...	0 to 19
60 (sexagesimal)	1, 60, 3600, 216 000, ...	0 to 59

Each place is the base times the place before it. A base-b system needs b different digits, including zero.

Three stories of numbers

The Babylonians (about 4000 years ago, in Mesopotamia, present-day Iraq) pressed wedge-shaped marks into clay tablets. They used a base-60 place-value system. For a long time they had no zero, so readers had to guess an empty place from the context. Later scribes used a separate mark to show an empty place. We still use base 60 today: 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a full turn.

The Maya of Mesoamerica (present-day Mexico and Central America) built a base-20 system using only three symbols: a dot for 1, a bar for 5, and a shell for zero. Numbers are written in levels, from the bottom up. The Maya used zero as a placeholder well over a thousand years ago, and used their numbers for detailed calendars and astronomy. (In calendar counts, Maya scribes made the third level worth 360 instead of 400, to fit the year. In this lesson we use the pure base-20 version, where every level is worth 20 times the one below.) Maya peoples today live in Mexico, Guatemala, Belize and Honduras, and millions speak Mayan languages.

Mathematicians in India developed the digits we use today, including zero as a number in its own right. In the year 628, the astronomer Brahmagupta wrote down rules for calculating with zero, such as "a number minus itself is zero." Persian and Arab scholars, such as al-Khwarizmi in Baghdad in the 800s, wrote books about these "Hindu numerals" and spread them. Our words algorithm (from al-Khwarizmi's name) and algebra (from al-jabr, a word in the title of his book) come from this story. In 1202, Fibonacci's book Liber Abaci helped make these digits popular in Europe.

Maya digits are built from dots (1) and bars (5), with a shell for zero. For example, 13 is two bars and three dots: 5 + 5 + 3. The largest digit is 19: three bars and four dots.

Worked example

Example 1: Reading a Maya number

A Maya number has two levels. The top level shows one bar and two dots. The bottom level shows three dots. What number is it in base 10?

Read each level as a digit. Top: one bar and two dots = 5 + 2 = 7. Bottom: three dots = 3.
Levels go up from the bottom: the bottom level is the 1s place, the next level up is the 20s place.
Value = 7 × 20 + 3 × 1 = 140 + 3 = 143.

143

Worked example

Example 2: Writing 2026 as a Maya number

Write 2026 in base 20.

List the base-20 place values: 1, 20, 400, 8000. 8000 is too big, so start with the 400s.
2026 ÷ 400 = 5 remainder 26. So there are 5 four-hundreds (5 × 400 = 2000), with 26 left.
26 ÷ 20 = 1 remainder 6. So there is 1 twenty, with 6 ones left.
Check: 5 × 400 + 1 × 20 + 6 × 1 = 2000 + 20 + 6 = 2026. ✓
In Maya digits: top level 5 (one bar), middle level 1 (one dot), bottom level 6 (one bar and one dot).

2026 = 5 | 1 | 6 in base 20

2026 in Maya numerals. Read from the top down: 5 four-hundreds, 1 twenty and 6 ones. The value of each digit depends on its level, just like the value of a digit in our numbers depends on its place.

Worked example

Example 3: Base 60 lives in your clock

A cross-country ski race took 1 h 14 min 35 s. Write the time in seconds, and explain the base-60 connection.

In base 60, the places for time are seconds (1), minutes (60 s) and hours (60 × 60 = 3600 s).
1 × 3600 = 3600 s; 14 × 60 = 840 s; 35 × 1 = 35 s.
Total: 3600 + 840 + 35 = 4475 s.

4475 s. Hours, minutes and seconds work exactly like a base-60 place-value number with digits 1, 14 and 35.

Worked example

Example 4: Binary, the base of computers

Convert 1011₂ (base 2) to base 10, and convert 13 to base 2.

Binary places from the right are 1, 2, 4, 8. So 1011₂ = 1 × 8 + 0 × 4 + 1 × 2 + 1 × 1 = 8 + 0 + 2 + 1 = 11.
For 13, find the largest place that fits: 8. 13 − 8 = 5. Then 4 fits: 5 − 4 = 1. 2 does not fit (write 0). 1 fits: 1 − 1 = 0.
Places 8, 4, 2, 1 get digits 1, 1, 0, 1.

1011₂ = 11, and 13 = 1101₂

Connections today

Zero and place value make written calculation, decimals and scientific notation possible.
Base 2 is how every phone and computer stores numbers, using only off (0) and on (1).
Base 60 survives in time, in angles, and in latitude and longitude (degrees, minutes, seconds).
Base 20 shows up in French: quatre-vingts ("four twenties") means 80, and quatre-vingt-dix means 90.
Iñupiaq students in Kaktovik, Alaska (the Iñupiat are Inuit who live in northern Alaska) created a set of base-20 numerals in the 1990s to match the base-20 counting of their language. Their Kaktovik numerals were added to Unicode, the system computers use for text, in 2022.

Researching a number concept

Curriculum expectation B1.1 asks you to research a number concept and tell its story. A good number story answers five questions:

What is the concept? (zero, place value, fractions, negative numbers, π, a counting system...)
Who developed or used it, where and when? Be as exact as your sources allow, and do not guess dates.
How did it work? Show at least one real example with numbers.
Why did that culture need it? (trade, calendars, building, astronomy, navigation...)
How is it relevant now? Connect it to something in today's world.

Use more than one reliable source (a museum, a university, a library database, an encyclopedia). When the story belongs to a living culture, such as First Nations, Métis or Inuit communities or Maya peoples, look for sources written by or with people from that community, and describe them in the present tense.

Hands-on activity

Build Maya numbers on the table

You need: about 20 small dried beans or buttons (dots), 10 pencils, toothpicks or craft sticks (bars), 3 coins or shells (zeros), a sheet of paper. Keep the beans away from young children, since they are a choking hazard.

Draw two horizontal lines across the paper to make three levels. Label them 400s (top), 20s (middle), 1s (bottom).
Build your age, then your house or apartment number, then the year you were born.
Have a family member build a mystery number for you to read.
Build 400. What do you need in the two lower levels? (A zero shell in each.)

Model answers: 15 = three bars in the 1s level. 2012 = 5 | 0 | 12 (five four-hundreds, a shell in the 20s level, and 12 = two bars and two dots in the 1s level).

Talk about it: Why does a base-20 system need 20 different digits but only three kinds of symbols?

Just for fun

Riddle: Why did the Roman numeral get lost on the way to the party?

It had no place to go. (Roman numerals have no place value!)

Your turn

Write MMXXVI and CDXLIV as base-10 numbers. (C = 100, D = 500, L = 50; a smaller letter before a larger one is subtracted.)

Write 307 in expanded form using powers of 10.

A Maya number has two bars in the top level and one shell in the bottom level. What is its value?

Write 85 in base 20.

Convert 2 h 30 min 15 s to seconds.

Convert 11001₂ to base 10.

Why is a symbol for zero needed in a place-value system?

Name one way base 60 is still used today.

More practice: warm-up

What is the value of the digit 4 in 6 4 0 3?

Draw (or describe) the Maya digit for 8.

What are the first four place values in base 2?

What is XIV in base 10?

Which culture developed the digits 0 to 9 that we use today?

More practice: on your own

Write 1000 in base 20.

Convert 25 to binary.

A movie is 8000 s long. Write this in hours, minutes and seconds.

A Babylonian number has the digits 2 and 15 (in that order, base 60). What is it in base 10?

Explain why the French word quatre-vingts is a clue to base 20.

Challenge

What is the largest number you can write with three levels of Maya digits? Show your reasoning.

In base 10, multiplying by 10 adds a 0 at the end (37 → 370). What does adding a 0 at the end of a binary number do? Test it with 101₂.

Without zero, a Babylonian reader saw the digits 1 and 30 with no gap marked. Give two different base-10 values it could mean, and explain.

Check your understanding

Quiz

In the base-10 number 5082, what does the 0 tell you?

What is the value of a two-level Maya number with three dots on top and one bar on the bottom?

Which system did the Babylonians use?

What is 10110₂ in base 10?

Who wrote rules for calculating with zero in the year 628?

Why do Roman numerals make long multiplication hard?

For parents and teachers

Big idea: Grade 9 expectation B1.1 asks students to research a number concept, tell the story of its development and use in a specific culture, and describe why it matters today. This lesson models that with place value and zero in three cultures, and gives the math (converting between bases) that makes the story real.

Common difficulties:

Reading Maya levels from the wrong end. The bottom level is the ones; values grow going up.
Thinking base 20 places are 1, 20, 40, 60. Each place is 20 times the one before: 1, 20, 400, 8000.
Treating research as copying. Ask your teen to explain the concept in their own words and to show a worked example.

How to help: Ask your teen to tell you the story of zero at dinner, or to convert the time on the oven clock into seconds since midnight. If they choose a First Nations, Métis or Inuit number concept for their research, encourage sources from those communities (for example, community language programs or Indigenous-led organizations), and avoid websites that present Indigenous knowledge only in the past tense.