The Story of Numbers · Mr. Scandrett's ClassroomOS

📜 Where Numbers Come From

Numbers weren't invented all at once. They evolved over thousands of years, each civilisation adding another piece. Here are the biggest milestones:

~30,000 BC

Counting
Tally bones. The Lebombo Bone found in Swaziland has 29 notches carved into it — likely a counting tool or lunar calendar. Humans were tracking quantities before writing fully existed.
~3,000 BC

Place Value
Babylonian base-60. The Babylonians counted in groups of 60 — which is why we still have 60 seconds in a minute and 360° in a circle. They were also early pioneers of place-value notation.
~300 BC

Proof & Primes
Greek number theory. Euclid proved there are infinitely many prime numbers. The Greeks had no symbol for zero — they considered it philosophically absurd to have a "number of nothing."

628 AD

The Concept of Zero

Zero becomes a number. Indian mathematician Brahmagupta was the first to treat zero as a real number with arithmetic rules — including what happens when you add, subtract, and multiply with it. Revolutionary.

Concept	Brahmagupta's framing
Positive numbers	"Property" or fortune
Negative numbers	"Debt"
Zero	Nothing / void

~1200 AD

Base-10 System
Hindu-Arabic numerals reach Europe. Fibonacci's Liber Abaci introduced Europeans to the 0–9 system we use today. Before this, Europe used Roman numerals — try doing long division with IV, XII, and XLVIII.
Today
Binary Logic
Binary powers the world. Every computer runs on just two digits: 0 and 1. Every image, video, and message you send is ultimately a very long sequence of zeros and ones.

🔢 The Number Zoo

Not all numbers are the same type. Mathematics organises them into families — and knowing which family a number belongs to tells you a lot about how it behaves. Click any card to learn more.

Integers

…−2, −1, 0, 1, 2…

Whole numbers — positive, negative, and zero. No decimals, no fractions.

Fractions

½, ¾, 22/7

Parts of a whole: one integer divided by another (denominator ≠ 0).

Decimals

0.5, 3.14, 0.333…

Fractions written in base-10 notation using a decimal point.

Ratios

3:1, 16:9, 1:√2

A comparison between two quantities — how much of one relative to another.

Select a number type above to explore it.

Integers Lab

Integers = whole counting numbers

Integers are whole numbers. We use them to count steps, blocks, apples, and days.

0, 1, 2, 3, 4, 5...

10 ten

Level 1 Counting by 1s Showing 11 values

Try counting by 5s. What pattern do you notice?

Fractions Lab

Fractions — The Partition Lab

A fraction shows how a whole is split into equal parts.

3/4 three fourths

Level 2 3 of 4 parts Value 0.75

How many equal parts is the whole split into? How many are filled?

Decimals Lab

Decimals — The Place Value Lab

Decimals show parts of a whole using tens.

0.25 twenty-five hundredths

25 hundredths 2 tenths 25/100

How many hundredths are filled?

Ratios Lab

Ratios — The Comparison Lab

A ratio compares two quantities.

3 : 5 three to five

Level 2 Simple form 3 : 5 Colors

For every 3 red blocks, there are 5 blue blocks.

How many red for every blue?

💡

These families nest inside each other. Every integer is also a fraction (5 = 5/1), and every terminating fraction is also a decimal (¼ = 0.25). Numbers are more related than they first appear.

⭐ Prime Numbers

A prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself. That makes primes the "atoms" of mathematics — every other whole number is built from primes multiplied together.

⚗️

Fundamental Theorem of Arithmetic: Every integer greater than 1 is either prime, or can be written as a unique product of primes. Example: 60 = 2 × 2 × 3 × 5. There is only one way to do this factorisation. Primes are irreducible — the atoms of numbers.

Click any number above to see its factors.

▶ Why primes matter

Encryption: When you buy online, your card is protected because factoring a huge number back into primes is computationally impossible — a 2048-bit key uses a ~600-digit number no computer can crack.

Nature: Cicadas emerge on 13- or 17-year prime cycles to minimise overlap with predator population spikes. Nature discovered primes before mathematicians did.

✖️ Factors & Multiples

Two sides of the same coin. Understanding them makes fractions, algebra, and even music theory much easier.

Factors

Factors of 12: 1, 2, 3, 4, 6, 12

Numbers that divide evenly into a given number. The "ingredients" of a number.

Multiples

Multiples of 4: 4, 8, 12, 16, 20…

What you get by multiplying a number by 1, 2, 3… They go on forever.

Factor Finder

Enter any number (2–10,000) to see all its factors and its prime factorisation.

🎵

Factors in music: A guitar string vibrates not just at its base frequency, but at all whole-number multiples of it (harmonics). That's why a piano and a guitar playing the same note sound different — it's which multiples dominate the sound.

🌀 Modular Arithmetic — Pattern Math

Clock math is just the beginning. When a number wraps around a circle and gets multiplied again and again, modular arithmetic turns into geometry. The result is string art made from pure remainders.

🔢

15 mod 12 = 3 — divide 15 by 12, the remainder is 3.
Written as: a mod n = the remainder when a is divided by n.

Modular String Art

Pick how many points live on the circle and how fast the multiplier changes. At multiplier 2, the pattern forms a cardioid. At 3, a nephroid appears. Arithmetic becomes visible.

Points on Circle 180 Multiplier 2.00 Animation Speed 0.35x

Rule: x → (x × m) mod n

Family: Cardioid threshold

x → (x × 2.00) mod 180

At multiplier 2, every point connects to double itself around the circle. The repeated wraparound makes a heart-like cardioid.

This is why modular arithmetic matters in computer graphics, music sequencing, cryptography, and digital pattern design: the same remainder rule can create rhythm, symmetry, and structure.

Mod in the Real World

🔐 Cryptography

RSA encryption — the system behind HTTPS — is built almost entirely on modular arithmetic with huge primes.

📅 Calendars

Day of the week is mod 7. "300 days from Tuesday?" — 300 mod 7 = 6, so 6 days after Tuesday = Monday.

🎮 Game Loops

Animation frame index = current_frame mod total_frames. The animation wraps seamlessly without any if-statements.

📦 Hash Tables

Computers store data using mod: slot = hash(item) mod table_size. Every dictionary in code uses this.

🔎 The Infinite Zoom Number Line

Between 0 and 1 there is no “next” fraction. Zoom in and the line keeps revealing new rational waypoints: halves, quarters, eighths, sixteenths. Density is the point.

Window: 0.0000 → 1.0000 Resolution: halves

Each zoom step halves the window around the centre. New fractions appear because number lines are not made of isolated pebbles; they are dense with structure.

💥 Why It All Matters: Division by Zero

Every rule in mathematics exists for a reason — and nowhere is this clearer than the rule you've probably heard since primary school: you cannot divide by zero.

Division asks: "How many times does this number fit into that one?" So 10 ÷ 2 = 5 means "2 fits into 10 exactly 5 times." Now ask: how many times does 0 fit into 10? Zero fits in infinitely many times — and infinity isn't a number, it's a concept. The question has no meaningful answer.

Watch What Happens

Drag the divisor toward zero. Watch the result grow without bound.

Divisor: 2.00

10 ÷ 2.00

= 5.00

⚠️

If division by zero were allowed, all of mathematics would collapse. Since 0 × a = 0 × b = 0 for any a and b, dividing both sides by zero would "prove" that a = b for all numbers. We could "prove" 1 = 2, or that 5 = 1,000,000. Every equation would be meaningless.

⚓

Real-world cost: In 1997, the USS Yorktown — a US Navy cruiser — was dead in the water for nearly 3 hours because a crew member entered zero into a database field. A divide-by-zero error cascaded through the ship's software and crashed the entire propulsion control system.

✏️ Practice Problems

1. What is the smallest prime number?

Hint: remember — 1 is not prime. The smallest prime is the first number divisible only by 1 and itself.

2. How many factors does 24 have? (count them all)

Hint: factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 — count them.

3. What is 17 mod 5?

Hint: 5 goes into 17 three times (= 15). What's left over?

4. Convert the fraction ¾ to a decimal.

Hint: divide 3 by 4.

5. It is 9 o'clock. What time will it be in 50 hours? (give a number 1–12)

Hint: (9 + 50) mod 12 = 59 mod 12. 12 × 4 = 48, so 59 − 48 = ?