# Overview

Most generally, number systems are the language we use to count. Our number system is a base-10 system, meaning that it has ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) and uses place values to represent amounts.

### Base systems

The Mayan Civilization used a base-20 system using a combination of dots and lines (a line being equivalent to five dots)

The Babylonians used a base-60 system

Computers use a base-2 (binary) system

### Place value

Unlike Roman numerals where a symbol will always represent the same thing, a '9' can represent many different quantities in our base-10 system. It can represent 9, 90, 900, etc, depending on its place value.

The number 25,387 can be thought of as $2\times10,000 + 5\times1,000 + 3\times 100 + 8\times 10 + 7\times 1$ or $2\times 10^4 + 5\times 10^3 + 3\times 10^2 + 8\times 10^1 + 7\times 10^0$

# Motivating questions and problems

Why do computers use binary? What is binary?

What is a winning strategy for a random Nim game?

Do other civilizations use different numbers?

How do people from other countries count? In French? Spanish? Chinese? (or whatever languages some student knows) What is the same, and what is different?

Where do our numbers come from?

Find $2^0+2^1+2^2+2^3+...+2^{10}$.

# Lesson plans

Specific ways to teach this topic, including:

## Exercises

Convert the following base-2 numbers into base-10.

Convert the following base-5 numbers into base-10.

Convert the following base-10 numbers into base-2.

Convert the following base-10 numbers into base-5.

## Problems

Find $2^0+2^1+2^2+2^3+...+2^{10}$.

What are the advantages and disadvantages of using a base-60 number system instead of a base-10 number system?

What are the advantages and disadvantages of using a base-10 number system instead of the Roman number system?

The number 53,298 is a multiple of 2 because the number ends in an even number. Find a similar divisibility rule for 2 in base-8. What about base-5?

## Extensions

History of different number systems

Operations in other base number systems

Fractions in other base number systems

What would a Fibonacci-base system look like?

What would a base-$\frac{1}{2}$ system look like?

# Motivating topics

The game on Nim

A deeper understanding of why our operation algorithms work