Intro to Number Systems

The technique to represent and work with numbers is called a number system. The denary number system (commonly called the decimal number system) is the one that we are most familiar with. Over the next few weeks, we will check out a few others as well.

To start with, let's make sure we understand the denary number system.

Firstly, let me ask you to count to ten (in any language)... go on...

Did you start at zero?

Did you stop at nine?

When you got to ten, actually, you have run out of digits to use, and you were using ones you had used before.

The denary number system, as we all already know is a positional value system. This means that the value of digits will depend on its position. If we look at an example, it'll be easier to understand.

If we look at 734, 917 and 207... the digit 7 is actually valued differently in each number:

• In 734, the digit 7 is valued at 700 or 7 x 100

• In 971, the digit 7 is valued at 70, or 7 x 10

• and in 207, the digit 7 is valued at 7 or 7 x 1

For our kids to visualise this, we often use blocks, so the number 2347 might look like this:

Where the blue blocks are worth 1000, the dark red blocks are worth 100, the green blocks are worth 10 and the orange blocks are worth 1 each.

What makes the denary number system what it is, is the fact that each place value is ten times the previous one.

INTRODUCING THE BINARY NUMBER SYSTEM

The video clip above from Khan Academy does a great job of explaining just why we are learning about a whole other number system!

You can cut it off at 3:46 because it goes on to talk about data representation. We will talk about data representation much more fully later on.

Exploring the Binary Number System

Now that we know why we would need a binary number system, let us explore more closely what a binary number system is.

Like the video said, a binary number system is a base-2 number system.

Where a base-10 number system has 10 digits (0,1,2,3,4,5,6,7,8,9) a base-2 number system only has 2 (0,1). The number system most humans count in is a base-10 number system, as we investigated in the first activity.

This activity makes it easy to learn how to convert a base-10 number system) to a base-2 number system.

Download and print Binary Grid and Binary Worksheet (The downloads are at the bottom of the page)

Let's start off easy. Get 5 counters. (or anything you can move around, like coins, lollies, bits of Plasticine etc) You know it is 5, because we can count to 5 in denary, or base-10.

Put the Binary Grid in front of you. You need to put your 5 counters into the tiny boxes. Which columns do the counters fit into?

You have two golden rules that you have to follow:

1. You must only fill from the left.

2. You are only allowed to put your counters into a column if you have enough counters to completely fill that column.

So, with 5 counters:

• ❌5 doesn’t fill into the first column because that has 16 gaps

• ❌5 doesn’t fill the second column because that has 8 gaps.

• ✅5 does fill the third column but there is 1 left over.

• ❌1 does not fill the fourth column, that has 2 gaps.

• ✅1 fills the last column completely

So, this means that the number 5 (base-10) is 00101 (base-2)

Try it again with 10 counters, what do you get?

binary worksheetS

Download the grid and worksheet (as screenshotted above) to run this activity with your students.

• Download the grid

• Download the worksheet

EXTRA FOR EXPERTS

Some of y'all might have already noticed the extra for experts at the bottom of the worksheets.

If you are teaching slightly older children, or children already advanced in numeracy, they will rocket thru the first activity. These are some suggested questions you can pose (with the answers!)

• If 31 in binary is 11111, and you only have five columns, how do you get 32, or 33 or 156!?

  • The answer is that you need another column. The first column we have has 16 boxes in it, every column to the left of that will double in size. (This is why this system is called base-2!) So the new first column will have 32 boxes in it, and then the one after that will have 64 boxes and so on and so forth.

  • So 32 in binary is 100000

  • So 33 in binary is 100001

  • and 156 in binary is 10011100

• How many squares is in the 6th column?

  • We answered this one already - 32 squares

• What is the largest column that we use to count in binary?

  • Just in like base-10 or any number system really, we can go on forever! Infinity is the same in any number system.

• Can you write your name?

  • We have left a box to the left of each number in the worksheet. For some younger students, we sometimes get them to write the alphabet in those boxes and therefore they can use binary as a 'secret code'. In one of the later lessons, we will talk about data representation for text, but this is a simplified version you can use with younger kids.