Lesson 4:
Numbers in binary
Year 8 – Data Representations – Representations: from clay to silicon
Discussion
Starter activity
Take a look at these strange coins (let’s call them ‘boins’).
You only have one of each.
Is there any amount that you won’t be able to pay with these?
You won’t be asked to pay for anything over 31.
Discuss this with the person next to you.
Starter activity
The answer is no.
We can include or exclude each of these boins to form any sum up to 31.
Example: 13
✔
✔
✔
✗
✗
Starter activity
The answer is no.
We can include or exclude each of these boins to form any sum up to 31.
Example: 26
✗
✗
✔
✔
✔
In this lesson, we will...
Objectives
Explore how a sequence of binary digits can represent numbers.
Convert between decimal and binary numbers.
You’ve seen this before
Activity 1
What do we call these symbols?
How many of them are there?
0 1 2 3 4 5 6 7 8 9
We call these symbols digits.
There are 10 of them.
A sequence of decimal digits represents a number.
A sequence of decimal digits represents a number.
Let’s think about this.
Activity 1
4
2
⨉
10
⨉
1
These are called
multipliers or weights
40
2
+
The sum of the products
is the number
Activity 1
3
1
4
⨉
100
⨉
10
⨉
1
300
10
4
+
+
Activity 1
2
7
1
8
⨉
1000
⨉
100
⨉
10
⨉
1
+
2000
700
10
8
+
+
⋯
Can you see a pattern
in the multipliers?
Activity 1
2
7
1
8
⨉
1000
⨉
100
⨉
10
⨉
1
+
2000
700
10
8
+
+
⋯
Multipliers are
powers of ten
⨉10
⨉10
⨉10
⨉10
Activity 1
We use 10 digits and the
decimal (base-10) system
for numbers.
This is probably because we have
10 fingers to count with.
‘digitus’ is Latin for ‘finger’
2
7
1
8
⨉
1000
⨉
100
⨉
10
⨉
1
+
2000
700
10
8
+
+
Enter binary
Activity 2
We will use two digits and the
binary (base-2) system
for numbers.
Same reasoning as in decimal.
Leibniz (1646–1716)
You’ve seen this before
Activity 2
What do we call these symbols?
How many of them are there?
0 1
We call these symbols binary digits.
There are only 2 of them.
A sequence of binary digits represents a number.
Let’s think about this.
Activity 2
1
0
0
1
⨉
◌
⨉
◌
⨉
◌
⨉
◌
In binary, there are also multipliers or weights
Activity 2
1
0
0
1
⨉
8
⨉
4
⨉
2
⨉
1
In binary, there are also multipliers or weights
Activity 2
0
8
0
1
1
0
0
1
⨉
8
⨉
4
⨉
2
⨉
1
⨉2
⨉2
⨉2
Multipliers are
powers of 2
+
+
+
The sum of the products
is the number
9
in decimal
Activity 2
1
0
0
1
⨉
8
⨉
4
⨉
2
⨉
1
9
in decimal
8
1
Activity 2
In binary, we use 2 digits and
the binary (base-2) system
for numbers.
It is convenient for systems
using switches.
1
0
0
1
⨉
8
⨉
4
⨉
2
⨉
1
9
in decimal
8
1
Activity 2
In a sense, binary digits act like switches:
Flip one to on, and the corresponding multiplier is included in the sum
4
2
8
1
1
0
0
1
9
in decimal
Activity 2
Let’s try another example.
0
1
0
1
1
Activity 2
Always start with the multipliers.
What will the next one be?
0
1
0
1
⨉
8
⨉
4
⨉
2
⨉
1
1
⨉
◌
Activity 2
Each multiplier is twice as big as the one before it.
0
1
0
1
⨉
8
⨉
4
⨉
2
⨉
1
1
⨉
16
⨉2
Activity 2
Can you see the decimal number yet?
0
1
0
1
⨉
8
⨉
4
⨉
2
⨉
1
1
⨉
16
Activity 2
When a binary digit equals 1, its multiplier is included in the sum.
0
1
0
1
⨉
8
⨉
4
⨉
2
⨉
1
1
⨉
16
4
0
0
1
+
+
+
16
+
Activity 2
When a binary digit equals 1, its multiplier is included in the sum.
0
1
0
1
⨉
8
⨉
4
⨉
2
⨉
1
1
⨉
16
16
1
4
Activity 2
Compute the sum:
this is the decimal number.
0
1
0
1
⨉
8
⨉
4
⨉
2
⨉
1
1
⨉
16
16
1
4
21
in decimal
Activity 2
Again, bits are like switches:
a value of 1 means that the multiplier is included in the sum.
0
1
0
1
1
16
1
4
21
in decimal
4
2
8
1
16
Convert binary to decimal: instructions
Activity 3
Write multipliers over the bits:
Start with 1 on the right, and double as you go from right to left.
For each bit set to 1, select its corresponding multiplier.
Add up the selected multipliers:
the sum is the decimal number.
1
0
1
0
1
⨉
4
⨉
2
⨉
1
⨉2
⨉2
⨉2
⨉
8
⨉
16
⨉2
16
2
8
26
in decimal
Convert binary to decimal
Activity 3
1
0
1
0
1
26
in decimal
4
2
8
1
16
Activity 3
Remember the boins?
1
0
1
0
1
4
2
8
1
16
Bits to numbers
Activity 3
Solve the problems!
‘Translate’ binary numbers back to the familiar decimal system.
Write the answers on your worksheet.
Bits to numbers: answers
Activity 3
Solve the problems!
‘Translate’ binary numbers back to the familiar decimal system.
Exchange worksheets with a classmate.
Check your peer’s answers and then discuss them together.
When you have finished, ask your teacher to check your work.
Whose cake is it?
Activity 3
You should now be able to understand this puzzle.
Whose cake is it?
Activity 3
There are 10 kinds of people in this world: those who understand binary and those who don’t.
Decimal to binary
Activity 4
Now, we will do the opposite: start with a decimal number and work out the corresponding binary number.
There are a few ways to do this.
We are only going to examine one of them.
Decimal to binary
Activity 4
Which binary digits do I set to 1?
Which multipliers do I select to ‘assemble’ a sum of 13?
◌
◌
◌
◌
⨉
8
⨉
4
⨉
2
⨉
1
◌
⨉
16
13
Decimal to binary
Activity 4
Start with the leftmost bit.
Go through the bits from left to right.
◌
◌
◌
◌
⨉
8
⨉
4
⨉
2
⨉
1
◌
⨉
16
13
Decimal to binary
Activity 4
Do I set this binary digit to 1?
Do I need to select multiplier 16
to ‘assemble’ a sum of 13?
No. It should be set to 0.
Setting it to 1 would include 16 in the sum,
i.e. the sum would exceed 13.
◌
◌
◌
◌
⨉
8
⨉
4
⨉
2
⨉
1
◌
⨉
16
13
0
Decimal to binary
Activity 4
Do I set this binary digit to 1?
Do I need to select multiplier 8
to ‘assemble’ a sum of 13?
Yes. It should be set to 1.
Setting it to 0 would exclude 8 from the sum, so the sum would never reach 13
(because the rest of the multipliers only add up to 7).
◌
◌
◌
◌
⨉
8
⨉
4
⨉
2
⨉
1
⨉
16
13
0
1
5
8
13-8=
Decimal to binary
Activity 4
Do I set this binary digit to 1?
Do I need to select multiplier 4
to ‘assemble’ a sum of 5?
Yes. It should be set to 1.
Setting it to 0 would exclude 4 from the sum, so the sum would never reach 5
(because the rest of the multipliers only add up to 3).
◌
◌
◌
⨉
8
⨉
4
⨉
2
⨉
1
⨉
16
0
1
5
1
1
4
8
5-4=
Decimal to binary
Activity 4
Do I set this binary digit to 1?
Do I need to select multiplier 2
to ‘assemble’ a sum of 1?
No. It should be set to 0.
Setting it to 1 would include 2 in the sum,
i.e. the sum would exceed 1.
◌
◌
⨉
8
⨉
4
⨉
2
⨉
1
⨉
16
0
1
1
1
4
8
0
Decimal to binary
Activity 4
Do I set this binary digit to 1?
Do I need to select multiplier 1
to ‘assemble’ a sum of 1?
Yes.
◌
⨉
8
⨉
4
⨉
2
⨉
1
⨉
16
0
1
1
1
4
8
0
1
1
13
in decimal
Decimal to binary
Activity 4
Let’s try another example.
Decimal to binary
Activity 4
Do I set this binary digit to 1?
Do I need to select multiplier 16
to ‘assemble’ a sum of 22?
Yes. It should be set to 1.
Setting it to 0 would exclude 16 from the sum, so the sum would never reach 22
(because the rest of the multipliers only add up to 15).
◌
◌
◌
◌
⨉
8
⨉
4
⨉
2
⨉
1
◌
⨉
16
22
1
16
6
22-16=
Decimal to binary
Activity 4
Do I set this binary digit to 1?
Do I need to select multiplier 8
to ‘assemble’ a sum of 6?
No. It should be set to 0.
Setting it to 1 would include 8 in the sum,
i.e. the sum would exceed 6.
◌
◌
◌
◌
⨉
8
⨉
4
⨉
2
⨉
1
⨉
16
6
1
0
16
Decimal to binary
Activity 4
Do I set this binary digit to 1?
Do I need to select multiplier 4
to ‘assemble’ a sum of 6?
Yes. It should be set to 1.
Setting it to 0 would exclude 4 from the sum, so the sum would never reach 6
(because the rest of the multipliers only add up to 3).
◌
◌
◌
⨉
8
⨉
4
⨉
2
⨉
1
⨉
16
1
0
6
1
2
4
16
6-4=
Decimal to binary
Activity 4
Do I set this binary digit to 1?
Do I need to select multiplier 2
to ‘assemble’ a sum of 2?
Yes. It should be set to 1.
That makes the sum exactly 2.
Set the rest of the digits to 0.
◌
◌
⨉
8
⨉
4
⨉
2
⨉
1
⨉
16
1
0
1
2
4
1
16
2
0
0
22
in decimal
2-2=
Numbers to bits
Activity 4
Now, you will be given some numbers in decimal.
Can you work out the corresponding binary numbers?
Write the answers on your worksheet.
Numbers to bits: answers
Activity 4
Exchange worksheets with a classmate.
Check your peer’s answers and then discuss them together.
When you have finished, ask your teacher to check your work.
Now, you will be given some numbers in decimal.
Can you work out the corresponding binary numbers?
Homework
Practise converting between binary and decimal
Explorer task: The Voyagers
Programming challenges
Plus: any activities that you did not complete during this lesson
Next lesson
Summary
In this lesson, we...
Explored how numbers can be represented as sequences of decimal and binary digits.
Converted between decimal and binary numbers.
Next lesson, we will...
Examine how we count the number of binary digits in sequences that are really long.
The Voyagers
Interlude
Have you heard of the Voyager spacecraft?
The Voyagers
Interlude
Both spacecrafts carry a golden phonograph record with sounds and images...
The Voyagers
Interlude
Both spacecrafts carry a golden phonograph record with sounds and images, along with instructions on how to decode them.
The Voyagers
Interlude
Take a look at some of the instructions.
Do the symbols look familiar?
The Voyagers
Interlude
Here is one of the images on the record.
It’s like a cheat sheet for reading numbers!
The Voyagers
Interlude
All the numbers on the instructions are encoded in binary, using ❙ and ━.
Why do you think this choice was made?