Faculty of Mathematics
Waterloo, Ontario N2L 3G1
Centre for Education in
Mathematics and Computing
Grade 6 Math Circles
October 25 & 26, 2016
Number Systems and Bases
Numbers are very important. Numbers are a part of our daily lives. We use numbers to
keep track of the time, count the money in our piggy banks, call and text our friends and
family, the list goes on and on and on.
Number System: The mathematical notation for representing a set of numbers using
digits or symbols
Base: The number of unique digits, including zero, used to represent numbers
In today’s society, our knowledge of numbers has grown so much that we have multiple
number systems. But where did they all originate from? There are many ancient number
systems but today we will learn about the ancient Egyptian number system.
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Ancient Egyptian Number System
Egyptian hieroglyphs was the formal writing system used by the ancient Egyptians. Not
only did they have hieroglyphs for words, they had hieroglyphs for numbers as well. Similar
to our decimal system, the ancient Egyptians also used a base 10 number system to represent
numbers as shown in the figure below.
Line
Heel bone
Coil of rope
Water lily
Finger
Tadpole
Man
1
10
100
1 000
10 000
100 000
1 000 000
Contrary to how we write left to right, the ancient Egyptians wrote their numbers from right
to left! To write a number using the Egyptian hieroglyphs, we write the largest value first
followed by smaller values in descending order from right to left.
For example, we would write 4622 as follows:
Example 1 Write 53 441 using the Egyptian hieroglyphs:
Answers may vary.
Example 2 Solve the following equation using Egyptian hieroglyphs:
+
486
+
=
734
=
2
1220
Decimal Numbers
As mentioned earlier, the decimal number system is the number we use to represent
numbers in our lives. Similar to the ancient Egyptians, our decimal number system is a base
10 number system because we have 10 unique digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to represent
numbers.
We can write numbers using these 10 digits or in words as shown in the example below.
4892 =⇒ four thousand eight hundred and ninety two
We can also write numbers using powers of 10! Here are a few powers of 10...
100
101
102
103
104
...
1
10
100
1 000
10 000
...
Below is an example of how we can write numbers in expanded form using powers of 10...
4892
=
4000 + 800 + 90 + 2
=
(4)(1000) + (8)(100) + (9)(10) + (2)(1)
=
(4)(103 ) + (8)(102 ) + (9)(101 ) + (2)(100 )
Example 1 Write 539 in expanded form:
539 = (5)(102 ) + (3)(101 ) + (9)(100 )
Example 2 Write the following in standard form (our usual digit representation):
(3)(103 ) + (4)(102 ) + (7)(101 ) + (5)(100 )
= 3475
3
Binary Numbers
A binary number is a number that is represented by 0s and 1s. For example, 11, 101, 1011
and 101101 are all binary numbers. The binary number system is another commonly used
number system. In fact, all of our electronic devices use binary numbers and we probably do
not even know it! This is a base 2 number system because we represent our numbers using
only two unique digits, 0 and 1.
Binary numbers are read a little differently.
101101 =⇒ one-zero-one-one-zero-one
Converting from Binary to Decimal
Earlier, we learned how to write decimal numbers in expanded form using powers of 10.
Similar to that, we can write binary numbers using powers of 2!
20
21
22
23
24
25
26
27
...
1
2
4
8
16
32
64
128
...
Let’s try an example and rewrite 101101...
101101
−→
(1)(25 ) + (0)(24 ) + (1)(23 ) + (1)(22 ) + (0)(21 ) + (1)(20 )
=
32 + 0 + 8 + 4 + 0 + 1
=
45
So, the binary number 101101 is equal to the decimal number 45 .
Example Convert 111011 into a decimal number:
111011
−→ (1)(25 ) + (1)(24 ) + (1)(23 ) + (0)(22 ) + (1)(21 ) + (1)(20 )
= 32 + 16 + 8 + 0 + 2 + 1
= 59
4
Converting from Decimal to Binary
So we have learned how to convert binary numbers into decimal numbers, but what about
the other way around? We can convert decimal numbers into binary numbers using division!
Recall:
quot
i
e
nt
di
vi
de
nd
di
vi
s
or
r
e
ma
i
nde
r
Suppose we want to convert 62 into a binary number. Since we are working in a base 2
number system, we will divide by 2.
Division
Remainder
62 ÷ 2 = 31
31 ÷ 2 = 15
15 ÷ 2 = 7
7÷2=3
3÷2=1
1÷2=0
0
1
1
1
1
1
If we read the remainder column from the bottom to the top, the decimal number 62 is
converted to the binary number 111110 . Don’t believe it? We can check by converting our
binary number back into a decimal number. Try it for yourself as an exercise!
Example Convert 37 from decimal to binary:
Division
Remainder
37 ÷ 2 = 18
1
18 ÷ 2 = 9
0
9÷2=4
1
4÷2=2
0
2÷2=1
0
1÷2=0
1
37 ⇒ 100101
5
Hexadecimal Numbers
Here is another commonly used number system. The hexadecimal number system is
used in accessing computer memory, in defining colours and more! It is widely used by
computer system designers and programmers. This is a base 16 number system and uses the
following numbers and letters to represent numbers:
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F)
A-F represents the numbers 10-15 respectively. For example, 2A1C is a hexadecimal number.
Converting from Hexadecimal to Decimal
We can convert hexadecimal numbers into decimal numbers in the same way we convert
binary numbers into decimal numbers except this time, we are working in base 16.
160
161
162
163
164
...
1
16
256
4096
65536
...
Rewrite A1C into a decimal number...
A1C
−→
(10)(162 ) + (1)(161 ) + (12)(160 )
=
2560 + 16 + 12
=
2588
The hexadecimal number A1C is equal to the decimal number 2588 .
Example Convert 2C5 from hexadecimal to decimal:
2C5
−→ (2)(162 ) + (12)(161 ) + (5)(20 )
= 512 + 192 + 5
= 709
6
Converting from Decimal to Hexadecimal
To convert a decimal number into a hexadecimal number, we can use the same method we
used to convert decimal numbers into binary!
For an example, let’s convert 11823 into a hexadecimal number. Since hexadecimal numbers
are in a base 16 number system, we will be dividing by 16.
Division
Remainder
11823 ÷ 16 = 738
738 ÷ 16 = 46
46 ÷ 16 = 2
2 ÷ 16 = 0
15 (F)
2
14 (E)
2
Reading the remainder column from the bottom to the top, the decimal number 11823 is
converted to the hexadecimal number 2E2F . Also like before, you check to see if this is
correct by converting the hexadecimal number back into a decimal number.
Example Convert 2548 from decimal into hexadecimal:
Division
Remainder
2548 ÷ 16 = 159
4
159 ÷ 16 = 9
15 (F)
9 ÷ 16 = 0
9
2548 ⇒ 9F4
Notation for Number Systems
There are so many different number systems with different bases, it can get a little confusing.
How can we tell the difference between numbers in different bases? We include the base as
a subscript when writing numbers.
Examples
* The decimal number 435 is written as 43510
* The binary number 110110 is written as 1101102
* 1011012 = 4510
* 2A1C16 = 1078010
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Other Base Number Systems
Decimal, binary, and hexadecimal number systems are the more commonly used systems
that we use. However, there are many different base number systems! For example, you can
have numbers in base 46. The ancient Babylonians used a base 60 number system!
So how do we work with these other base number systems? We can pretty much deal
with any base number system the way we have been dealing with the binary, decimal, and
hexadecimal number systems.
Converting from One Base Number to Another
Suppose we want to convert 14205 into a base 7 number. There is no immediate direct way
of converting from base 5 to base 7. To do this, we do the following steps:
We can convert 14205 into a decimal number as seen below...
14205
=
(1)(53 ) + (4)(52 ) + (2)(51 ) + (0)(50 )
=
125 + 100 + 10 + 0
=
23510
Next, convert 23510 into a base 7 number as we did for the previous few base number systems.
Since we want our number to be in base 7, we will divide by 7!
Division
Remainder
235 ÷ 7 = 33
4
33 ÷ 7 = 4
5
4÷7=0
4
23510 = 4547
Reading the remainder column from the bottom to the top, the correct answer is 4547 .
Example Convert the 10103 into a base 6 number.
10103
= (1)(33 ) + (0)(32 ) + (1)(31 ) + (0)(30 )
= 27 + 0 + 3 + 0
= 3010
⇒ 10103 = 506
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Division
Remainder
30 ÷ 6 = 5
0
5÷6=0
5
3010 = 506
Problem Set Solutions
1. Write the following numbers using Egyptian hieroglyphs: Answers may vary.
(a) 82 073
(b) 123 456
(c) 1 240 518
2. Solve the following using Egyptian hieroglyphs: Answers may vary.
(a)
(b)
+
+
=
=
(c)
+
=
3. What integers are allowed to be digits in the base 7 number system?
0, 1, 2, 3, 4, 5, 6
9
4. Write the following numbers in expanded form:
(a) 622110 = (6)(103 ) + (2)(102 ) + (2)(101 ) + (1)(100 )
(b) 1011012 = (1)(25 ) + (0)(24 ) + (1)(23 ) + (1)(22 ) + (0)(21 ) + (1)(20 )
(c) F4A116 = (15)(163 ) + (4)(162 ) + (10)(161 ) + (1)(160 )
(d) 246078 = (2)(84 ) + (4)(83 ) + (6)(82 ) + (0)(81 ) + (7)(80 )
5. Write the following in standard form:
(a) (2)(103 ) + (1)(102 ) + (0)(101 ) + (6)(100 ) = 210610
(b) (8)(162 ) + (11)(161 ) + (3)(160 ) = 8B316
(c) (1)(24 ) + (1)(23 ) + (0)(22 ) + (1)(21 ) + (0)(20 ) = 110102
(d) (10)(183 ) + (6)(182 ) + (17)(181 ) + (2)(180 ) = A6H218
6. Why do you think we use a base 10 number system?
We learn how to count using our hands and we have a total of 10 fingers, 5 on each
hand. This is a simple reason as to why we use a base 10 number system (better
known as our decimal system). If we excluded our thumbs, we would be using a base
8 number system!
7. Convert the following numbers into a decimal number:
(a) 10112 = 1110
(b) 1EA716 = 784710
(c) 33214 = 24910
8. Convert the following numbers from decimal to binary:
(a) 8810 = 10110002
(b) 52410 = 10000011002
(c) 101510 = 11111101112
9. Convert the following numbers from decimal to hexadecimal:
(a) 48810 = 1E816
(b) 1003510 = 273316
(c) 1245110 = 30A316
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10. Convert the following numbers into the base number given in parentheses:
(a) 1223 (base 7) ⇒ 237
(b) 8429 (base 5) ⇒ 102215
(c) EG4217 (base 15) ⇒ 16B8615
*11. We can use number systems to encrypt secret messages as well!
(a) Convert the alphabet into binary numbers. (ex. A → 1, B → 10, C → 11, ...)
A
B
C
D
E
F
G
H
I
J
K
L
M
N
1
10
11
100
101
110
111
1000
1001
1010
1011
1100
1101
1110
O
P
Q
R
S
T
U
V
W
X
Y
Z
1111
10000
10001
10010
10011
10100
10101
10110
10111
11000
11001
11010
(b) Decrypt the message below:
“1-1100-1100 1111-10101-10010 100-10010-101-1-1101-10011 11-1-1110
11-1111-1101-101 10100-10010-10101-101, 1001-110 10111-101
1000-1-10110-101 10100-1000-101 11-1111-10101-10010-1-111-101
101000-1111 10000-10101-10010-10011-10101-101 10100-1000-101-1101."
- Walt Disney
“All our dreams can come true, if we have the courage to pursue them.”
- Walt Disney
**12. Do the following binary calculations: (Hint: It might help to convert the numbers into
decimal numbers. But remember your answer must be a binary number!)
0
(a) + 1
10
(b) + 1
11
(c) + 10
111
(d) + 101
1
11
101
1000
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**13. (a) How many possible binary numbers with 6 digits are there?
There are a few possible solutions. Here is one solution: Each digit can be 0
or 1. Since there are two possible values for each digit, there are 26 numbers.
However, if the first digit is 0, the number becomes a 5 digit number which we
cannot include in our count. We need to remove the 25 numbers with 5 digits. So
we have 26 − 25 = 64 − 32 = 32.
⇒ There are 32 binary numbers with 6 digits.
(b) How many possible 6 digit binary numbers are there where the last digit is 1?
Notice that all binary numbers whose last digit is 0 are even, and all binary
numbers whose last digit is 1 are odd. Since there are 32 binary numbers with 6
digits, half of them must be odd.
⇒ Therefore, there are 16 binary numbers with 6 digits whose last digit is 1.
***14. For any given length n ≥ 2
(a) How many possible binary numbers with n digits are there? 2n−1
(b) How many possible numbers are there where the last digit is 1? 2n−2
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