We will see how to
count using any
number system.
Counting (number systems)
We will see how to count and represent quantities
using positional number systems, like our own
decimal system.
Skills: counting using any number system
Concepts: positional number systems, base (radix)
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3.0 License.
We will present
number system
concepts.
We count objects –
soldiers, words,
carrots, stones, etc.
Where does this topic fit?
This presentation presents number system
concepts.
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I’m not sure if it is true or not, but I’ve heard that
Roman generals estimated the size of their armies
by having each soldier put a stone on a pile.
The bigger the pile, the bigger the army.
The Romans wrote numbers like this:
MMX
Those Roman numerals were easy to read, but not
suitable for doing arithmetic.
For arithmetic, we needed positional numbers.
Let’s use positional numbers to count stones.
We have all learned
to count using the
decimal or base 10
number system.
Using decimal numbers to count stones
When you were small, you learned to count – you
learned the concept of quantity.
Later, you learned to write ten 10 symbols or
glyphs to represent the quantities 0 through 9.
0
1
2
3
Here is the symbol
or “glyph” for the
quantity five in six
different cultures.
…
5
9
European
Arabic-Indic
Persian and Urdu
Hindi
We can count up to nine stones using our decimal
or base 10 number system since it uses ten
glyphs.
Our glyphs, our “Arabic numerals” are relatively
new.
Here we some ancient glyphs for the quantity
five.
Mayan numbers date back to the fourth century.
Tamil
__
In our decimal
system, what
happens when you
add the tenth stone?
Mayan
Let’s get back to our familiar decimal numbers.
One more stone
What happens when you have nine stones and add
one more?
How can we write that quantity?
?
We move to the
tens position..
Zero, carry 1
There is no glyph for the quantity ten, so we add a
position.
We “carry” one into the tens position, and reset
the ones position to zero.
10
What happens if we put a 2 in the tens position?
That gives us the number 20.
To write the
number for twenty
stones, we need a 2
in the tens position.
How about a two in the tens position and a 3 in
the ones position?
20
That’s 23.
Twenty three has a
2 in the tens
position and a 3 in
the ones position.
You get the idea.
23
What comes after 99?
What comes after
99?
99
The ones and tens position revert to zero and we
carry one to the one hundreds position.
The end result is
100?
This is what happens when the odometer in your
car goes from 99 miles to 100 miles.
100
The counting pattern continues indefinitely.
The pattern
continues with 228
stones.
Here we have 228 stones.
Eight in the ones position, and two in the
hundreds and tens positions.
228
Interpreting the
numbers in each
position.
100
10
1
2
2
8
2 x 100 = 200
2 x 10 = 20
8 x 1
=
8
228
Do you see the pattern in the number positions?
1
10
100
1,000
…
10,000
Number positions
100,000
In the decimal
(base 10) system,
the positions go up
by powers of 10.
As shown here, we can think of 228 as 2 hundreds
+ 2 tens + 8 ones.
228
Each position is ten times the one to its right
because we have ten glyphs in the decimal
number system.
What will be the next position? And the one after
that?
The Mayans used a
base 20 number
system with 20
glyphs – back when
the Romans were
stuck with Roman
numerals.
Mayan base 20 glyphs
We use a decimal number system with 10 glyphs,
but it turns out you can use any base or radix for a
numbering system.
The Mayans used a base 20 number system.
Their 20 glyphs are shown here.
The symbol for zero is a seashell.
They date back to the fourth century.
Computers and
other IT devices
use binary (base 2)
numbers and
programmers often
use base 16
(hexadecimal
numbers).
Today’s IT systems use binary -- base 2 -numbers.
And programmers often use hexadecimal -- base
16 – numbers.
Binary: base 2
Hexadecimal: base 16
Binary numbers are written using only two
glyphs.
Here are the binary
and hexadecimal
glyphs.
For hexadecimal numbers, we need 16 glyphs.
0 1
Here we are
counting in binary,
decimal and
hexadecimal, but
you can use any
base for a number
system.
0 1 2 3 4 5 6 7
8 9 A B C D E F
Counting works the same regardless of the
number system base (radix). When you run out
of combinations, a carry is generated, and you
keep going.
Since there are more glyphs, hexadecimal
numbers are generally shorter than decimal
numbers, which are shorter than binary numbers.
Think how fast the odometer on your car would
overflow to the next position if there were only
two, binary glyphs.
The bigger the base, the shorter the numbers, but
the harder it is to learn to do arithmetic. (We will
see how arithmetic works in a subsequent
presentation).
Binary, decimal and hexadecimal are three
common number systems, but we can count using
any integer as a base.
Binary number
positions increase
as powers of two.
Do you see the pattern in the binary number
positions?
1
2
4
8
32
…
16
Binary number positions
Each position is two times the one to its right
because we have two glyphs in the binary number
system.
What will be the next position? And the one after
that?
A binary number
example
32
16
8
4
2
1
1
0
1
1
0
1
1
0
1
1
0
1
x 32 = 32
x 16 = 0
x 8 = 8
x 4 = 4
x 2 = 0
x 1 = 1
45
1
16
256
4,096
65,536
…
101101 is just a different way to write 45 – they
both signify the same number of stones or
whatever we are counting.
Do you see the pattern in the hexadecimal number
positions?
Hexadecimal number positions
1,048,576
Hexadecimal
number positions
follow the same
pattern, but they
increase as powers
of 16.
Converting the binary number 101101 to decimal
gives 45.
A binary example
Each position is 16 times the one to its right
because we have 16 glyphs in the hexadecimal
number system.
What will be the next position? And the one after
that?
A hexadecimal
number example
We’ve seen how
positional numbers
of any base can be
used to count.
256
16
1
A hexadecimal example
3
6
B
3 x 256 = 768
6 x 16 = 96
B x
1 = 11
875
Summary
Base 20
Converting the hexadecimal number 36B to
decimal gives 875.
36B is just a different way to write 875 – they
both signify the same number of stones or
whatever we are counting.
This presentation focused on positional number
systems.
Base 10
We’ve seen how they worked, using binary,
decimal and hexadecimal numbers as examples.
Base 2
Base 16
All positional number systems are equivalent –
you can express a given quantity using any
number system.
Programmers often communicate in
hexadecimals, today’s IT systems use binary, and
we are used to decimal.
The Mayan number system was base 20.
Self-study
questions
Self study questions
1. If I am counting in binary, what is 11111 plus 1?
2. If I am counting in binary, what is 11111 plus 11?
3. If I am counting in hexadecimal, what is FFFFF plus 1?
4. If I am counting in hexadecimal, what is aa3 plus 3?
5. If I am counting in hexadecimal, what is FFFFF plus F?
6. Write the number 12 (base 10) in binary, hexadecimal and trinary (base 3)
7. Write the number 1101011 (binary) in decimal, hexadecimal and trinary.
8. If I were a Mayan, how would I write the number 550 (base 10)?
9. True or false – all odd binary numbers end in 1?
Links
Links
Mayan numbers:
http://en.wikipedia.org/wiki/Maya_numerals
Hindu-Arabic number system:
http://en.wikipedia.org/wiki/Hindu-Arabic_numeral_system