Download Bases Slides - Dr Frost Maths

KS3: Bases
Dr J Frost ([email protected])
Objectives:
1. To appreciate how we can have different number systems using
different ‘bases’.
2. Count in different bases.
3. To convert numbers from decimal to another base.
4. To convert numbers from any base to decimal.
Last modified: 22nd June 2015
Starter
What 3 numbers comes next in each sequence?
Base 10
Base 2
Base 3
Base 5
Base 2
Base 16
5
6
7
8
9
10
11
12
0
1
10
11 ?
100
2
10
11
12
344
400
401?
402
111
1000
1001?
1010
99
9A
9B ?
9C
?
?
Follow up questions:
• What do you think it means for our ‘normal’ number system to be base 10?
Each digit has 10 ?
possible values.
• What is the rule for counting (if say in ‘base 6’)
Once the digit goes past 5, we jump back ?
to 0 (and the digit to the left goes up by 1)
Base
!
The base of a number system is the number of possible values
for each digit.
Values for each
digit
0 to 9
0 to 1
0 to F
(A=10, B=11, ... F=15)
Base
10
2
16
?
?
?
Name of
number system
Decimal?
Binary ?
Hexadecimal
?
Counting Game!
Everyone stand up. Take it in turns to count in ternary (base 3), starting at 0. If you get it
wrong, you sit down.
0
1
2
10
11
12
20
21
22
100
101
102
110
111
112
120
121
122
200
201
202
210
211
212
220
221
222
1000
1001
1002
1010
1011
1012
1020
1021
1022
1100
1101
1102
1110
1111
1112
1120
1121
1122
1200
1201
1202
1210
1211
1212
1220
1221
1222
2000
2001
2002
2010
2011
2012
2020
Exercise 1
1
Write out the first ten numbers in each of these bases, starting at 1.
a
Base 2: 1, 10, 11, 100, 101, 110, ?
111, 1000, 1001, 1010
b
Base 5: 1, 2, 3, 4, 10, 11, 12, 13, ?
14, 20
c
Base 4: 1, 2, 3, 10, 11, 12, 13, 20,?21, 22
2
a
b
What number comes after 555 in:
Base 6? 1000
?
Base 7? 556
?
3
How many times does the digit 0 occur if you write out the numbers 1 to 111111
in binary? (Hint: consider all two-digit numbers, then three, and so on)
2 digit numbers: 1 occurrence
3 digit numbers: 4 occurrences
4 digit numbers: 12 occurrences
?
5 digit numbers: 36 occurrences
6 digit number: 108 occurrences
Total = 161
Any Base → Decimal
If we were to write out the digits of the decimal
number “2493”, what is the value of each digit?
(Hint: Think primary school!)
1000 100? 10
multiply
1
2 4 9 310
?
2000 +400 +90?+ 3 = 2493
This means number
is in base 10. We
don’t include it if
the base is obvious
from the context.
Any Base → Decimal
Now suppose we had a number in base 5 instead.
How do we convert it to decimal?
125 25 ? 5
multiply
1
4 3 0 15
500 + 75 + 0 ?+ 1 = 576?
Test Your Understanding
Copy and complete in your book.
8
4
?2
64 16 ? 4
1
1 0 1 12
8 + 0 +? 2 + 1 = 11
27 9 ?3
1
3 3 0 24
192 + 48 + ? 0 + 2 = 242
1
1 2 2 03
27 + 18 +? 6 + 0 = 51
The Maya numeral system is base 20
(“vigesimal”).
Use the approach you used for
converting other bases to decimal to
vote for the correct number.
Example
20
60
3
30
1
+
0
= 0
60
300
Q1
20
120
66
126
1
+
6
= 6
105
156
Q2
123
243
53
223
Q3
123
243
53
223
Q4
239
144
129
1
Q5
400
400
121
20
+
211
0
1
+
11 = 411
111
411
Q6
490
1180
1980
1380
Q7
8000
152000
157784
400
+
5600
582984
20
+
180
=
396884
1
+4
196884
Exercise 2
1
Convert the following numbers from
the indicated base to decimal.
11012
1112
1100112
10223
7348
2335
5306
2
13
7
51
35
476
68
198
?
?
?
?
?
?
?
What is the following Mayan number
in decimal?
4
?
5
When the number “a036” in base 7
is converted to decimal, the value is
1742. Determine the value of the
digit 𝑎.
𝟑𝟒𝟑𝒂 + 𝟎 + 𝟐𝟏 + 𝟔 = 𝟏𝟕𝟒𝟐
𝒂=𝟓
?
In general, what is the largest number in decimal that
can be represented by 𝑛 binary digits? Give your
answer in terms of 𝑛.
𝟐𝒏 − 𝟏
?
6
A three-digit number is 100 in decimal. What’s the
smallest the base can be?
In base 4 the biggest number in decimal is 𝟒𝟑 − 𝟏
= 𝟔𝟑, whereas in base 5 it’s 𝟓𝟑 − 𝟏 = 𝟏𝟐𝟒. So 5 is
the smallest base.
?
N
160?001
3
In computing, a byte consists of 8 bits, where each bit
is a binary digit. What is the largest possible number
in decimal that a byte can represent?
𝟐𝟓𝟓
The number with digits "𝑎1𝑏", where 𝑎 and 𝑏 are
unknown digits, is 107 in decimal if the number was
originally in base 5, and 205 in decimal if it was
originally in base 7. Determine 𝑎 and 𝑏.
𝟐𝟓𝒂 + 𝟓 + 𝒃 = 𝟏𝟎𝟕
𝟒𝟗𝒂 + 𝟕 + 𝒃 = 𝟐𝟎𝟒
Solving, 𝒂 = 𝟒, 𝒃 = 𝟐.
?
N
432 is in an unknown base, but when converted to
decimal, gives 164. Determine the base.
Let the base be 𝒃. Then 𝟒𝒃𝟐 + 𝟑𝒃 + 𝟐 = 𝟏𝟔𝟒.
Solving this quadratic equation gives 𝒃 = 𝟔.
?
Summary So Far
We have learnt that the numbers we use in everyday life are in
“base 10”.
?
But numbers can be in any ‘base’ such as base 2 (binary).
?
The base of a number system is
the number of possible
? values for each digit.
To convert a number to decimal, we just consider the value of
each digit, just like in decimal each digit represents “units”, “tens”,
“hundreds” and so on.
14035 = 𝟏𝟐𝟓 + 𝟏𝟎𝟎?+ 𝟑 = 𝟐𝟐𝟖
Decimal → Any Base
Do the opposite! Convert 18 from decimal to binary.
16
8
?4
2
1
1? 0? 0? 1 0 2
?
?
16 + 0 + 0 + 2 + 0 = 18
Bro Tip: Start with the highest multiple possible of the
highest power (in this case 16). Then see what’s left and
continue to get the digits.
Another Example
Convert 272 for decimal to base 5.
125
25
?
5
1
2? 0 4 2
?
?
?
5
250+ 0 + 20+ 2= 272
Test Your Understanding
Convert 100 from decimal to base 4.
?
64
16
4
1
1?
2? 1? 0?
4
64 +32 + 4 + 0 = 100
Decimal → Any Base
It can help to write out multiples of your various powers. Below is base 6.
Multiples of 6
x1
x2
x3
x4
x5
6
12
18
24
30
c. We can only
have 1 lot of 6.
Multiples of 62
Multiples of 63
36
72
108
144
180
216
432
648
864
1080
b. We can have
4 lots of 62.
Therefore what is 800 is base 6?
?
3412
a. We can have
3 lots of 63.
Exercise 3
1
Copy and complete the
following table.
Decimal Binary (Base 2)
3
?
8
10
77
102
105?
1365
2
11
?
1010 ?
1001101
?
1100110
?
1101001
?
10101010101
?
1000
3
?
12 ?
14 ?
205?
250?
N
10153
?
a horizontal line means 5, a dot 1 and a shell 0).
1000 is a four-digit decimal number
whose first digit one. In what other
bases can this can be converted to such
that we still have a four-digit number
which starts with 1?
If the base is 𝒃, 𝒃𝟑 has to be between
500 and 1000 (if it were less, the first
digit wouldn’t be 1). Only 8 and 9
satisfy this.
?
253
Convert 123 in decimal to Mayan
numerals (recall that Mayan is base 20, and that
?
?
Base 6
3
The decimal number “7a2” is 10322 in
base 5. Determine the digit 𝑎.
𝒂=𝟏
N
Prove that there is no base 𝑏 such that
123 in decimal can be converted to:
i. 45 in that base.
𝟒𝒃 + 𝟓 = 𝟏𝟐𝟑
𝟓𝟗
𝒃=
𝟐
ii. 456 in that base.
𝟒𝒃𝟐 + 𝟓𝒃 + 𝟔 = 𝟏𝟐𝟑
Solving gives 𝒃 = 𝟒. 𝟖𝟐, −𝟔. 𝟎𝟕,
neither or which are integers.
?
?
Decimal → Hexadecimal
The most well-known usage of hexadecimal is to represent colours.
Each colour can be
composed of red, green and
blue light, each of intensity
varying between 0 and 255.
...which can be represented
using just 6 digits in
hexadecimal, 2 for each of the
three colour components.
A means 10, B means 11, ...
F means 15
Multiples
of 16:
0:
1:
2:
3:
4:
5:
6:
7:
8:
9:
A:
B:
C:
D:
E:
F:
0
16
32
48
64
80
96
112
128
144
160
176
192
208
224
240
RED
GREEN
BLUE
HEXADECIMAL
255
255
255
FF, FF, FF
0?
0?
0?
? 00
00, 00,
0?
?
255
0?
? 00
00, FF,
?
255
?
255
0?
? 00
FF, FF,
?
75
?
172
?
198
? C6
4B, AC,
?
255
?
128
0?
? 00
FF, 80,
Adding in decimal
+
2
3
0
6
5
1
3
9
3 + 9 = 12
We’d use the 2 then carry the 1.
Adding in other bases
+
1
?
1
1
0
?
0
1
1
?
0
0
1
?
1
1
1
0
?
Another Example
+ 1
1 0
1
1
0
?
0
0
1
1
1
0
0
1
1
Test Your Understanding
1
+
1
0
1
1
1?
0
1
0
1
1
0
2
+
3
3
2
0?
0
2
3
35
45
2
Exercise 4
Convert the following to hexadecimal.
QQQ Time
1a The number of possible
values each ?digit can
have.
4
3900
?
5
11011
?
6
2400
?
7
100100
?
8
a = 2, b =?4
1b Because each digit must
be between 0 and one
less than the? base/the
digits must be less than
the base.
2a 2
?
2b 178
?
551
?
3