Ζ Bases Dr Frost Objectives: 1.To appreciate how we can have different number systems using different ‘bases’. 2.To convert numbers from decimal to another.
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ζBasesDr Frost
Objectives: 1. To appreciate how we can have different number systems using
different ‘bases’.2. To convert numbers from decimal to another base.3. To convert numbers from any base to decimal.
1001100101
11010001
96E854
FFBA4D
000000
1295823
8409284
These are all examples of numbers!What connects each group of numbers?
10
942
Base
The base of a number system is the number of possible values for each digit.
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Values for each digit
Base Name of number system
0 to 9 10 Decimal0 to 1 2 Binary0 to F(A=10, B=11, ... F=15)
16 Hexadecimal
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Numbers in decimal
If we were to write out 2493, what is the value of each digit?
2 4 9 31000 100 10 1
2000 +400 +90 + 3 = 2493
multiply
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Numbers in decimal
Now suppose we had a number in base 5 instead. How do we convert it to decimal?
4 3 0 1 125 25 5 1
500 + 75 + 0 + 1 = 576
multiply
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Numbers in decimal
Copy and complete in your book.
1 0 1 12
8 4 2 1
8 + 0 + 2 + 1 = 11?
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3 3 0 24
64 16 4 1
192 + 48 + 0 + 2 = 242
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The small number indicates the base.
1 2 2 03
27 9 3 1
27 + 18 + 6 + 0 = 51
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I’m a Mayan, Get
Me Out Of Here!
(Switch to ‘I’m a Mayan’ slides)
Exercises
Original number In base 10 (decimal)11012 13
1112 7
1100112 51
10223 35
7348 476
2345 69
5306 198
Base 2 ?
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Converting FROM decimal to other bases
Do the opposite! Convert 18 to binary.
1 0 0 1 0 2
16 8 4 2 1?
16 + 0 + 0 + 2 + 0 = 18? ? ? ? ?
Converting FROM decimal to other bases
Convert 272 to base 5.
2 0 4 2 5
125 25 5 1?
250+ 0 + 20+ 2= 272? ? ? ?
Converting FROM decimal to other bases
It can help to write out multiples of your various powers. Below is base 6.
Multiples of 6 Multiples of 62 Multiples of 63
612182430
3672108144180
2164326488641080
x 1
x 2
x 3
x 4
x 5
Therefore what is 800 is base 6?3412?
Exercises
Decimal Binary Base 63 11 38 100 1210 1010 1477 1001101 205102 1100110 250105 1101001 2531365 10101010101 10153
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Decimal to 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
0 255 0
0 0 0
255 255 255
RED GREEN BLUE
255 255 0
75 172 198
255 128 0
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HEXADECIMAL
FF, FF, FF
00, 00, 00
00, FF, 00
FF, FF, 00
4B, AC, C6
FF, 80, 00
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0: 01: 162: 323: 484: 645: 806: 967: 1128: 1289: 144A: 160B: 176C: 192D: 208E: 224F: 240
Multiples of 16:
Exercises
Provided worksheet.
Adding in other bases
1 0 0 1+ 1 1 0 1
Adding in other bases
1 0 1 0+ 1 1 0 1 1
Multiplying in other bases
1 0 1 0 x 1 0 1
QQQ Time
The number of possible values each digit can have.
1a
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
3 551
4
5
6
7
8
3900
11011
240
100100
a = 2, b = 4
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