Week 3 - Other bases and their applications Flashcards
Positional Notation
Positional notation is a numeral system in which each position is related to the next by a constant multiplier called the base of that numeral system.
Positional Notation advantages
Only a limited number of numerals are required
The same numeral can be used to represent
a different number by just changing the position of the numeral
E.g. 2, 20, 200, 2000
Babylonians Base Number
60
Mayans Base Number
20
Hindu-Arabic Number System base number
10
Importance of 0
0 is both a value and a placeholder
12 vs 102 vs 120
Babylonians Maths System
- Mesopotamia
- Base 60 (sexagesimal)
60 seconds, minutes, hours, degrees
60 is a highly composite number
- More than 2 numbers fit into 60
- Lots of numbers fit into 60
- Think about time, 60 mins in an hour (15, 30, 45)
For us, we would write, 10^3, 10^2, 10^1 etc
For them, they would write 60^3, 60^2, 60^1
Mayans Maths System
Base 20 (vigesimal)
3 symbols: shell, dot and bar
Shell = 0
Bar = 5
Dot = 1
They work in level, so you start at the bottom and work your way up.
Level 1 - 1 (20^0)
Level 2 - 20 (20^1)
Level 3 - 400 (20^2)
Level 4 - 8000 (20^3)
Level 5 - 16000 (20^4)
Conversion from Base 10 (decimal) to Base 2 (binary)
In a table, the top row should include 10 to the power of (starting from the right), 0 = 1, 1 = 10, 2 = 100, 3 = 1000, 4 = 10,000 and 5 = 100,000
Below you enter the number of each one.
E.g. 494
There is no 100,000s, 10,000s or 1000s in this number so skip those columns
There are 4 hundreds so add a 4 into that column (10^2)
There are 9 tens so add 9 into that column (10^1)
Finally, there are 4 ones, so add 4 into that column (10^0)
Then you continue to the next row, which is base 2. So instead of having 10^0, it’s 2^0. Repeat process.
1, 2, 4, 8, 16, 32
Conversion from Base 10 (decimal) to Base 3 (ternary)
In a table, the top row should include 10 to the power of (starting from the right), 0 = 1, 1 = 10, 2 = 100, 3 = 1000, 4 = 10,000 and 5 = 100,000
Fill the numbers into the correct columns.
The next row should contain numbers with a base of 3 instead of 10. E.g. 3^0 instead of 10^0
Fill numbers into table again
1, 3, 9, 27, 81, 243
Converting decimal to binary example
E.g. Convert 342 (base 10) into base 2 number
Fill table with powers of 2.
2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
342 goes into 256 once so add a 1 into that column.
342 - 256 = 86
86 does into 64 once so add a 1 into that column.
Repeat process
Binary (2) to base 7
E.g. Concert 110001 to base 7
Fill table with powers of 2 and add the numbers given
This tells us that there is one 32, one 16, and one 1
Add these up to get 49
Then make a table with powers of 7
E.g. 7^0 = 1
7^1 = 7
7^2 = 49
7^3 = 243
49 fits perfectly into this so you add a 1 in that column, and you get a Base 7 number of 100.
Base 10 (decimal) to Base 5
Convert 3036 into base 5
Fill table with powers of 5
5^0 = 1
5^1 = 5
etc
Add 3036 into the table and you get 44121
