devise a number system based on base 11.
a)draw the number line
b)show examples of addition and subtraction
c)develop a representation of negative numbers based on elevens complement
To devise a number system based on base 11, follow these steps:
a) Drawing the number line:
The number line in a base 11 system would consist of the digits 0-9 and an additional symbol to represent 10. Here's a representation of the number line in base 11:
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
A B C D E F G 0 1 2 3 4 5 6 7
The symbols A-G represent the digits 10-16 in the base 11 system.
b) Examples of addition and subtraction:
Let's consider the addition and subtraction of two numbers in base 11.
Example 1: Addition
To add 6 and 9 in base 11, you would start by adding the units place:
6
+ 9
----
B (11 in base 10)
As B represents 11, you carry it over to the next column. Then, you add the carry with the next column's digits:
1 (carry)
6
+ 9
----
1B (B in base 11 = 11 in base 10)
So, 6 + 9 in base 11 is equal to 1B.
Example 2: Subtraction
To subtract 5 from 9 in base 11, you compare the digits:
9
- 5
----
4 (4 in base 10)
Since 9 is larger than 5, you subtract them as usual and get 4.
c) Representation of negative numbers based on elevens complement:
In base 11, the representation of negative numbers can be done using elevens complement.
To represent a negative number, you take the elevens complement of its positive counterpart. Here's how it works:
1. Write down the positive number representation in base 11.
2. Replace each digit with the complement to 11. To find the complement, subtract the digit from 10.
3. Add 1 to the result obtained in step 2.
For example, let's find the elevens complement of -4:
Positive representation of 4: 4
Complement to 11: 6
Add 1: 7
So, the elevens complement of -4 is 7 in base 11.
Remember, negative numbers are usually represented with a negative sign (-) in front in most number systems, but in this case, elevens complement is used for representation.