Number Systems and Bases Flashcards
(28 cards)
What are ordinal numbers?
Numbers used for explaining the position of a number in a well ordered set
What numbers are used for counting and measurement?
- Natural numbers for counting
- Real numbers for measurement
What is a bit? What is a byte?
The most fundamental unit of information. 8 bits make up a byte
How can you tell how many numbers can be represented by a binary number? What is the maximum value?
- 2^n where n is the number of bits used
- 2^n - 1 is the maximum value
Binary prefixes
Kibi - 2^10
Mebi - 2^20
Gibi - 2^30
Tebi - 2^40
Decimal prefixes
kilo - 10^3
mega - 10^6
giga - 10^9
tera - 10^12
Why is hexadecimal used?
- Easy to convert from binary
- Easily read by humans
- More concise
Where is hexadecimal used?
memory addresses, error message codes, HTML colour codes
What is pure binary?
Unsigned binary
How does binary shift work?
Shift to right is double, shift to left is half
Describe signed binary numbers.
- Sign is represented through MSB (most significant bit)
- 1 in sign bit represents a negative number, 0 is positive
- Range is same as unsigned binary
How to convert from binary to hexadecimal?
Consider each nibble and convert to hexadecimal equivalent 0 to F
How to use two’s complement to find negative binary numbers? How is a carry bit used?
- write number out in binary
- if number is positive, add a sign bit of 0 and leave it
- if negative, change to 1’s complement (switch all 0’s and 1’s) AND add 1 (ignoring carry bit at the sign bit)
- carry bit is used for checking overflow
What is an overflow error? What is an underflow error?
- Overflow: a number is too large to represent within the number or bits available
- Underflow: a number is too small e.g. 0.000001 to be represented in number of bits available
solved by
- store overflow bits elsewhere to prevent error
- set flag bit to indicate error
How is underflow or overflow error dealt with?
Checking for a carry out from the MSB or LSB (right and left) in the arithmetic logic circuit and setting a “flag” to state an error has occurred.
Consequences of underflow/overflow
- Erroneous calculations
- Data corruption
- Program crashes
How to multiply binary numbers? How to subtract binary numbers?
Multiplication:
- convert to 2^n and perform n shifts
- do long multiplication
Subtraction:
- convert to two’s complement
- add together
How do fixed point binary numbers work?
- there is an imaginary radix point which separates the fractional and integer parts
- it works same as positive, e.g. 1/2, 1/4, 1/8
- more bits to the right of radix means more range
- more bits to left of radix means more precision
How do floating point binary numbers work?
- 2 parts, mantissa and exponent
- both have sign bit and then numbers to represent value
- mantissa by default has radix point after sign bit
- exponent tells how much we do binary shift by
What is the mantissa?
The mantissa represents the significant digits of the number.
What is the exponent?
The exponent represents the power of 2 by which to multiply the mantissa.
What is the advantage and disadvantages of floating point binary as compared to fixed point binary?
Floating point binary provides a larger range for large numbers and a greater accuracy for small numbers for a given number of bits
- More **difficult and slower to perform calculations with **(processed more slowly than fixed point)
What is the advantage and disadvantages of fixed point binary?
+ processed more quickly as there is no movement of the binary point
+ Hardware required to process it is much less complicated making it smaller and less energy intensive
- Fixed and limited precision
-Offers less range of numbers than floating point for a given number of bits
