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5.1 Binary and Hexadecimal Numbers

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Oli Howson

on 7 January 2014

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Transcript of 5.1 Binary and Hexadecimal Numbers

Introduction
Binary & Denary
Negative Numbers
Hexadecimal
Real Numbers
Objectives
Starters
Storage locations in computers are built using electrical circuits
These can exist in one of two states - low voltage and high voltage
We replace these states with two symbols - 0 for low voltage and 1 for high voltage
This is known as a binary system
Digital computers use the binary system for storing instructions (programs) and data of all types
A binary digit, or bit, is a 0 or 1
A group of 8 bits is known as a byte
A group of bytes is known as a word
How many bytes make up a word depends on the type of computer. For example, a 64-bit machine has a word size of 64 bits = 8 bytes
Data Representation
Binary and Hexadecimal Numbers
Some key terms to remember
A number system that uses two digits,
0 and 1.
Binary
A binary digit; it can be 0 or 1
Bit
A group of 8 bits
Byte


Word


Denary number system
A group of bytes
A number system that uses the
digits 0 to 9
Let's remind ourselves
how the decimal or denary number system works.
It uses the 10 digits from 0 to 9.
Consider the number 146. These three digits represent one hundred, four tens and six ones.

100 10 1
1 4 6
As we move from right to left each digit is worth ten times as much as the previous one. This is known as base 10, the number base specifying how many digits are used and how much each digit is multiplied by as we move from right to left .
The binary system is a base 2 sytem, using only the digits 0 and 1
As we move from right to left, each digit is worth twice as much as the previous one. The place values are powers of 2.
The binary number 10010010 can be set out under place values as follows:
128 64 32 16 8 4 2 1
1 0 0 1 0 0 1 0
This represents 128 + 16 + 2 = 146.
Using only one byte to hold a number would limit the computer to storing only small numbers. Usually four or more consecutive bytes are used to store numbers
Converting from Binary to Denary
Put each binary digit under the correct heading in the table.
eg To translate 01001001 into decimal, arrange the digits in the table as follows
128 64 32 16 8 4 2 1
0 1 0 0 1 0 0 1
Add them up - 64+8+1 = 73
Activity
Convert the following
binary numbers to decimal
(a) 0100
(b) 0101
(c) 1010
(d) 01000010
(e) 01011001
(a) 4
(b) 5
(c) 10
(d) 66
(e) 89
Converting from Denary to Binary
To convert 21 from denary to binary, write down the place values
Then find the largest place value that is less than or equal to 21
That value is 16. Write a 1 in the column headed 16
Subtract 16 from 2
and you get 5
Now find the largest place value that is less than or equal to 5. It is 4.
Write a 1 in the column headed 4
Subtract 4 from 5
and you get 1.
Write a 1 in the column headed by 1
Fill in the empty
columns with 0
You end up with:
128 64 32 16 8 4 2 1
0 0 0 1 0 1 0 1
So 21 in binary is 00010101
Activity
1) Convert the number 227 to binary
2) Convert the following numbers to binary
a) 3 b) 9 c) 19
d) 28 e) 76 f) 129
1) Denary 227 is binary 11100011
2) a) 000 0011 b) 0000 1001
c) 0001 0011 d) 0001 1100
e) 0100 1100 f) 1000 0001
Binary Addition
Let's remind us how we add two denary numbers
When we add two digits which produce a result greater than 9, we use place values and carry a 1 to the next column on the left. Binary addition works in the same way
Here are the rules of binary
0 + 0 = 0 addition
0 + 1 = 1 + 0 = 1
1 + 1 = 1 (10 and carry 1)
1 + 1 + 1 = 11 (1 and carry 1)
Example
00011001 25
+ 00011010 + 26
00110011 51
Activity
1) a) Add 0101 and 0010
b) Add 01101010 and 01000011
c) Add 01010101 and 01111110
Check your results by converting to denary
2) Add 01010111 and 10101001
Comment on your results
Binary Multiplication
Let's remind ourselves how we multiply two numbers in denary.
If we want to multiply 7 by 4, we could interpret this as adding 7 four times
This is fine for small numbers, but how do we do multiplication with large numbers?
We use the place values of digits
Multiplying 7 by 10 gives 70
We have just moved the 7 into the next column on the left
Multiplying 7 by 100 gives 700;
We move the 7 two coumns to the left.
Binary multiplication works in the same way
Here are the rules of binary multiplication
0 x 0 = 0
0 x 1 = 1 x 0 = 0
1 x 1 = 1
1 x 10 = 10
If we want to multiply a binary number by 2 we move the binary pattern one place to the left and put a zero in the vacated binary place
If we want to multiply a binary number by 4, we move the binary pattern two places to the left and put a zero in each of the vacated binary places.
Let's do an
example
11000
x 10001
24
x 17
110000000
11000
Top number timesed by 1
Top number timesed by 10000
168
240
Top number timesed by 7
Top number timesed by 10
110000000
11000
110011000
168
240
408
Activity
Check that the rule is true by multiplying the following bit patterns by 2 and converting the numbers before and after multiplication to denary:
101 110 1001 10101 1100 1000
This causes bit overflow, an extra bit would be required to store the result.
101 multiplied by 2: 1010
In denary: 5 x 2 = 10
110 multiplied by 2: 1100
In denary: 6 x 2 = 12
1001 multiplied by 2: 10010
In denary 9 x 2 = 18
10101 multiplied by 2: 101010
In denary 21 x 2 = 42
1100 multiplied by 2: 11000
In denary 12 x 2 = 24
1000 multiplied by 2: 10000
In denary 8 x 2 = 16
a) Multiply 0101 and 0010
b) Multiply 01001010 and 00000011
c) Multiply 01000001 and 00000110
Check your results
Questions
1) What is the largest binary number that can be held in:
(a) 8 bits
(b) 16 bits
(c) 32 bits
(d) 64 bits
2) How many bits make on byte?
3) What are the possible values that one bit can take?
4) Convert the decimal numbers 20 and 10 into binary. Add the binary numbers for 20 and 10. Show your working. Then convert the result back into denary to check your answer
5) Convert the decimal numbers 5 and 6 into binary. Multiply the binary numbers for 5 and 6. Show your working. Then convert the result back to denary to check your answer.
Answers
1) What is the largest binary number that can be held in:
(a) 8 bits
(b) 16 bits
(c) 32 bits
(d) 64 bits
a) 11111111 = denary 255 =28 -1
b) 11111111 11111111 = denary 65,535 = 216 -1
c) 11111111 11111111 11111111 11111111 = 429,4967,295 = denary 232 -1
d) 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111
= denary 264 -1
2) How many bits make on byte?
8
3) What are the possible values that one bit can take?
0 or 1
4) Convert the decimal numbers 20 and 10 into binary. Add the binary numbers for 20 and 10. Show your working. Then convert the result back into denary to check your answer
5) Convert the decimal numbers 5 and 6 into binary. Multiply the binary numbers for 5 and 6. Show your working. Then convert the result back to denary to check your answer.
Two's Complement
Negative numbers can be represented using a system called
two's complement
Imagine the milometer of a car set at 00000
If the car goes forward one mile, the reading becomes 00001
If the milometer is turned back one mile, the reading becomes 99999 miles.
This could be interpreted as -1 mile
Key term to remember
Two's complement: A system for representing negative numbers
Worked Example
When we add the binary equivalents of 5 and -5, we obtain:
1011
+ 0101
(1)0000 We ignore the carry of 1
Notice that in two's complement representation if a number starts with a 1, it represents a negative number
Here is the procedure for converting a negative denary number to two's complement binary:
1 Find the binary value of the equivalent positive denary number
2 Change all the 0's to 1's and all the 1's to 0's
3 Add 1 to the result
And here is a method for changing the sign of a binary number:
1 Starting from the right, leave all the digits alone up to and including the first 1
2 Change all the other digits from 0 to 1 or from 1 to 0; for example -00101100 becomes 11010100
Activity
Convert the following to two's complement binary using 8 bits to represent your number
a) -5 b) -10 c) -20 d) 5
Converting a negative two's complement number to denary
Method 1
Flip the bits, add 1, then work out the denary equivalent
Method 2
Use the place values. Write the binary digits down under the place values 1,2,4,8,etc
Note that the leftmost bit has a negative place value. Add up all the place values which ahve a 1 under them.
Example
Convert the two's complement number 10011010 to denary
-128 64 32 16 8 4 2 1
1 0 0 1 1 0 1 0
This gives -128+16+8+2 = -102
Activity
Convert the following two's complement numbers to denary:
(a) 10101010 (b) 10000001
(c) 11111111 (d) 00010111
Binary Subtraction
The easiest way of performing binary subtraction is to convert the number to be subtracted to a negative number and then add it
Subtract 13 from 17 using 1 byte for each number
00001101 13
11110011 -13
00010001 + 17
00000100 4
Activity
1) Subtract 25 from 28 in binary and show your working
2) Subtract 23 from 123 in binary and show your working
Answers
a) 11111011 b) 11110110
c) 11101100 d) 00000101
Answers
a) -86 b) -127
c) -1 d) 23
Answer
25 is 00011001
so -25 is 11100111
28 is 00011100
00011100 28
11100111 -25
(1)00000011 3
Answer
23 is 00010111
so -23 is 11101001
123 is 01111011
01111011 123
11101001 -23
(1)01100100 100
The Most or Least Significant Bit
A positive number always has 0 as the most significant bit (MSB)
A negative number always has 1 as the most significant bit (MSB)
An even number always has 0 as the least significant bit (LSB)
An odd number always has 1 as the least significant bit (LSB)
-1 is represented by a 1 in every bit, whatever the word size
Questions
1. What is the largest negative number that can be held in 8 bits using two's complement?
2. Convert the decimal numbers 11 and 9 to binary. Using 8 bits for each number, show your working in binary of subtracting 9 from 11
3. Convert the decimal numbers 17 and 15 to binary. Using 8 bits for each number, show your working in binary of subtracting 17 from 15
4. Convert the decimal numbers 96 and 40 to binary. Using 8 bits for each number, show the binary pattern for -96 and -40. Commend on your result of adding these two patterns together
-128
00001001 in denary 9

11110111 -9
00001011 11
(1)00000010 2
00010001 in denary 17

11101111 -17
00001111 15
11111110 -2
Overflow!

10111111 -96
11011000 -40
(1)10010111 -136
01100000 in denary 96
10111111 in denary -96

00101000 in denary 40
11011000 in denary -40
Fixed-Point Binary Numbers
So far we have only been able to represent whole numbers
Calculations often have fractional results, do we need to be able to store real numbers
There are an infinate number of fractions...
...but there are only a limited number of bits to represent them.
Trying to map an infinite number of values onto a limited number of bit patterns results in a loss of precision.
We can easily extend the system we used for integers to include fractions. Look at the place values of denary numbers
100 10 1 . 1/10 1/100
0 6 8 . 2 5
The number 68.25 represents 6 tens, 8 ones, 2 tenths and 5 hundredths.
In binary the equivalent place values are...
126 64 32 16 8 4 2 1 . 1/2 1/4 1/8 1/6
The number 68.25 can be expressed as 68 + 1/4 andusing the place values converts to...
126 64 32 16 8 4 2 1 . 1/2 1/4 1/8 1/6
0 1 0 0 0 1 0 0 . 0 1 0 0
Notice the first digit after the point (known as the binary point) is worth one half. The equivalent in denary is worth one tenth. With the same number of digits after the point the binary system is less precise!
If we write down an amount in pounds sterling to 2 decimal places the amount is accurate to the nearest penny.
If we convert the amount to binary and allow only two digits after the binary point, we can hold .00 (0 pence), .01 (25 pence), .10 (50 pence) or .11 (75 pence) as the fractional part, so the amount is only accurate to the nearest 25 pence.
Fixed point notation assumes a binary point in a set position as there is no third symbol to store it explicitly.
Some decimal fractions
and their binary equivalents
Advantage
Disadvantage
Arithmetic is simple. It's the same as integer arithmetic. So processing is fast!
Of limited range. Increasing the number of bits after the binary points for precision decreases the range, and vice versa, in a fixed number of bits
There is an alternative, floating point numbers. This is covered in the A2 part of the course.
Questions
1. Using one byte to hold each number, with an imaginary binary point after the fourth digit, convert the following decimal numbers to binary.
a) 3.75 b) 5.1875
c) 7.562 d) 7.5627
2. Convert the following numbers to decimal, assuming 4 bits after the point.
i) 0000000001011000
ii) 0000000000110010
3. What is (i) the largest positive number and (ii) the smalles positive number that can be held in two bytes, assuming four bits after the point?
(ii) what is the difference between two consecutive values?
4. Using one byte to hold each number, with an imaginary binary point fixed after the fourth digit, (i) what is the largest number that can be stored?
(iii) How could the precision of storing a fractional number be improved? What effect would this have on the range of numbers that could be stored?
1. Using one byte to hold each number, with an imaginary binary point after the fourth digit, convert the following decimal numbers to binary.
a) 3.75 b) 5.1875
c) 7.562 d) 7.5627
2. Convert the following numbers to decimal, assuming 4 bits after the point.
i) 0000000001011000
ii) 0000000000110010
3. What is (i) the largest positive number and (ii) the smalles positive number that can be held in two bytes, assuming four bits after the point?
(ii) what is the difference between two consecutive values?
4. Using one byte to hold each number, with an imaginary binary point fixed after the fourth digit, (i) what is the largest number that can be stored?
(iii) How could the precision of storing a fractional number be improved? What effect would this have on the range of numbers that could be stored?
a) 0011.1100 b) 0101.0011
c) 0111.1001 (note: these are really the
d) 0111.1001 value 7.5625)
i) 5.5
ii) 3.125
i) 011111111111.1111
ii) 000000000000.0001
i) 0.111.1111 = denary 7.9375
ii) 0.0625
iii) Precision can be improved by using more bits after the binary point, but then the range of numbers is reduced for a given number of bits
Digital computers represent everything using the binary system, but binary patterns are very inconvenient for humans to read due to the size of them
Even a small number such as 254 requires eight digits to represent it in pure binary!
To make it easier for a human to read the contents of main memory or a computer file, the binary numbers are commonly put into groups of four bits
These are then displayed using the hexadecimal number system
The hexadecimal number system uses the digits 0 to 9 and the letters A to F. It is a shorthand notation for binary number patterns
Hexadecimal number are base-16
It is worth remembering that 1010 in binary is 10 in denary and A in hexadecimal. It's also worth remembering that 1111 is 15 in denary and F in hexadecimal. You can work out the rest as required
Let's convert decimal 213 to hex. First we convert it to binary
128 64 32 16 8 4 2 1
1 1 0 1 0 1 0 1
Divide the binary into groups of four starting from
the right
128 64 32 16 8 4 2 1
1 1 0 1 0 1 0 1
Now convert the groups into hexadecimal
1101 = 13 = D
0101 = 5 = 5
D5
Activity
Convert denary 126 into (a) binary and (b) hexadeximal
126 is 01111110 in
binary and 7E in hexadecimal
Converting from Hex to Decimal
Method 1: Convert each hex digit to its binary equivalent
A 3
8 4 2 1 8 4 2 1
1 0 1 0 0 0 1 1
128 64 32 16 8 4 2 1
1 0 1 0 0 0 1 1
So the hexadecimal number A3 equals 128+32+2+1 = 163 in denary
Method 2: Use the hexadecimal place values. To translate the hexadecimal number A3 to denary, do 10x16 + 3 = 160 +3 = 163 in denary
Activity
1. Convert the binary number 0110 1101 into (a) denary and (b) hexadecimal
2. Convert the following binary patterns into hexadecimal numbers: 1111 1110 1110 1101 1101 1010 1101. What is the message?
3. The following hexadecimal numbers represent 16-bit two's complement numbers: B13C, 7010, 1FFF, 8A1E, 9000, F73F. Can you tell which ones represent negative numbers? Convert them and check
4. Where have you seen hexadecimal numbers used? What were they used for?
B13C negative 1011 0001 0011 1100
7010 positive 0111 0000 0001 0000
1FFF positive 0001 1111 1111 1111
8A1E negative 1000 1010 0001 1110
9000 negative 1001 0000 0000 0000
F73F negative 1111 0111 0011 1111
If the first digit is greater than 7 then the number is negative
Examples: colour codes in paint software.
Memory dump, contents of memory locations.
0110 1101 is (a) 109 in denary and (b) 6D in hexadecimal
1111 1110 1110 1101
F E E D

1101 1010 1101
D A D
Starter 1
We are going to be numbers!
Three of you are going to be binary numbers
Four of your are going to be denary numbers
Denary numbers, you will stand up to show 9, lie down for 1, hide for 0, and be somewhere in-between for other digits.
Binary numbers, you will stand up to show 1, and sit to be 0
Familiarise yourself with these instructions
Everyone except the denary numbers sit with your back to the board. Get ready to write down what you think the numbers are.
Demonstrate 269
Demonstrate 504
Demonstrate 734
Everyone except the binary numbers sit with your back to the board. Get ready to write down what you think the numbers are.
Demonstrate 1001
Demonstrate 0110
Demonstrate 1101
What is the point?
Activity
Full transcript