Transcript Natural Numbers

The Binary Number System
Data Representation
What is a number?
What is a number?
A number is a unit of an abstract
mathematical system subject to the “Laws
of Arithmetic.”
The Laws of Arithmetic

Succession

Addition

Multiplication
Number Categories

Natural (Whole)


Negative


The counting numbers
Less than 0
Rational

An integer, or the quotient of 2 integers
Succession
Positional Notation

The Decimal system is based on the number of digits we
have.


Positional Notation allows us to count past 10 by organizing
numeric digits in columns.

Each column of a number represents a power of the base.


The base is 10.
The exponent is the order of magnitude for the column.
Positional Notation
3
10
1
1000
2
10
1
100
1
10
1
10
0
10
1
•The exponent is the order of magnitude for the column.
•The Least Significant digit is in the right-most column.
•The Most Significant digit is in the left-most column.
1
Positional Notation
3
2
10
1
1000
10
1
100
The base is 10.
1
10
1
10
0
10
1
1
Positional Notation
3
10
1
1000
2
10
1
100
1
10
1
10
The magnitude of the column is base
0
10
1
exponent
1
Positional Notation
104 103 102 101
10000 1000 100 10
2
7
9
1
20000+7000 +900 +10
=27916


Consider a number like the one above.
How many does it represent?
100
1
6
+6
Positional Notation
104 103 102 101
10000 1000 100 10
2
7
9
1
20000+7000 +900 +10
=27916

100
1
6
+6
The size of a number is determined by
multiplying the magnitude of the column by the
digit in the column and summing the products.
Positional Notation
104 103 102 101
10000 1000 100 10
2
7
9
1
20000+7000 +900 +10
=27916

100
1
6
+6
The columns are labelled with their exponents.
Positional Notation
104 103 102 101
10000 1000 100 10
2
7
9
1
20000+7000 +900 +10
=27916

The base of the system is 10.
100
1
6
+6
Positional Notation
104 103 102 101
10000 1000 100 10
2
7
9
1
20000+7000 +900 +10
=27916

The magnitude of the column is base
100
1
6
+6
exponent
Positional Notation
104 103 102 101
10000 1000 100 10
*2
*7
*9 *1
20000+7000 +900 +10
=27916

100
1
*6
+6
Multiply the magnitude of the column by the digit
in the column.
Positional Notation
104 103 102 101
10000 1000 100 10
*2
*7
*9 *1
20000+7000 +900 +10
100
1
*6
+6
27 thousand, 9 hundred, sixteen

Sum the products.
Binary Numbers
The binary number system is a means of
representing quantities using only 2 digits:
0 and 1.
Like other number systems it’s based on
Positional Notation.
Positional Notation
In Binary, the columns have the expected exponents,
3
2
2
2
1
2
0
2
81
41
21
11
Positional Notation
In Binary, the columns have the expected exponents,
but the base of the system is 2.
3
2
2
2
1
2
0
2
81
41
21
11
Positional Notation
In Binary, the columns have the expected exponents,
but the base of the system is 2.
So the column magnitudes are powers of 2.
3
2
2
2
1
2
0
2
81
41
21
11
Binary
Rather than referring to each of the numbers as
a binary digit, we shorten the term to bit.
To facilitate addressing, binary values are
typically stored in units of 8 bits, which is
called a byte.
Large values occupy multiple bytes.
A Single Byte
27 26 25 24 23 22
128 64 32 16 8 4
1
1
1
1 1 1
128 +64 +32 +16 +8 +4 +
=255
21 20
2
1
1
1
2 + 1
A Single Byte
27 26 25 24 23 22
128 64 32 16 8 4
1
1
1
1 1 1
128 +64 +32 +16 +8 +4 +
=255
21 20
2
1
1
1
2 + 1
A Single Byte
27 26 25 24 23 22
128 64 32 16 8 4
1
1
1
1 1 1
128 +64 +32 +16 +8 +4 +
=255
21 20
2
1
1
1
2 + 1
A Single Byte
27 26 25 24 23 22
128 64 32 16 8 4
1
1
1
1 1 1
128 +64 +32 +16 +8 +4 +
=255
21 20
2
1
1
1
2 + 1
is the largest decimal value that can be expressed in 8 bits.
How many different patterns are there?
A Single Byte
27 26
128 64
0
0
0 +0
25
32
0
+0
24
16
0
+0
=0
23 22
8 4
0 0
+0 +0 +
21 20
2
1
0
0
0 + 0
There is also a representation for zero, making 256 (28)
combinations of 0 and 1, in 8 bits.
Natural Numbers in Binary
Consider the pattern:
10010101
To calculate the Decimal equivalent:
1.
multiply each digit by the value of the column
2.
sum the products.
Natural Numbers in Binary
27 26
128 64
1
0
25
32
0
24 23
16 8
1 0
22
4
1
21
2
0
20
1
1
Natural Numbers in Binary
27 26
128 64
1
0
25
32
0
24 23
16 8
1 0
22
4
1
21
2
0
20
1
1
Natural Numbers in Binary
27 26
128 64
1
0
25
32
0
24 23
16 8
1 0
22
4
1
21
2
0
20
1
1
Natural Numbers in Binary
27 26 25 24 23 22
128 64 32 16 8 4
1
0
0
1 0 1
128 + 0 + 0 +16 +0 +4 +
21 20
2
1
0
1
0 + 1
Natural Numbers in Binary
27 26 25 24 23 22
128 64 32 16 8 4
1
0
0
1 0 1
128 + 0 + 0 +16 +0 +4 +
=149
21 20
2
1
0
1
0 + 1
Natural Numbers in Binary
Conversion from Decimal to Binary uses the
same technique, in reverse.
Consider the value 73.
In base 10, this is 7 units of 10, plus 3 units of 1.
Natural Numbers in Binary
We need to express the value in terms of powers of 2.
27
26
25
24
23
22
21
20
128
64
32
16
8
4
2
1
0
1
Natural Numbers in Binary
What is the largest power of 2 that is included in 73?
27
26
25
24
23
22
21
20
128
64
32
16
8
4
2
1
0
1
Natural Numbers in Binary
64 is the largest power of 2 that is included in 73, so a
1 is needed in that position
27
26
25
24
23
22
21
20
128
64
32
16
8
4
2
1
0
1
Natural Numbers in Binary
Subtracting 64 from 73 leaves 9, which cannot include
32, or 16, but does include 8.
27
26
25
24
23
22
21
20
128
64
32
16
8
4
2
1
0
1
0
0
1
Natural Numbers in Binary
Subtracting 8 from 9 leaves 1, which cannot include 4,
or 2, but does include 1.
27
26
25
24
23
22
21
20
128
64
32
16
8
4
2
1
0
1
0
0
1
0
0
1
Natural Numbers in Binary
So the 8 bit binary representation of 73 is:
01001001
Short Forms
Longer Numbers
Since 255 is the largest number that can be
represented in 8 bits, larger values simply
require longer numbers.
For example, 27916 is represented by:
0110110100001100
Longer Numbers
Since 255 is the largest number that can be
represented in 8 bits, larger values simply
require longer numbers.
For example, 27916 is represented by:
0011011010000110
Can you remember the Binary representation?
Short Forms for Binary
Because large numbers require long strings of
Binary digits, short forms have been
developed to help deal with them.
An early system was called Octal.
It’s based on the 8 patterns in 3 bits.
Short Forms for Binary - Octal
111
7
110
6
101
5
100
4
011
3
010
2
001
1
000
0
0011011010000110
can be short-formed by
dividing the number into 3 bit
chunks (starting from the
least significant bit) and
replacing each with a single
Octal digit.
Short Forms for Binary - Octal
111
7
110
6
101
5
100
4
011
3
010
2
001
1
000
0
000011011010000110
0
added
3
3
2
0
6
Short Forms for Binary - Hexadecimal
0111
7
1111
F
0110
6
1110
E
0101
5
1101
D
0100
4
1100
C
0011
3
1011
B
0010
2
1010
A
0001
1
1001
9
0000
0
1000
8
It was later determined that
using base 16 and 4 bit
patterns would be more
efficient.
But since there are only 10
numeric digits, 6 letters
were borrowed to complete
the set of hexadecimal
digits.
Short Forms for Binary - Hexadecimal
0111
7
1111
F
0110
6
1110
E
0101
5
1101
D
0100
4
1100
C
0011
3
1011
B
0010
2
1010
A
0001
1
1001
9
0000
0
1000
8
0011011010000110
can be short-formed by
dividing the number into 4bit chunks (starting from the
least significant bit) and
replacing each with a single
Hexadecimal digit.
Short Forms for Binary - Hexadecimal
0111
7
1111
F
0110
6
1110
E
0101
5
1101
D
0100
4
1100
C
0011
3
1011
B
0010
2
1010
A
0001
1
1001
9
0000
0
1000
8
0011011010000110
3
6
8
6
Short Forms for Binary
Octal and Hexadecimal are number systems.
It is possible to perform arithmetic in both.
2
There are 64 (8 ) rules of octal addition, and
2
256 (16 ) rules of hexadecimal addition.
But why design a machine with so many rules
when conversion to Binary is simple and
there are only 4 rules of Binary addition?