The Binary Number System Data Representation. What is a number?
-
Upload
byron-jackson -
Category
Documents
-
view
254 -
download
1
Transcript of The Binary Number System Data Representation. What is a number?
![Page 1: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/1.jpg)
The Binary Number System
Data Representation
![Page 2: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/2.jpg)
What is a number?
![Page 3: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/3.jpg)
What is a number?
A number is a unit of an abstract mathematical system subject to the “Laws of Arithmetic.”
![Page 4: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/4.jpg)
The Laws of Arithmetic
Succession
Addition
Multiplication
![Page 5: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/5.jpg)
Number Categories
Natural (Whole) The counting numbers
Negative Less than 0
Rational An integer, or the quotient of 2 integers
![Page 6: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/6.jpg)
Succession
![Page 7: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/7.jpg)
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.
![Page 8: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/8.jpg)
Positional Notation
103 102 101 100
10001 1001 101 11
•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.
![Page 9: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/9.jpg)
Positional Notation
103 102 101 100
10001 1001 101 11
The base is 10.
![Page 10: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/10.jpg)
Positional Notation
103 102 101 100
10001 1001 101 11
The magnitude of the column is base exponent
![Page 11: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/11.jpg)
Positional Notation
104 103 102 101 100
10000 1000 100 10 1 2 7 9 1 6 20000+7000 +900 +10 +6
=27916 Consider a number like the one above. How many does it represent?
![Page 12: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/12.jpg)
Positional Notation
104 103 102 101 100
10000 1000 100 10 1 2 7 9 1 6 20000+7000 +900 +10 +6
=27916 The size of a number is determined by
multiplying the magnitude of the column by the digit in the column and summing the products.
![Page 13: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/13.jpg)
Positional Notation
104 103 102 101 100
10000 1000 100 10 1 2 7 9 1 6 20000+7000 +900 +10 +6
=27916 The columns are labelled with their exponents.
![Page 14: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/14.jpg)
Positional Notation
104 103 102 101 100
10000 1000 100 10 1 2 7 9 1 6 20000+7000 +900 +10 +6
=27916 The base of the system is 10.
![Page 15: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/15.jpg)
Positional Notation
104 103 102 101 100
10000 1000 100 10 1 2 7 9 1 6 20000+7000 +900 +10 +6
=27916 The magnitude of the column is base exponent
![Page 16: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/16.jpg)
Positional Notation
104 103 102 101 100
10000 1000 100 10 1 *2 *7 *9 *1 *6 20000+7000 +900 +10 +6
=27916 Multiply the magnitude of the column by the digit
in the column.
![Page 17: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/17.jpg)
Positional Notation
104 103 102 101 100
10000 1000 100 10 1 *2 *7 *9 *1 *6 20000+7000 +900 +10 +6 27 thousand, 9 hundred, sixteen
Sum the products.
![Page 18: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/18.jpg)
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.
![Page 19: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/19.jpg)
Positional Notation
In Binary, the columns have the expected exponents,
23 22 21 20
81 41 21 11
![Page 20: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/20.jpg)
Positional Notation
In Binary, the columns have the expected exponents,
but the base of the system is 2.
23 22 21 20
81 41 21 11
![Page 21: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/21.jpg)
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.
23 22 21 20
81 41 21 11
![Page 22: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/22.jpg)
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.
![Page 23: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/23.jpg)
A Single Byte
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
1 1 1 1 1 1 1 1
128 +64 +32 +16 +8 +4 + 2 + 1
=255
![Page 24: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/24.jpg)
A Single Byte
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
1 1 1 1 1 1 1 1
128 +64 +32 +16 +8 +4 + 2 + 1
=255
![Page 25: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/25.jpg)
A Single Byte
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
1 1 1 1 1 1 1 1
128 +64 +32 +16 +8 +4 + 2 + 1
=255
![Page 26: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/26.jpg)
A Single Byte
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
1 1 1 1 1 1 1 1
128 +64 +32 +16 +8 +4 + 2 + 1
=255is the largest decimal value that can be expressed in 8 bits.
How many different patterns are there?
![Page 27: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/27.jpg)
A Single Byte
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
0 0 0 0 0 0 0 0
0 +0 +0 +0 +0 +0 + 0 + 0
=0There is also a representation for zero, making 256 (28)
combinations of 0 and 1, in 8 bits.
![Page 28: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/28.jpg)
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.
![Page 29: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/29.jpg)
Natural Numbers in Binary
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1 1 0 0 1 0 1 0 1
![Page 30: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/30.jpg)
Natural Numbers in Binary
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1 1 0 0 1 0 1 0 1
![Page 31: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/31.jpg)
Natural Numbers in Binary
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1 1 0 0 1 0 1 0 1
![Page 32: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/32.jpg)
Natural Numbers in Binary
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1 1 0 0 1 0 1 0 1128 + 0 + 0 +16 +0 +4 + 0 + 1
![Page 33: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/33.jpg)
Natural Numbers in Binary
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1 1 0 0 1 0 1 0 1128 + 0 + 0 +16 +0 +4 + 0 + 1
=149
![Page 34: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/34.jpg)
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.
![Page 35: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/35.jpg)
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
![Page 36: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/36.jpg)
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
![Page 37: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/37.jpg)
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
![Page 38: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/38.jpg)
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
![Page 39: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/39.jpg)
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
![Page 40: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/40.jpg)
Natural Numbers in Binary
So the 8 bit binary representation of 73 is:
01001001
![Page 41: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/41.jpg)
Short Forms
![Page 42: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/42.jpg)
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
![Page 43: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/43.jpg)
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?
![Page 44: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/44.jpg)
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.
![Page 45: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/45.jpg)
Short Forms for Binary - Octal111 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.
![Page 46: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/46.jpg)
Short Forms for Binary - Octal111 7
110 6
101 5
100 4
011 3
010 2
001 1
000 0
000011011010000110
0 3 3 2 0 6added
![Page 47: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/47.jpg)
Short Forms for Binary - Hexadecimal0111 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.
![Page 48: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/48.jpg)
Short Forms for Binary - Hexadecimal0111 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 4-bit chunks (starting from the least significant bit) and replacing each with a single Hexadecimal digit.
![Page 49: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/49.jpg)
Short Forms for Binary - Hexadecimal0111 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
![Page 50: The Binary Number System Data Representation. What is a number?](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649d985503460f94a82b7e/html5/thumbnails/50.jpg)
Short Forms for Binary
Octal and Hexadecimal are number systems.
It is possible to perform arithmetic in both.
There are 64 (82) rules of octal addition, and 256 (162) 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?