Data Representation Using Binary Digits

Binary
Hexadecimal
0	0000	0x0
1	0001	0x1
2	0010	0x2
3	0011	0x3
4	0100	0x4
5	0101	0x5
6	0110	0x6
7	0111	0x7
8	1000	0x8
9	1001	0x9
10	1010	0xA
11	1011	0xB
12	1100	0xC
13	1101	0xD
14	1110	0xE
15	1111	0xF

Table 5 Decimal, binary, and hexadecimal conversions
Let us convert 0x3B to decimal via binary. The first step is to find out what the individual hexadecimal number represents in binary. Replace the hexadecimal number with the binary number. Therefore, 0x3B becomes 00111011₂. As you may recall from the previous section on binary, converting numbers from binary to decimal is much easier than with other systems, since multiplying by binary digits of 0 and 1 are trivial.
0x3B=
00111011₂=
32 + 16 + 8 + 2 + 1 =
59
A byte of data (eight bits) can be written as just two hex digits. For example, the character "N" in extended ASCII code has the binary representation 01001110. If we write this as two groups of four bits each, we get 0100.1110. Using table 5 above, we find that 0100 is 0x4 and 1110 is 0xE. Therefore, the corresponding hexadecimal code for 0100.1110 is 0x4E.
When setting up or maintaining a computer system, you will sometimes encounter hexadecimal numbers as representations of memory addresses, network addresses, or other hardware-related qualities. You may encounter them in operating system-related contexts as well, such as when a machine "crashes" and displays a failure report.
Question 1. Decimal, binary, and hexadecimal conversions
a. Complete the following chart by converting the numbers given in one of the notations to the other two. Show calculations.

Binary	Decimal	Hexadecimal
10001101	140	0x8D
1110011	115	0x3E
11111101	253	0x FD
10101011	171	0xAB
000110111101	445	0x1BD

1×2^7+0×2^6+0×2^5+0×2^4+1×2^3+1×2^2+0×2^1+1×2^0=1×128+0+0+0+1×8+1×4+0=128+8+4=140
1×2^7+1×2^6+1×2^5+1×2^4+1×2^3+1×2^2+0×2^1+1×2^0=1×128+1×64+1×32+1×16+1×8+1×4+0+1×1=128+64+32+16+8+4+1=253

b. What is the minimum number of bits that can be used to represent the decimal number 213? (Hint: convert 252 to binary.)
c. What is the minimum number of bytes that can be used to represent the decimal number 213? 2×10^2+1×10^2+3×10^0=2×100+1×10+3×1=200+10+3=213

d. What is a purpose of using hexadecimal notation? Hexadecimal (base16), or «hex,» is most likely the largest numbering system that you will work with. In the modern decimal system, the Arabic number set 0-9 has to be supplemented by additional values to represent the decimal equivalents of 10, 11, 12, 13, 14, and 15. Instead of inventing new symbols to represent these numbers, the letters A-F are used. Hexadecimal is represented by the set of numbers 0-F. While both lower case and upper case letters can be used in hexadecimal for A-F. In this course, we will use upper case A-F. Hexadecimal, however, is not usually represented by appending a 16 as a subscript to the number. There are two differing formats for representing hexadecimal numbers: prepending 0x or appending h. We will use 0x to denote hexadecimal numbers.
e. What is ASCII code? All data, including audio, visual, and program instruction data can be represented and stored using a sequence of binary digits, or a sequence of bytes. Recall that a file is a collection of data. In some files, the bits of data directly encode individual letters, numbers, and punctuation symbols that make up words and sentences. In other files such as an image file, the bits of data need to be computed in order for them to transform to images that we can interpret.

Question 2. Data Representations in a Computer System
Computer systems are made up of electrical components that are either on, or off, representing 1’s and 0’s, also called binary numbers. All data, such as numbers, are converted into binary representations in the computer system. The Extended ASCII chart below shows characters and their binary representations. Use the chart below and a calculator to answer the following questions:

Figure 1 ASCII code
You can obtain the binary representation of a word by concatenating the ASCII values of each letter. For example, “SSD2” in binary representation is 01010011 01010011 01000100 00110010.
a. “Hex” in binary representation is 01001000 01100101 01111000
b. “Read” in binary representation is _01010010 01100101 01100001 01100100, which is equivalent to in decimal representation.
c. 01010011 01110100 01110101 01000100 01100101 01101110 01110100 is the binary representation of the ASCII string StuDent.
d. 0x47524F5550 is a hexadecimal representation of the ASCII string 00110000 01111000 00110100 00110111 00110101 00110010 00110100 01000110 _00110101 00110101 00110101 00110000
e. 1229542485 is the decimal representation of the ASCII string _00110001 00110010 00110010 00111001 00110101 00110100 00110010 00110100 00111000 00110101_.