Question

a. Write the number 100 by representing 1 and 0 in ASCII. b. Write the number 255 in binary representation.

Short answer

a. ASCII for 100 is 00110001 00110000 00110000. b. 255 in binary is 11111111.

Step-by-step solution

Understanding ASCII Code

ASCII stands for American Standard Code for Information Interchange, which assigns a number to each character. In ASCII, the number '1' is represented by the decimal number 49, and the number '0' is represented by the decimal number 48.

Writing Number 100 in ASCII

To write the number 100 in ASCII using binary, we represent the digits '1', '0', and '0' in ASCII. The ASCII code for '1' is 49 and for '0' is 48. In binary, 49 is represented as 00110001 and 48 is represented as 00110000. Therefore, the number 100 in ASCII binary will be: 00110001 (for '1'), 00110000 (for '0'), 00110000 (for '0').

Understanding Binary Representation

Binary is a base-2 numeral system that uses two numbers, 0 and 1. Each binary digit is called a 'bit'. Larger numbers can be represented in binary by combining bits.

Converting Number 255 to Binary

To convert the decimal number 255 to binary, we start by finding the largest power of 2 less than or equal to 255. 255 in binary is 11111111. This is because 255 is the sum of all bits set to 1 in an 8-bit binary number (from 2^7 to 2^0), which are 128, 64, 32, 16, 8, 4, 2, and 1.

Key concepts

Binary Representation

Binary representation is a way of displaying numbers using only two digits: 0 and 1. This system is the foundation of all digital computing. Every piece of data in a computer, whether it’s a number, a letter, or a symbol, is ultimately represented in binary.

In a binary system, each digit is called a "bit", with each bit having two states, 0 or 1. The position of each bit represents a power of 2, much like each digit in our familiar decimal system represents a power of 10. Thus, to understand binary representation is to see numbers as sequences of powers of 2. For example, the binary number 101 stands for:

• 1 (in the 2^2 position) = 4
• 0 (in the 2^1 position) = 0
• 1 (in the 2^0 position) = 1

So, 101 is the decimal number 5. The principle of binary representation is about how different combinations of bits can be used to create all possible numbers - the same way we use digits from 0 to 9 in decimal.

Decimal to Binary Conversion

Converting from decimal to binary involves determining which powers of 2 sum to the original number. This method relies on understanding each power of 2 and how many times it fits into the remainder of the number to be converted.

To convert the decimal number 255 to binary, start by identifying the largest power of 2 less than or equal to 255, which is 2^7 = 128.

• Subtract 128 from 255 to get 127.
• The largest power of 2 that fits into 127 is 2^6 = 64.
• Subtract 64 from 127 to get 63, and continue this pattern.
• Proceed with 2^5 = 32, 2^4 = 16, down to 2^0 = 1.

Each power of 2 used corresponds to a "1" in the binary representation, and each power of 2 not used is a "0". Therefore, 255 equates to all bits being "1" in binary, resulting in 11111111, reflecting that each bit represents a power of 2 from 2^7 down to 2^0, respectively.

Character Encoding

Character encoding is a way to convert text data into a form that computers can understand. This process involves assigning a numerical value to each character.
One of the most well-known encoding systems is ASCII (American Standard Code for Information Interchange). ASCII assigns a decimal number to each character in the English alphabet, special symbols, and control characters.

For instance, according to ASCII, the number 49 represents the character '1', while 48 represents '0'. When encoding a sequence such as the number 100, each character is mapped to its ASCII number and then transformed into binary for storage or processing on computers:

• The character '1' is represented by 49, which in binary is 00110001.
• Each '0' is represented by 48, which translates to binary as 00110000.

Character encoding systems, including ASCII, are essential for ensuring proper data communication and storage, allowing computers to interchange information accurately among different systems and languages.