Binary Number Representation

Binary Representation

• 78 (decimal) can be represented in binary as 01001110, using the place values 64 + 8 + 4 + 2 = 78.
• A negative number in binary representation has the first (most significant) binary digit as 1, for example, -95 (decimal) can be represented as 10010111, using the place values -128 + 32 + 1 = -95.

Two's Complement Method

• To represent a negative number using two's complement, first take one's complement (invert all the values – 0's to 1 and 1's to 0) for the given binary number.
• Then, add 1 to the result, for example, the binary representation of -35 is 11011101.

Data Storage

• All types of data, including numbers, text, images, and sound, must be held in binary form inside the computer.
• Whole numbers are called integers, and one byte of storage can store all the numbers from 0 to 255.
• When you want to represent the next number, i.e., ten, you use a combination of these digits, i.e., 10.

Binary System

• Binary is a base two system, with only two different digits: 0 and 1.
• Binary starts off in the same way as a decimal: 0 (zero), 1 (one), using a combination of 0's and 1's to represent different values.
• The columns in binary represent successive powers of two, just like the columns in decimal represent different powers of ten.

Binary Conversion

• A decimal number can be expressed in binary form using a large number of digits, for example, the number 65 can be expressed in binary form as 1000001.
• These binary codes can be represented using hexadecimal values or decimal values.

Hexadecimal Representation

• Hexadecimal numbers are used in assembly code and machine code, as they are easier, faster, and less error-prone to write code compared to binary.
• Hexadecimal numbers can be used to represent URLs or web addresses, for example, www.hodder.co.uk becomes %77%77%77%2E%68%6F%64%64%65%72%2E%63%6F%2E%75%6B.

Binary Integers

• Whole numbers such as 5, 7, 12, and 3988 are called integers.
• Unsigned integers have positive values by definition, while signed integers can be positive or negative.
• Unsigned integers in binary are represented as 2^n, where n is the number of bits.
• The minimum value that can be represented with n bits is 0, and the maximum value is 2^n-1.