1.4. Hexadecimal Number System

The Hexadecimal Number System is a base-16 system that uses 16 symbols: 0 to 9 and A to F (where A=10, B=11, C=12, D=13, E=14, F=15). Hexadecimal is often used in computing as a shorthand for binary because one hexadecimal digit represents four binary digits (bits).

The hexadecimal number system operates on a base of 16, which means it has 16 unique symbols to represent values. These symbols include the standard digits from 0 to 9, which represent values from zero to nine, and the letters A to F, which represent the values ten to fifteen. The reason hexadecimal is important in computing is that it simplifies binary representation. Each hex digit corresponds to four binary digits (bits). For instance, the hexadecimal number A is represented in binary as 1010, allowing us to write larger binary numbers more compactly.

Imagine you're working with two languages: a long, verbose one (like binary) and a short, efficient one (like hexadecimal). If you want to say '10' in the verbose language, it would take a lot of space to write it out. However, in the short language, you simply use a single character. This is similar to how hexadecimal allows us to write long binary numbers more succinctly.

Hexadecimal numbers can be converted to decimal by expanding the number using powers of 16:

Example: Convert 2F3 (hexadecimal) to decimal. 2F316=2×162+15×161+3×160=512+240+3=75510

To convert a hexadecimal number to decimal, each digit in the hexadecimal number must be multiplied by 16 raised to the power of its position, counting from right to left starting from 0. For the example of 2F3, we break it down: the '2' is in the 256 place (16^2), the 'F' represents 15 and is in the 16 place (16^1), and the '3' is in the 1 place (16^0). Thus, the calculation follows as: 2 * 16^2 + 15 * 16^1 + 3 * 16^0, which simplifies to 512 + 240 + 3, yielding the decimal number 755.

Think of hex to decimal conversion like converting a recipe measurement. If you know that 1 cup is 16 tablespoons, and you need to figure out how much is in 2 cups, you calculate 2 cups * 16 tablespoons per cup = 32 tablespoons. Similarly, in converting hex values, you're essentially figuring out how to express a quantity in a different measurement system.

Decimal numbers can be converted to hexadecimal using successive division by 16, noting the remainders.

Example: Convert decimal number 755 to hexadecimal. 755÷16=47 remainder 3 47÷16=2 remainder 15(F) 2÷16=0 remainder 2

Reading the remainders from bottom to top gives 2F3.

To convert a decimal number to hexadecimal, you repeatedly divide the decimal number by 16. Each time you divide, you take note of the remainder. These remainders represent the hexadecimal digits, starting from the least significant digit (rightmost). Once you can no longer divide (when the quotient hits zero), you read the remainders upwards to get the hexadecimal result. In our example with the number 755, the steps give us remainders of 3, 15 (represented as F), and 2, which when read from bottom to top gives us the hexadecimal number 2F3.

Imagine you’re packing items into boxes of certain sizes to represent different values. Each time you fill a box, it can only hold a limited amount (like the number 16). So, you keep dividing your total number of items until you can’t fill any more boxes. The leftover items become your remainders, and when you stack the boxes (read them from the last filled to the first filled) you get the complete picture of how things are packed. This is akin to how you derive the hexadecimal value from decimal.

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