Binary Numbers and Decimal Numbers

💡 Introduction Binary numbers can initially appear intimidating because they differ greatly from decimal numbers. However, when it comes to the fundamentals, the mathematical operations of counting, adding, or subtracting binary numbers are exactly the same as with decimal numbers. It's important to note that there aren't different types of numbers; numbers are universal. The only difference lies in the notations used to represent them.

🔢 Decimal Numbers Humans, primarily due to having 10 fingers and 10 toes, adopted a decimal system that utilizes 10 individual numerals (0-9) to represent all numbers. By combining these numerals, we can represent any whole number. This system is often referred to as base 10 because it uses 10 numerals.

🔀 Binary Numbers Computers, on the other hand, find it easier to think in terms of 0 and 1 due to the way logic gates work inside processors. This binary system, also known as base 2, uses only two numerals: 0 and 1. Binary numbers can represent all whole numbers, just like decimal numbers, but their representation appears different.

🔄 Counting in Decimal and Binary Counting in decimal involves moving through all the numerals from 0 to 9, then adding a new column with higher significance once you reach 9. This process repeats to count all whole numbers. Counting in binary is similar, except you only have two numerals: 0 and 1. You start with 0, then increment to 1. Once you reach 1, you add a new column and start over at 0 in the original column. This process continues to count all whole numbers in binary.

🔀🔢 Number Representation with Bits In computing, you often encounter the concept of bits, which represent either 1 or 0. To determine how many decimal numbers can be represented by a certain number of bits, you can use the formula 2^N, where N is the number of bits. For example, an eight-bit number can represent 2^8, which equals 256 decimal numbers (0-255). Similarly, a four-bit number can represent 2^4, or 16 total numbers.

🔀🔄 Number Systems and Base The trick mentioned earlier works for any number system. By swapping out the base (10 for decimal, 2 for binary), you can calculate the total numbers representable with a given number of columns. For example, a base 10 number with two columns of digits corresponds to 10^2, which equals 100 (representing the numbers 0-99).

➕ Binary Addition Binary addition is simpler than addition in other bases because there are only four possible scenarios. 0+0=0, 0+1=1, 1+0=1 (similar to decimal), and 1+1=10 (carry a digit to the next column). The carry concept is similar to carrying a digit when adding in decimal.

🔒 Operators: OR and AND In computer logic, the numbers 1 and 0 represent true and false, respectively. The OR operator evaluates each digit and returns true if either of them is true. The AND operator returns true only if both values are true. These operators play a crucial role in computer calculations.

🌐🔒🔀🔢