A number system is a set of symbols and rules used to represent numerical values. The key characteristic of a number system is its base, or radix, which is the number of unique digits it uses, including zero. The value of a digit depends on its position within the number, which is a power of the base. For example, in the decimal system, which uses base 10, the number 2537 represents 2 thousands, 5 hundreds, 3 tens, and 7 units, or 2 times 10 to the power of 3, plus 5 times 10 squared, plus 3 times 10, plus 7.
Let's explore the most common number systems. The decimal system, which is base 10, uses 10 digits from 0 to 9. This is the system we use in everyday life. Each position represents a power of 10, such as units, tens, hundreds, and so on. For example, the number 2537 in decimal equals 2 times 10 cubed, plus 5 times 10 squared, plus 3 times 10, plus 7. The binary system, which is base 2, uses only two digits: 0 and 1. This is the fundamental system used by computers because it can be easily represented by electrical states like on or off. Each position represents a power of 2. For instance, the binary number 101101 equals 1 times 2 to the power of 5, plus 0 times 2 to the power of 4, and so on, which equals 45 in decimal.
Now let's look at two other important number systems. The octal system, which is base 8, uses 8 digits from 0 to 7. It was sometimes used in computing as a compact way to represent binary numbers, since 8 is 2 to the power of 3. Each position in octal represents a power of 8. To convert from binary to octal, we group binary digits in sets of three, starting from the right. For example, the binary number 1011 0111 can be grouped as 001 011 011 111, which gives us 1337 in octal. The hexadecimal system, which is base 16, uses 16 symbols: the digits 0 through 9 and the letters A through F, where A equals 10, B equals 11, and so on up to F equals 15. Hexadecimal is widely used in computing for representing memory addresses, colors, and other values because it's a convenient way to represent groups of 4 binary digits, since 16 is 2 to the power of 4. To convert from binary to hexadecimal, we group binary digits in sets of four. So 1011 0111 becomes B7 in hexadecimal.
Let's explore how to convert between different number systems. To convert from decimal to binary, we divide the decimal number by 2 repeatedly and collect the remainders from bottom to top. For example, to convert 25 to binary, we first divide 25 by 2, which gives 12 with a remainder of 1. Then we divide 12 by 2, giving 6 with a remainder of 0. Continuing this process, we get the binary representation 11001. To convert from binary to decimal, we multiply each digit by its place value, which is a power of 2, and sum the results. For the binary number 11001, we calculate 1 times 2 to the power of 4, plus 1 times 2 to the power of 3, plus 0 times 2 squared, plus 0 times 2, plus 1 times 1, which equals 16 plus 8 plus 0 plus 0 plus 1, giving us 25 in decimal. Different number systems have specific applications. Binary is fundamental for computer processing and storage. Hexadecimal is commonly used for memory addresses and color representations in RGB format. Octal was used in some legacy computing systems.
To summarize what we've learned about number systems: First, a number system is defined by its base, which is the number of unique digits it uses. Second, the value of a digit depends on its position within the number, which represents powers of the base. Third, the decimal system, which is base 10, is intuitive for humans and used in our everyday life. Fourth, the binary system, which is base 2, is fundamental to computing and digital systems because it can be easily represented by electrical states. Finally, the hexadecimal system, which is base 16, provides an efficient way to represent binary data in computing applications. Understanding these different number systems and how to convert between them is essential for computer science, digital electronics, and many other technical fields.