Students, do you know what the base system is? The base system is a counting system that dictates how many different symbols are used to represent different numerical values. We usually use the decimal system, that is, we use ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to represent all numbers. However, in addition to decimal, there are other decimal systems, such as binary, octal, and hexadecimal. So, why do we use a different base?
In fact, different base systems have different characteristics and uses. Binary uses only two symbols (0,1), which is the simplest switching state, so it is very suitable for representing electronic signals and is the basic language of computers. The octal has eight symbols (0, 1, 2, 3, 4, 5, 6, 7), and it can be conveniently converted to binary because each octal number can be represented by three binary numbers.
For example, an octal 7 is a binary 111, and an octal 5 is a binary 101. The hexadecimal system has sixteen symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f), and it can also be conveniently converted to binary because each hexadecimal number can be represented by four binary numbers. For example, the hexadecimal f is the binary 1111, and the hexadecimal a is the binary 1010. One advantage of hexadecimal is that it can represent larger numbers with fewer symbols, so it is often used to represent information such as colors, memory addresses, etc.
So, how do you convert between different decimal systems? In fact, we can use division and remainder to achieve the conversion of the base system. Division is the division of one number by another to get the quotient and remainder. The remainder method is to take only the remainder of the division and ignore the quotient. For example, if we want to convert 25 in decimal to binary, we can do this:
25 divided by 2, the quotient is 12, the remainder is 1, we write down the remainder and get 1
Divide 12 by 2, the quotient is 6, the remainder is 0, we write down the remainder and get 01
6 divided by 2, the quotient is 3, the remainder is 0, we write the remainder down to get 001
3 divided by 2, the quotient is 1, the remainder is 1, we write down the remainder to get 1001
1 is divided by 2, the quotient is 0, the remainder is 1, we write down the remainder and we get 11001
When the quotient is 0, we stop dividing and join all the remainders from bottom to top to get the binary result. So, 25 in the decimal is equal to 11001 in the binary. In the same way, we can also use this method to convert decimal to octal or hexadecimal by replacing the divisor with the corresponding base of the base. For example, the base of octal is 8 and the base of hexadecimal is 16. If we want to convert other decimal systems to decimal system, we can do so by multiplication and addition. Multiplication is the process of multiplying one number by another to get the product. Addition is the addition of multiple numbers to get the sum. For example, if we want to convert binary 11001 to decimal, we can do this:
11001 from right to left, each number multiplied by the corresponding power of 2, to get 1*20, 0*21, 0*22, 1*23, 1*24, add all the products to get 1+0+0+8+16=25
So, 11001 in binary is equal to 25 in decimal. In the same way, we can also use this method to convert octal or hexadecimal to decimal by replacing the multiplier with the corresponding base of the decimal system. For example, the base of octal is 8 and the base of hexadecimal is 16.
Through this article, we learned what a base system is, why we use a different base system, and how to convert between different base systems. We learned about the characteristics and uses of binary, octal, decimal and hexadecimal, and how to use division and remainder, or multiplication and addition, to achieve base conversion. We also used some vivid metaphors to help students remember and understand the concept of the base system. It is hoped that students can master the knowledge of the base system, use the skills of the base system, and solve practical problems.