Understand the number systems decimal, binary, octal, and hexadecimal
Number systems are essential for representing and processing data in the
world of mathematics and computers. The decimal, binary, octal, and hexadecimal
number systems will all be covered in this article along with the successive
division method for converting between them.
The Decimal System: The Basis
The one we are most accustomed to is the decimal system. When you hear
the word "decimal," think of the prefix "deci," which
denotes a tenth of a whole. Base 10 is a system with ten distinct digits: 0, 1,
2, 3, 4, 5, 6, 7, 8, and 9. For ordinary counting, we represent numbers like
12, 36, or 468 using this approach.
The Computer's Language is Binary
The binary system, on the other hand, is crucial to computing. In
binary, the prefix "bi" stands for "two," and it uses the
base 2 system. There are only two digits in binary: 0 and 1. A lot of computers
use this system. Digital computers use the numbers 0 and 1 to represent
"on" and "off," respectively.
Base Eight System: Octal System
The octal system, which uses base 8, is denoted by the prefix
"octa" (which means eight). There are eight distinct numbers in it:
0, 1, 2, 3, 4, 5, 6, and 7. Digital circuits and many computer applications
benefit greatly from octal numbers.
The Power of Sixteen in the Hexadecimal System
Base 16 is used in the hexadecimal system, where the prefix
"hexa" stands for six. It uses six letters (A, B, C, D, E, and F) and
10 decimal digits (0–9) to represent numerals. The mapping is as follows: A =
10, B = 11, C = 12, D = 13, E = 14, and F = 15. This method is frequently
utilized in digital applications and programming, where binary representations
might be cumbersome.
Successive Division in Decimal to Binary Conversion
By using successive division, we can change a decimal integer into a
binary one. Let's use the binary representation of the decimal number 348 as an
example:
348 divided by 2 gives 174 (remainder 0)
174 divided by 2 gives 87 (remainder 0)
87 divided by 2 gives 43 (remainder 1)
43 divided by 2 gives 21 (remainder 1)
21 divided by 2 gives 10 (remainder 1)
10 divided by 2 gives 5 (remainder 0)
5 divided by 2 gives 2 (remainder 1)
2 divided by 2 gives 1 (remainder 0)
1 divided by 2 gives 0 (remainder 1)
Reading from bottom to top, the binary
representation of 348 is 101011100.
Octal and Hexadecimal to Decimal Conversion
The procedure of successive division is used
to convert to octal and hexadecimal but with different bases. In base 8 octal:
348 divided by 8 gives 43 (remainder 4)
43 divided by 8 gives 5 (remainder 3)
5 divided by 8 gives 0 (remainder 5)
Reading from bottom to top, 348 in octal is
534.
For hexadecimal (base 16):
348 divided by 16 gives 21 (remainder 12)
21 divided by 16 gives 1 (remainder 5)
1 divided by 16 gives 0 (remainder 1)
Converting the remainders to hexadecimal
values, 348 in hexadecimal is 15C.
For many uses in computer science and
mathematics, knowing these number systems and how to convert between them is
crucial. Knowing these principles can be immensely useful whether you're a
programmer, engineer, or just interested in the digital world.