Monday 12 February 2024

conversion of numbers from one radix to another radix


We have learned and use the decimal numbering system simply because humans are born with ten fingers! Hence, the numeric system we is the decimal number system, but this system is not convenient for machines since the information is handled codified in the shape of ON or OFF bits.

This means, we have to learn the binary system in addition to the decimal system. We also will discuss the octal and hexadecimal systems because conversion to/from binary is easy and numbers in these systems are easier to read than binary numbers for humans. 

This way of codifying takes us to the necessity of knowing the positional methods of calculation which will allow us to express a number in any base where we need it.

A base of a number system or radix defines the range of values that a digit may have.

Converting numbers from one radix (base) to another is a fundamental concept in computer science and mathematics. Here's an overview of the different methods for conversion:

1. Integer Conversion:

a) Decimal to Another Radix:

  1. Divide the number repeatedly by the target radix, noting the remainders from each division (in reverse order). These remainders become the digits in the new base.
  2. If the number is negative, convert the absolute value first, then add a negative sign at the end.

Example: Convert 235 (decimal) to binary.

  1. 235 / 2 = 117 R 1 -> 1 (least significant digit)
  2. 117 / 2 = 58 R 1 -> 1
  3. 58 / 2 = 29 R 0 -> 0
  4. 29 / 2 = 14 R 1 -> 1
  5. 14 / 2 = 7 R 0 -> 0
  6. 7 / 2 = 3 R 1 -> 1
  7. 3 / 2 = 1 R 1 -> 1
  8. 1 / 2 = 0 R 1 -> 1

Therefore, 235 (decimal) = 11101011 (binary).


Conversion of decimal number to octal
Now, let's express the same decimal number 1341 in octal notation. 

Conversion of decimal number to hexadecimal
Let's express the same decimal number 1341 in hexadecimal notation. 

The easiest way to convert fixed point numbers to any base is to convert each part separately. We begin by separating the number into its integer and fractional part. The integer part is converted using the remainder method, by using a successive division of the number by the base until a zero is obtained. At each division, the reminder is kept and then the new number in the base r is obtained by reading the remainder from the lat remainder upwards.

b) Another Radix to Decimal:

  1. Multiply each digit by its corresponding place value (power of the original radix) and sum the results.

2. Fractional Conversion:

a) Decimal to Another Radix:

  1. Multiply the fractional part by the target radix repeatedly. Note the integer part of each product (these become digits in the new base).
  2. For the remaining fractional part, repeat step 1 until a repeating pattern emerges or a desired precision is reached.

Example: Convert 0.625 (decimal) to binary.

  1. 0.625 * 2 = 1.25 -> 1 (integer part)
  2. 0.25 * 2 = 0.5 -> 0 (integer part)
  3. 0.5 * 2 = 1.0 -> 1 (integer part)

Therefore, 0.625 (decimal) = 0.101 (binary) (repeating pattern).

b) Another Radix to Decimal:

  1. Sum the values of each digit multiplied by its corresponding place value (a fraction of the original radix).

 Example: Convert 234.14 expressed in an octal notation to decimal.



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