Number System Conversion

What is a Number System?

In the simplest terms, a number system can be described as a system representing numbers. There are different types of number systems in mathematics, including binary number systems and decimal number systems.

In this article, readers will be able to understand the concept of the number system. They will also be able to learn the method of solving number system conversion questions. There is a number conversion table for a better understanding of these concepts.

As mentioned above, a number system can be explained as a system of writing that expresses numbers. It can also be defined as the mathematical notation for representing various numbers of a given set. This is done by using digits or other symbols in a very consistent manner. Students should use this knowledge to work on number system conversion with solutions.

It should be noted that number system conversion online also provides a very unique representation of every single number. It also represented the arithmetic and algebraic structures of the figures. It also allows an individual to operate different arithmetic operations like subtraction, addition, and division.

If a student has to solve questions related to converting any base to decimal, then he or she should know that the value of any digit in a number can also be determined by:

• The digit

• The position of the number

• The base of the number system

Types of Number Systems

There are many different types of number systems. In this section, readers will learn about those different types of number systems. This discussion is carried out below.

• The Decimal Number System or the Base 10 Number System

The decimal number system is characterized by having a base of 10. This is because in this number system ten digits are used from 0 to 9. Further, the positions successive to the left of the decimal point represent units, tens, hundreds, thousands, and other numbers. This system can be described in terms of decimal numbers.

Every position in a decimal number system is used for highlighting a particular power of the base (1). For example, the decimal number 1457 consists of the digit 7 in the units place, 5 in the tens place, 4 in the hundreds place, and 1 in the thousands place. This value can also be written as:

(1 x 103) + (4 x 102) + (5 x 101) + (7 x 100)

(1 x 1000) + (4 x 100) + (5 x 10) + (7 x 1)

1000 + 400 + 50 + 7

= 1457

• Binary Number System or the Base 2 Number System

The binary number system or the base 2 number system has only two binary digits, including 0 and 1. It should be noted that the usual base 2 is a radix of 2. The digits that are described under the binary number system are simply known as binary numbers, which are combinations of 0 and 1. For example, 101001110 is a binary number. It is possible to convert any number system into a binary system. This is also true vice versa.

• Octal Number System and Base 8 Number System

In the base 8 number system or the octal number system, the numbers from 0 to 7 are used for representing numbers. Students must remember that octal numbers are usually used in computer applications. Also, converting an octal number to decimal is similar to the conversion of the decimal number system. To understand this in a clearer manner, let’s look at the example that is given below.

Convert 2158 into decimal.

2158 = 2 x 82 + 1 x 81 + 5 x 80

= 2 x 64 + 1 x 8 + 5 x 1

= 128 + 8 + 5

= 14110

• Hexadecimal Number System or Base 16 Number System

As the name indicates, the hexadecimal number system is represented or written with a base of 16. In this number system, the numbers are usually represented just like a decimal system, which is from 0 to 9. After that, the numbers are represented by using alphabets from A to F.

To summarize all of this, students should go through a number system chart. Luckily, we have compiled a similar chart and we are attaching that chart below.

Number System Conversion

As it was mentioned before, any number that is represented in a number system type can be converted into a number of other systems. There are several methods that can be used for solving these types of questions. Apart from that, students can also refer to a number system conversion table.

Using a number system conversion table can be very useful. Keeping this in mind, we have mentioned a similar table below. Students should go through this table to solve questions related to number system conversions.

Now, let’s look at the other methods of converting numbers into other systems. This is done on the basis of the base of the numbers. For example, if an individual wants to change a decimal number with a base 10 to a binary number with base 2, then we must start with the basic representation of the number system base conversion. The general form of any base number is:

(Number)_b = d_{n - 1}, d_{n - 2}, --, d₁ d₀. D_-1 d_-2 ---- d_-m

In this expression, d_{n -1} d_{n -2} --- d₁ d₀ represents the values of the integer part, and d_-1 d_-2 -- d_m represents the fractional part of the equation.

Also, d_n-1 is also known as the Most Significant Bit (MSB) and dm is the Least Significant Bit (LSB). Let’s move on to specific cases of number system conversion. These specific cases are discussed in a list below.

• Decimal to Other Bases

For converting a decimal number to some other base number, we have to divide the decimal number after converting the value of the new base. This might seem complex but is quite easy. Let’s suppose that if we have to convert decimal to binary number system, then we need to divide the decimal number by 2. For example, let’s try to convert (25)₁₀ to binary numbers.

To get a better handle on the answer, we will look at the answer in the form of a table. This table is mentioned below.

Hence, from this table, we can conclude that

(25)₁₀ = (11001)₂

• Decimal to Octal Number

If you want to convert decimal to octal number, then we need to start by dividing the given original number by 8. This should be done in a manner so that the base 10 changes to a base 8. For example, if we have to convert (128)₁₀ to an octal number, then, we can arrive at the answer with the help of the table that is mentioned below.

Hence, it can be said that the equivalent octal number is (200)₈.

• Decimal to Hexadecimal

When it comes to the conversion of decimal to hexadecimal conversion, we need to divide the given decimal number by the number 16. For example, if we have to convert (128)₁₀ to a hexadecimal number, then, the answer can be arrived at with the help of the table that is mentioned below.

Hence, the equivalent hexadecimal number is (80)₁₆.

Did You Know?

• The Hindu-Arabic numeral system is often called the decimal number system.

• There are almost 200 different number systems like mix radic, non-integer bases, negative bases, etc.

• There is only one even prime number.

• Humans have been using the number system since 35,000 BCE.