Decimal Positional Notation

Positional Number System and Examples (Updated 2025)

Positional Number System and Examples (Updated 2025)

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In a positional number system, each symbol represents a different value depending on the position it occupies in a number, and each system has a value that relates to the number directly next to it. The total value of a positional number is the total of the resultant value of all positions. The decimal number system is known as a positional number system because the value of the number depends on the position of the digits. For example, 12345 has a very different value than 54321, although the same digits are used in both numbers. The position in which the digit appears affects the value of that digit.

In a positional number system, the value of each digit is determined by which place it appears in the total number. The lowest place value is the rightmost position, and each successive position to the left has a higher place value. So, the rightmost position represents the “ones” column, the next position represents the “tens” column, the next position represents “hundreds,” and the next position represents the “thousand” and then ten thousand, etc.

Therefore, the number 54321 represents 5x ten thousand, 4x one thousand, 3x hundred, 2 tens, and 1 x one, whereas the number 12345 represents 1x ten thousand, 2 x one thousand 3 hundred, 4 x tens, and 5 ones.  The binary, octal, and hexadecimal are all examples of a Positional Number System. Following is the calculator for conversion between all number systems, and then you can study how to calculate without a calculator.

Number System Converter

Number System Converter

Positional vs. Non-Positional Systems

AspectPositional (Decimal)Non-Positional (Roman Numerals)
ZeroSupports zeroNo zero symbol
ArithmeticSimplifies calculationsComplex and limited
ScalabilityHandles large numbersCumbersome for large values

Historical Evolution of Number Systems

Babylonians (2000 BCE): A base-60 (sexagesimal) system using wedge-shaped cuneiform symbols, pioneering positional notation for astronomy and trade, with a lasting legacy in time (60 minutes) and geometry (360 degrees).

Mayans (300 BCE): A base-20 (vigesimal) system with dots, bars, and a shell glyph for zero, enabling precise calendrical and astronomical calculations central to their rituals and cosmology.

Indian-Arabic System (500 CE): The decimal (base-10) system with digits 0-9, perfected by Arab scholars, revolutionized global mathematics through positional notation and zero’s dual role as placeholder and numeral.

Decimal Positional Notation

The decimal positional notation system works as described in the table in the figure below. To use the positional number system, match a given number to its positional value. The arrow at the left shows that the table can be expanded according to position and can be calculated.

Positional Number System: An educational image explaining the concept of decimal position notation in the base-10 numeral system. The table showcases five columns, each corresponding to a decimal digit's positional value, labeled as positions 4, 3, 2, 1, and 0. It visually illustrates positional value with rows detailing radix (10), position, calculation (powers of 10), and numerical value (e.g., 10,000, 1,000, 100). The image includes a labeled arrow highlighting 'Continue' to suggest further positions exist.

Radix

The Term Radix describes the number of digits utilized in the positional number system to identify the system’s base. The decimal notation system is based on 10, so the radix is 10.

Position in

It shows the position of the decimal number. The position starts from the right and moves to the left. So 0 is the 1st position 1 is the 2nd position, 2 is the 3rd position, and so on. The numbers also represent the exponential value that will be used to calculate the positional value of the 4th row.

Calculate

The third row calculates the positional value by taking the radix and raising it to the power of its position. Importantly, n0 is always = 1.

Position Value

The first row shows the number base or radix. From left to right, the values listed represent units of thousands, hundreds, tens, and ones.

Example – 12345 and 54321

A clear representation of the number 12345, dissected into its positional values through a table format. The table is divided into columns labeled Ten Thousands, Thousands, Hundreds, Tens, and Ones. Each column outlines the position, decimal number, and positional value, demonstrating how each digit contributes to the total value of 12345. Mathematical calculations are shown to illustrate the breakdown, including positional multiplication and summation.
An educational image showcasing the breakdown of the number 54321 into positional values in a base-10 numeral system. The table explains the process of decomposing the number into units of Ten Thousands, Thousands, Hundreds, Tens, and Ones. Each column demonstrates the positional values, multiplications, and their sums, illustrating how the total value of 54321 is achieved. The table emphasises the addition of the individual values in each column to form the number 54321.

Binary Positional Notation

The binary positional notation also operates similarly to the decimal number system, as shown in the figure below. It also has a Radix, Position, which is from right to left; Calculation and positional value as described for decimal. The Radix for Binary is 2. The example is also shown below

A table titled 'Binary Position Notation' illustrating the breakdown of binary values. The table consists of four rows and eight columns. The rows are labeled Radix, Position, Calculation, and Positional Value. Each column corresponds to a binary position from 7 to 0. The 'Radix' row displays the number 2 across all columns. The 'Position' row lists positions from 7 to 0 in descending order. The 'Calculation' row shows powers of 2 for each position, such as 2 7 , 2 6 , and so on. The 'Positional Value' row provides the resulting values, including 128, 64, 32, 16, 8, 4, 2, and 1. This image helps visualise how binary positional notation operates, with each binary digit's positional value based on powers of 2.

Examples

"An educational table explaining the conversion of the binary number 11101101 into its decimal equivalent. The table is organized into columns titled Position, Binary Number, Calculate, and Positional Value. The positional values are derived using the powers of 2, starting from 2^7 (128) down to 2^0 (1). The binary number is broken down as follows: 1 in the position of 128, 64, and 32; 0 in the position of 16; 1 in the position of 8, 4, and 1; and 0 in the position of 2. The calculations demonstrate multiplication of each positional value by the respective binary digit, with a final summation showing the decimal result of 237."
"An educational table demonstrating the conversion of the binary number 10001101 into its decimal equivalent. The table is divided into rows that include Position (128, 64, 32, 16, 8, 4, 2, 1), Binary Number (1, 0, 0, 0, 1, 1, 0, 1), Calculation (e.g., 128×1, 64×0, and so on), and Positional Value (e.g., 128, 0, 0, 0, 8, 4, 0, 1). A summation of all positional values provides the decimal result, which is 141. This visualisation helps explain binary-to-decimal conversion concepts.

Converting Between Decimal and Binary

Converting binary to decimal (base-2 to base-10) numbers and back is an essential concept for understanding the binary numbering system. Binary number systems form the basis for all computers, digital systems, and programming languages, and they are also essential for comprehending IP address and IP address subnetting.

Binary to Decimal Conversion

To understand IP addressing and subnetting, the network technician must understand how to convert from binary to decimal. There are several ways to convert from binary to decimal. Here, we will realize conversion from binary to decimal and then from decimal to binary.

Binary to Decimal Using Positional Notation

Positional notation is one of the most used methods for converting binary and decimal, so if we have a binary number 110001102 and want to convert this number to decimal base 10 system using the positional system notation.

The figure below illustrates the conversion of the binary to decimal. Write down the positional value from right to left and right down the binary number in the next row below the positional values. Next, multiply the binary numbers in the second row with their corresponding positional value in the first row and add them up to get the decimal number 199.

A mathematical table explaining the conversion of the binary number 11000110 to its decimal equivalent. The table dissects each binary bit into positional values in powers of 2, starting from 128 (2^7) to 1 (2^0). Each position includes the respective binary digit (1 or 0), the multiplication of positional values, and the resulting values. Calculations shown are as follows: 128×1, 64×1, 32×0, 16×0, 8×0, 4×1, 2×1, and 1×0. The sum of these positional values—128, 64, 0, 0, 0, 4, 2, and 0—yields a decimal value of 198. This breakdown illustrates how binary digits contribute to forming a decimal number.

The positional value doubles from right to left. For example, from right to left, the first positional value is 1, the next position value is 2, double 1 (the first position value), and the third place value is 4, double the second position value. We can write positional values using this method according to the binary digits.

The following binary value is 10111100. So,  write the positional value according to the number of digits in the binary value. Then, write the binary numbers below the positional values and map them up. If the digit is a 1, write its corresponding positional value below the line, under the digit. If the digit is a 0, write a 0 below the line, under the digit, and add up the numbers written below the line. This is the decimal equivalent of the binary number. See figure the figure below.

"An educational image explaining the conversion of the binary number 10111100 to its decimal equivalent. The image features a table with four rows labeled Position (128 to 1), Binary Digits (1, 0, 1, 1, 1, 1, 0, 0), Calculations (e.g., 128×1, 64×0), and Positional Value (e.g., 128, 0, 32). Each column demonstrates the relationship between binary digits and positional values, with calculations summing up to provide the final decimal result, 188.

Now that we understand the basic conversion from binary to decimal, we can try to convert a binary IPv4 address to its dotted-decimal equal. First, divide 32 bits of the IPv4 address into four 8-bit octets. Then, calculate the binary positional value of the first octet binary number and all others.

For example, consider that 11110110.10111111.11101011.11011011 is the binary IPv4 address. As shown in the Figure below, you must convert the binary address to decimal, starting from the first octet. Enter the 8-bit binary number under the positional value of row 1 and then calculate to produce the decimal number for the first octet of the dotted decimal notation. Next, convert the remaining octets. Write down the full IP address using dotted decimal notation.

An educational image showcasing the binary-to-decimal conversion of an IP address. The binary IP address is divided into four octets: 11110110, 10111111, 11101011, and 11011011. Each octet is explained through step-by-step calculations, with the positional values for binary digits converted to decimal equivalents. The results for each octet are shown as follows: the first octet totals 246, the second 191, the third 235, and the fourth 219. The final decimal IP address is displayed as 246.191.235.219, emphasising the conversion process clearly.

 Decimal to Binary Conversion

Decimal to binary conversion is also essential to understand; we can convert a dotted decimal IPv4 address to binary. We can also use a positional value table to convert from decimal to binary. The figure below illustrates the decimal to the binary conversion process. For example, we want to convert Figure 246 into binary. The steps for converting from decimal to binary are the following:

A step-by-step visual explanation of converting the decimal number 242 into binary. The image features a process using subtraction of powers of 2, starting from 128 down to 1. Each power of 2 is represented in a row, showing whether it is included (marked with a binary digit '1') or excluded (marked with a '0'). Calculations include steps like 242 - 128 = 114 and so on until the remainder is zero. The resulting binary number is displayed as 11110010. This diagram highlights the logic and process behind decimal-to-binary conversion.

Step-1

Write down the positional value from left to right, as shown in the figure below. If the decimal number (242 in our case) is equal to or greater than the most significant bit (128), add a binary 1 below the 128 positional value and subtract 128 from the decimal number (242 -128 =114 in our case). If not, enter binary 0 below the 128 positional value.

Step-2

Compare if the remainder (114) equals or exceeds the next most significant bit (64). If yes, add a binary 1 below the 64 positional value and subtract the positional value 64 from the decimal number (114 -64 =50  in our case). If no, then enter binary 0 below the 64 positional value.

Step-3

Compare if the remainder (50) equals or exceeds the next most significant bit (32). If yes, add a binary 1 below the 32 positional value and subtract the positional value 32 from the decimal number (50 -32 =18  in our case). If no, then enter binary 0 below the 64 positional value.

Step-4

Compare if the remainder (18) equals or exceeds the next most significant bit (16). If yes, add a binary 1 below the 16 positional value and subtract the positional value 16 from the decimal number (18 -16 =2  in our case). If no, then enter binary 0 below the 16 positional value.

Step-5

Compare the reminder (2) with the following -most significant bit, you can see that the remainder is less than the considerable value (80), so add a binary 0 in the 8 positional value and step forward to the next positional value, which is (4), the remainder is also less than this positional value, so, add 0 below this position value and step forward to next value. The next value is positional value is (2), So, Subtract the reminder 2 from the positional value (2-2=0), put the 0 in the last position.

Where Are Positional Number Systems Used?

Binary in Computers:

  • Transistors use binary (0=off, 1=on) to perform calculations.
  • Example: RGB colors like #FF0000 (pure red) use binary values (0-255) for intensity.

Hexadecimal in Web Design:

  • Hex codes simplify color selection (e.g., #1A3FBC = blue shade).
  • Memory addresses in programming (e.g., 0x1A3F).

Base-60 in Timekeeping:

  • Babylonian legacy: 60 seconds/minute, 60 minutes/hour.

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