What is Decimal in Number System?

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What is Decimal in Number System? Explore the fundamentals of decimals in the number system and how they play a crucial role in everyday mathematics.

What is Decimal in the Number System?

In the number system, decimal refers to the base-10 numeral system. It is the most widely used number system in the world and is the one that most people are familiar with. In the decimal system, there are ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These digits are used to represent numbers, and the position of each digit in a number determines its value.

Decimal numbers are written using a positional notation system, where each digit’s position represents a power of 10. The rightmost digit represents the ones place, the digit to its left represents the tens place, the next digit to the left represents the hundreds place, and so on. Each digit is multiplied by the corresponding power of 10 and then summed to get the value of the number.

For example, in the decimal number 1234:

The digit 4 is in the ones place, so its value is 4.

The digit 3 is in the tens place, so its value is 3 * 10 = 30.

The digit 2 is in the hundreds place, so its value is 2 * 100 = 200.

The digit 1 is in the thousands place, so its value is 1 * 1000 = 1000.

Adding these values together, you get the decimal number 1234, which represents the quantity one thousand two hundred thirty-four.

Decimal numbers are used for everyday counting and arithmetic operations, and they form the basis of our numerical system in mathematics and everyday life.

What is the Decimal Number System with Example?

The decimal number system, also known as the base-10 system, is the most commonly used numeral system in the world. It uses ten distinct symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to represent numbers. Each digit’s position in a decimal number represents a power of 10, with the rightmost digit representing 10^0 (1), the next digit to the left representing 10^1 (10), the next one representing 10^2 (100), and so on.

Here’s an example of a decimal number and how it works:

Decimal Number: 375.62

The rightmost digit is 2, which represents 2 multiplied by 10^0, so it’s just 2.

Moving to the left, the next digit is 6, which represents 6 multiplied by 10^1, so it’s 6 * 10 = 60.

The next digit is 5, which represents 5 multiplied by 10^2, so it’s 5 * 100 = 500.

The decimal point separates the whole number part from the fractional part.

After the decimal point, we have the digit 7, which represents 7 multiplied by 10^(-1), so it’s 7 * 0.1 = 0.7.

Finally, the last digit is 3, which represents 3 multiplied by 10^(-2), so it’s 3 * 0.01 = 0.03.

Now, we add all these parts together:

2 + 60 + 500 + 0.7 + 0.03 = 562.73

So, the decimal number 375.62, when expanded, is equal to 562.73 in decimal form. The decimal system is used extensively in everyday life for various mathematical calculations and representations of quantities.

Decimal Number System Formula

The decimal number system, also known as the base-10 number system, is the most commonly used number system in everyday life. It uses 10 digits (0 through 9) to represent numbers. Each digit’s position in a decimal number represents a power of 10. The formula to express a decimal number in terms of its digits and positions is as follows:

For a decimal number N with n digits:

• N = dn-1 * 10^(n-1) + dn-2 * 10^(n-2) + … + d1 * 10^1 + d0 * 10^0

Where:

N is the decimal number.

dn-1, dn-2, …, d1, d0 are the individual digits of the number (from left to right).

n is the total number of digits in the number.

10^(n-1), 10^(n-2), …, 10^1, and 10^0 are the powers of 10 corresponding to each digit’s position.

For example, let’s take the number 723:

d2 = 7 (the leftmost digit).

d1 = 2 (the middle digit).

d0 = 3 (the rightmost digit).

n = 3 (there are 3 digits).

Using the formula:

N = 7 * 10^2 + 2 * 10^1 + 3 * 10^0

N = 700 + 20 + 3

N = 723

So, the number 723 can be expressed in the decimal number system using this formula.

Conversion From Other Bases to Decimal Number System

Converting numbers from other bases to the decimal number system involves understanding the positional value of each digit in the given number and then performing the necessary calculations. The decimal number system is base 10, meaning it uses 10 different digits (0 through 9) to represent numbers. Here are the steps to convert numbers from other bases to decimal:

Identify the base of the given number: Determine the base of the number you want to convert. Common bases include binary (base 2), octal (base 8), and hexadecimal (base 16), but other bases are possible as well.

Write down the number in its original form: Record the given number in its original base. For example, if you have a binary number, write down all the binary digits.

Assign positional values: Starting from the rightmost digit, assign positional values to each digit based on its position in the number. For example, in the decimal system, the rightmost digit has a positional value of 1, the next digit to the left has a positional value of 10, the next one has a positional value of 100, and so on. In other bases, the positional values are powers of the base itself.

In binary (base 2), the rightmost digit has a positional value of 2^0 (1), the next one has 2^1 (2), the next 2^2 (4), and so on.

In octal (base 8), the rightmost digit has a positional value of 8^0 (1), the next one has 8^1 (8), the next 8^2 (64), and so on.

In hexadecimal (base 16), the rightmost digit has a positional value of 16^0 (1), the next one has 16^1 (16), the next 16^2 (256), and so on.

Multiply each digit by its positional value: Multiply each digit by its corresponding positional value and then sum all these products to get the decimal equivalent.

Add the products together: Sum up all the products obtained in the previous step to get the decimal equivalent.

Here are some examples:

Example 1: Convert the binary number 1101 to decimal.

Positional values: 2^0, 2^1, 2^2, 2^3

Calculation: (1 * 2^0) + (0 * 2^1) + (1 * 2^2) + (1 * 2^3) = 1 + 0 + 4 + 8 = 13 (in decimal)

Example 2: Convert the octal number 46 to decimal.

Positional values: 8^0, 8^1

Calculation: (6 * 8^0) + (4 * 8^1) = 6 + 32 = 38 (in decimal)

Example 3: Convert the hexadecimal number 1A7 to decimal.

Positional values: 16^0, 16^1, 16^2

Calculation: (7 * 16^0) + (10 * 16^1) + (1 * 16^2) = 7 + 160 + 256 = 423 (in decimal)

These steps should help you convert numbers from other bases to the decimal number system.

Conversion From Decimal Number System to Others

Converting a decimal number to another number system, such as binary, octal, or hexadecimal, involves dividing the decimal number by the base of the target number system and recording the remainders at each step until the quotient becomes zero. Here are the steps for converting a decimal number to binary, octal, and hexadecimal:

Binary (Base 2):

Start with the decimal number you want to convert.

Divide the decimal number by 2.

Record the remainder (either 0 or 1).

Continue dividing the quotient by 2 and recording the remainders until the quotient becomes 0.

Read the remainders in reverse order to get the binary equivalent.

For example, let’s convert the decimal number 25 to binary:

25 divided by 2 equals 12 with a remainder of 1.

12 divided by 2 equals 6 with a remainder of 0.

6 divided by 2 equals 3 with a remainder of 0.

3 divided by 2 equals 1 with a remainder of 1.

1 divided by 2 equals 0 with a remainder of 1.

Reading the remainders in reverse order, we get the binary equivalent: 11001.

Octal (Base 8):

Follow the same steps as for binary conversion, but divide the decimal number by 8 instead of 2.

Record the remainders.

Read the remainders in reverse order to get the octal equivalent.

For example, let’s convert the decimal number 134 to octal:

134 divided by 8 equals 16 with a remainder of 6.

16 divided by 8 equals 2 with a remainder of 0.

2 divided by 8 equals 0 with a remainder of 2.

Reading the remainders in reverse order, we get the octal equivalent: 206.

Hexadecimal (Base 16):

Divide the decimal number by 16.

Record the remainders. For remainders 10 through 15, use the letters A through F.

Read the remainders in reverse order to get the hexadecimal equivalent.

For example, let’s convert the decimal number 255 to hexadecimal:

255 divided by 16 equals 15 with a remainder of 15 (which is represented as F in hexadecimal).

15 divided by 16 equals 0 with a remainder of 15 (also represented as F).

Reading the remainders in reverse order, we get the hexadecimal equivalent: FF.

These are the basic steps for converting a decimal number to binary, octal, and hexadecimal. You can use these methods to convert decimal numbers to other number systems with different bases as well.

Rules on Decimal Number System

The decimal number system, also known as the base-10 number system, is the most commonly used numeral system worldwide. It uses ten distinct symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to represent numbers. Here are some fundamental rules and concepts related to the decimal number system:

1. Place Value: The value of a digit in a decimal number depends on its position within the number. Each position has a place value that is a power of 10. Starting from the right, the place values increase by a factor of 10 for each position to the left. For example:

• The rightmost digit has a place value of 1.
• The next digit to the left has a place value of 10.
• The next digit has a place value of 100.
• And so on.

2. Digits: Decimal numbers consist of digits 0 through 9. Any other symbol is not valid in the decimal number system.

3. Leading Zeros: Leading zeros (zeros to the left of the first non-zero digit) do not change the value of a decimal number. For example, 007 is the same as 7.

4. Decimal Point: The decimal point separates the whole part (left of the decimal point) from the fractional part (right of the decimal point) of a decimal number. For example, in the number 123.45, 123 is the whole part, and 45 is the fractional part.

5. Negative Numbers: Negative numbers in the decimal system are indicated by placing a minus sign (-) in front of the number. For example, -5 represents the negative number five.

6. Addition and Subtraction: When adding or subtracting decimal numbers, align the decimal points, and then perform the operation as you would with whole numbers. Ensure that the result is properly aligned with the decimal point.

7. Multiplication and Division: When multiplying or dividing decimal numbers, ignore the decimal points initially and perform the operation as if they were whole numbers. After the operation, count the total decimal places in both the original numbers and place the decimal point in the result accordingly. The number of decimal places in the result is equal to the sum of the decimal places in the original numbers.

8. Rounding: Decimal numbers can be rounded to a specified number of decimal places. Common rounding methods include rounding up, rounding down, and rounding to the nearest value. The specific rounding method used depends on the context and the desired precision.

9. Scientific Notation: In scientific notation, decimal numbers are expressed as a product of a number between 1 and 10 and a power of 10. For example, 6.02 x 10^23 represents Avogadro’s number.

10. Place Value System: The decimal number system is a place value system, which means that the position of each digit determines its value. Moving a digit one place to the left multiplies its value by 10, while moving it one place to the right divides its value by 10.

Understanding these rules and concepts is essential for working with decimal numbers effectively in mathematics and everyday life. Decimal numbers are integral to various fields, including science, finance, and engineering.

Some Solved Problems on Decimal Number System

Problem 1: Convert the decimal number 123.456 to binary.

Solution:

Divide the decimal number by 2, and write down the remainders.

123.456 / 2 = 61 with a remainder of 1

Continue dividing the quotient by 2 and writing down the remainders until the quotient is 0.

61 / 2 = 30 with a remainder of 1

30 / 2 = 15 with a remainder of 0

15 / 2 = 7 with a remainder of 1

7 / 2 = 3 with a remainder of 1

3 / 2 = 1 with a remainder of 1

1 / 2 = 0 with a remainder of 1

The binary representation of the decimal number is the sequence of remainders, starting from the bottom (MSB) to the top (LSB).

Binary representation: 1110111.1

Problem 2: Find the decimal equivalent of the binary number 1101001.

Solution:

Write down the binary number, starting from the MSB.

Binary number: 1101001

Multiply each digit by the corresponding power of 2, and add the products together.

Decimal equivalent: (1 * 2^5) + (1 * 2^4) + (0 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 97

Problem 3: Add the following two decimal numbers:

123.456 + 78.903

Solution:

Line up the two numbers, with the decimal points aligned.

123.456 + 78.903

Add each column, starting from the rightmost column. If the sum of two digits is greater than 9, carry over the 1 to the next column.

123.456 + 78.903

———-

202.359

Problem 4: Multiply the following two decimal numbers:

123.456 * 78.903

Solution:

Multiply each digit of the first number by each digit of the second number, and add the products together.

123.456 * 78.903

———-

1091112.5936

These are just a few examples of solved problems on the decimal number system. There are many other problems that can be solved using the same principles.

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