Decimals

Decimals are an essential part of our everyday lives and play a crucial role in various aspects of mathematics and science. In this chapter, we will go through the world of decimals, exploring what they are, how they work and their practical uses.

What are Decimals?

Decimals are a numerical system used to represent parts of a whole or fractions in a more precise and convenient manner. They are often denoted by a dot or period (.) which separates the whole number from its fractional part.

An example of a decimal number is 38.4. The decimal number 38.4 is read as thirty-eight point four.

In this decimal number "243.978", "243" represents the whole number while "978" represents the fraction.

The decimal number 243.978 is read as two hundred forty-three point nine seven eight.

Decimal Fractions

Decimal fractions are fractions where the denominator is either 10 or a multiple of 10 such as 100, 1000, 10000 and so on. They represent parts of a whole in a decimal form.

Examples of Decimal Fractions:

Decimal fractions include numbers like 1/10, 2/10, 3/10, 4/10, 7/10, 9/10, 11/10, 13/10, 1/100, 2/100, 15/100, 1/1000, etc. These fractions are expressed in decimal notation, shown as:

Decimal Places

Decimal places are determined by the number of digits found in the fractional or decimal part of a number.

Examples of decimal places:

In the number 12.37, there are two decimal places.
In the number 4.245, there are three decimal places.

The count of digits to the right of the decimal point signifies the number of decimal places a number has.

Like and Unlike Decimals

Decimals with an equal number of digits after the decimal point are referred to as "like decimals" while decimals with varying numbers of digits after the decimal point are termed "unlike decimals".

1.25, 3.76 and 7.93 are like decimals because they have two decimal places after the decimal point.

3.7, 4.98 and 9.763 are unlike decimals because they do not have the same number of digits after the decimal point.

Note:

Adding any number of zeros to the end (the extreme right) of the decimal part of a decimal number does not alter its value.

Example:

Decimals 0.7 is represented as:

0.7 = 0.70 = 0.700 = 0.7000

All decimals represent the same value.

Converting Unlike Decimals into Like Decimals

To convert unlike decimals into like decimals, zeros are added to the right of the last digit. It is important to understand that these extra zeros are placeholders and do not change the actual value of the decimal part.

Example: 143.15, 25.019 and 3472.8 are unlike decimals.

143.150, 25.019 and 3472.800 are like decimals converted by adding extra zeroes to the right of the last digit. Each with three decimal places.

Comparing Decimals

To compare two decimals, we begin by examining the digits to the left of the decimal point which is the whole number part. The decimal number with the larger whole number part is considered greater. If the whole number is the same then compare the digits after the decimal point.

Example:

Some more examples are as follows:

Converting Decimals into Fractions

To convert a decimal into a fraction, follow these steps:

Step 1: Begin by writing the given decimal as the numerator of the fraction and set the denominator as 1.

Step 2: Remove the decimal point by multiplying both the numerator and denominator by 10, 100, 1000 and so on which depend on the number of digits after the decimal point. If there is one digit after the decimal, multiply by 10, if there are two digits, multiply by 100, if there are three digits, multiply by 1000 and so on.

Step 3: Simplify the fraction obtained and reduce it to its lowest form. If necessary, you can further convert it into a mixed fraction.

Example: Converting 10.125 into a Fraction.

Step 1: Begin by creating a fraction using the decimal number 10.125 as the numerator and 1 as the denominator:

Step 2: To eliminate the decimal places, multiply both the numerator and denominator by a suitable power of 10. Since there are three decimal places in the given number, multiply by 1000:

Step 3: Simplify the fraction by finding the Highest Common Factor (HCF) of the numerator and denominator and dividing both by the HCF as 25:

Step 4: If possible, simplify the remaining fraction into a mixed fraction:

Hence, the decimal number 10.125 can be expressed as a fraction or as a mixed fraction 10¹⁄₈.

Converting Fractions into Decimals

To convert a fraction into a decimal number, you can achieve this by dividing the numerator by the denominator.

Example: 1/8 = 0.125

The conversion using division is shown as:

In the given example, the numerator (1) is smaller than the denominator (8) which is indicated as 1 < 8. To facilitate division, we insert a decimal point in both the dividend and the quotient. Therefore, 1 can be expressed as 1.000 and we proceed with the division as usual.

It is important to note that decimals are essentially fractions with denominators such as 10, 100, 1000 and so on.

Another method for converting decimals into fractions:

Step 1: If the fraction is not equivalent then begin by converting the fraction into an equivalent fraction with a denominator of 10, 100 or 1000.

For example:

Step 2: Take the numerator of the fraction and place the decimal point after two places, moving from right to left, corresponding to the number of zeroes in the denominator.

Note: The number of digits in the decimal part matches the number of zeroes in the denominator.

Examples:

Addition of Decimals

To add two or more decimals together, follow these systematic steps:

Step 1: Begin by converting the given decimals into like decimals if they are not already.

Step 2: Arrange the digits of the given numbers vertically and ensure that all the decimal points are perfectly aligned in a single vertical line.

Step 3: Add the digits as you would when adding whole numbers. Start from the rightmost digit after the decimal point and work your way to the left.

Step 4: In the result, place the decimal point in the same position as it appears in the numbers above it. This ensures that the decimal places are correctly aligned in the final result.

Example: Add 97.03, 485.8 and 364.062.

Subtraction of Decimals

To subtract two or more decimals, follow these systematic steps:

Step 1: Ensure that the given decimals are in the same format (like decimals) for accurate subtraction.

Step 2: Organize the digits of the provided numbers vertically and align all the decimal points precisely in a single vertical line.

Step 3: Perform the subtraction as you would when subtracting whole numbers. Begin from the rightmost digit after the decimal point and progress toward the left.

Step 4: Place the decimal point in the answer in the same position where it appears in the numbers above it. This ensures the proper alignment of decimal places in the final result.

Example: Subtract 69.42 from 79

Example 1: Which is smallest among the decimals 43.03, 4.30, 4.030, 4.3, 4.03, 4.003, 4.05, 4.330?

a) 4.03
b) 4.030
c) 4.003
d) 4.0003

Answer: c) 4.003

Explanation: The increasing order of decimal numbers:

4.003 < 4.030 = 4.03 < 4.30 = 4.3 < 4.330 < 43.03

4.003 is the smallest of all.

Example 2: By how much does the sum of 93.274 and 27.345 exceed the sum of
98.42 and 17.561?

a) 4.238
b) 4.368
c) 4.538
d) 4.638

Answer: d) 4.638

Explanation: Sum of 93.274 and 27.345 = 93.274 + 27.345 = 120.619

Sum of 98.427 and 17.561 = 98.42 + 17.561 = 98.420 + 17.561 = 115.981

Difference (exceed by) = 120.619 − 115.981 = 4.638