Chapter 8 Decimals (Concepts)

Decimal Number & Place Value

We have already learned about fractions, which are used to represent parts of a whole by dividing it into a certain number of equal parts. Decimals are another way of representing parts of a whole, and they are particularly useful when the whole is divided into $10, 100, 1000,$ or other powers of $10$ equal parts. Decimal numbers are an extension of our place value system to include values less than one.

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Decimal Numbers

A decimal number is a number that has two parts: a whole number part and a fractional part. These two parts are separated by a dot, which is called the decimal point.

Example: Consider the number $34.56$.

• The part to the left of the decimal point (34) is the whole number part.
• The part to the right of the decimal point (56) is the fractional part (also called the decimal part).
• The dot '.' between 34 and 56 is the decimal point.

How do we read decimal numbers? We read the whole number part as we normally would. When we reach the decimal point, we say "point". Then, we read each digit in the fractional part separately.

Example: $34.56$ is read as "thirty-four point five six". It is incorrect to read the decimal part as a whole number, like "thirty-four point fifty-six".

Example: $2.7$ is read as "two point seven".

Example: $105.08$ is read as "one hundred five point zero eight".

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Place Value in Decimal Numbers

The place value system we use for whole numbers (Ones, Tens, Hundreds, Thousands, etc.) extends to the right of the decimal point to represent the fractional part. As we move to the right of the decimal point, the place value of each digit becomes one-tenth of the value of the place to its left.

• The first place to the right of the decimal point is the Tenths place ($\frac{1}{10}$ or $0.1$).
• The second place to the right is the Hundredths place ($\frac{1}{100}$ or $0.01$).
• The third place to the right is the Thousandths place ($\frac{1}{1000}$ or $0.001$).
• And so on (Ten-thousandths, Hundred-thousandths, etc.).

Let's look at the place values for the number $123.456$:

Hundreds (100)	Tens (10)	Ones (1)	.	Tenths ($\frac{1}{10}$)	Hundredths ($\frac{1}{100}$)	Thousandths ($\frac{1}{1000}$)
1	2	3	.	4	5	6
Value: $1 \times 100 = 100$	Value: $2 \times 10 = 20$	Value: $3 \times 1 = 3$	.	Value: $4 \times \frac{1}{10} = \frac{4}{10}$	Value: $5 \times \frac{1}{100} = \frac{5}{100}$	Value: $6 \times \frac{1}{1000} = \frac{6}{1000}$

From the table, we can see:

• The digit $1$ is in the Hundreds place, so its value is $1 \times 100 = 100$.
• The digit $2$ is in the Tens place, so its value is $2 \times 10 = 20$.
• The digit $3$ is in the Ones place, so its value is $3 \times 1 = 3$.
• The digit $4$ is in the Tenths place, so its value is $4 \times \frac{1}{10} = \frac{4}{10}$ (which is $0.4$).
• The digit $5$ is in the Hundredths place, so its value is $5 \times \frac{1}{100} = \frac{5}{100}$ (which is $0.05$).
• The digit $6$ is in the Thousandths place, so its value is $6 \times \frac{1}{1000} = \frac{6}{1000}$ (which is $0.006$).

Expanded Form of a Decimal Number

A decimal number can be written in expanded form by writing the sum of the place values of all its digits.

For the number $123.456$, the expanded form is:

Whole number part: $1 \times 100 + 2 \times 10 + 3 \times 1$

Fractional part: $4 \times \frac{1}{10} + 5 \times \frac{1}{100} + 6 \times \frac{1}{1000}$

So, the expanded form of $123.456$ is:

Or, using decimal values for the fractional part:

Adding these values gives back the original number: $100 + 20 + 3 + 0.4 + 0.05 + 0.006 = 123.456$.

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Representation of Decimal Numbers on a Number Line

We have seen how whole numbers, integers, and fractions can be placed on a number line. Decimal numbers, which are essentially another way of writing fractions with denominators that are powers of 10, can also be represented on a number line. The number line provides a visual understanding of the value of a decimal number and its position relative to other numbers.

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Representing Decimals between Whole Numbers (Tenths)

A decimal number like $0.7$ has one digit after the decimal point, which is in the tenths place. This means the whole unit (the segment between two consecutive whole numbers) is divided into 10 equal parts. Each part represents one-tenth ($0.1$).

To represent a decimal like $0.7$ on a number line:

1. Draw a number line. Identify the two consecutive whole numbers between which the decimal lies. Since $0.7$ is greater than $0$ and less than $1$, it lies between $0$ and $1$. Mark $0$ and $1$ on the number line.
2. Divide the segment between these two whole numbers ($0$ and $1$) into $10$ equal parts. These divisions represent $0.1, 0.2, 0.3, ..., 0.9$.
3. Starting from the left whole number ($0$), count the number of divisions indicated by the digit in the tenths place. In $0.7$, the digit in the tenths place is $7$. So, count $7$ divisions to the right from $0$.
4. The point at the end of the $7$th division represents the decimal number $0.7$.

Consider representing a decimal number like $1.2$ on a number line.

1. The decimal $1.2$ has a whole number part $1$ and a decimal part $0.2$. It lies between the whole numbers $1$ and $2$. Mark $1$ and $2$ on the number line.
2. Divide the segment between $1$ and $2$ into $10$ equal parts (tenths). These divisions represent $1.1, 1.2, 1.3, ..., 1.9$.
3. Starting from the left whole number ($1$), count $2$ divisions to the right (as the tenths digit is 2).
4. The point at the end of the $2$nd division from $1$ represents the decimal number $1.2$.

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Representing Decimals with More Decimal Places (Hundredths, Thousandths, etc.)

To represent decimals with two or more decimal places, we use the concept of subdividing the smaller intervals. For example, a decimal with two decimal places (like $2.45$) involves tenths and hundredths. It lies between two tenths values (like $2.4$ and $2.5$).

To represent a decimal like $2.45$ on a number line:

1. First, locate the interval of tenths it falls into. $2.45$ is between $2.4$ and $2.5$.
2. Draw a number line and mark the whole number $2$. Divide the segment between $2$ and $3$ into $10$ equal parts to locate the tenths $2.1, 2.2, ..., 2.9$. Mark $2.4$ and $2.5$ on this line.
3. Now, consider the segment between $2.4$ and $2.5$. This smaller segment represents one-tenth ($0.1$). We need to divide this segment into $10$ equal parts to represent hundredths (since $0.1$ divided into $10$ parts is $0.01$). Each part represents $0.01$.
4. Starting from the left value of this interval ($2.4$), count the number of divisions indicated by the digit in the hundredths place. In $2.45$, the digit in the hundredths place is $5$. So, count $5$ divisions to the right from $2.4$.
5. The point at the end of the $5$th division from $2.4$ represents the decimal number $2.45$.

This process of zooming in and subdividing intervals allows us to pinpoint the position of any decimal number on the number line, no matter how many decimal places it has. We divide the relevant tenths interval into 10 parts for hundredths, a hundredths interval into 10 parts for thousandths, and so on.

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Fraction & Decimal Numbers

In the previous section, we learned about fractions, which are used to represent parts of a whole. We also introduced decimal numbers as another way to represent parts of a whole, particularly those parts that are based on divisions into powers of 10 (tenths, hundredths, thousandths, etc.). Fractions and decimal numbers are simply two different forms of writing the same value, and we can convert between these forms.

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Converting Fractions to Decimals

To convert a fraction into a decimal number, the goal is to express the fraction in a way that its denominator is a power of 10 (like 10, 100, 1000, etc.). Once the denominator is a power of 10, converting it to a decimal is straightforward.

Case 1: The denominator is already a power of 10.

If the denominator of a fraction is 10, 100, 1000, or any other power of 10, the conversion is simple. The number of zeros in the denominator tells us how many places to the left we need to move the decimal point from the right end of the numerator.

• If the denominator is 10 (one zero), move the decimal point 1 place to the left.
• If the denominator is 100 (two zeros), move the decimal point 2 places to the left.
• If the denominator is 1000 (three zeros), move the decimal point 3 places to the left, and so on.

Example: Convert $\frac{3}{10}$ to a decimal.

The denominator is 10, which has one zero. The numerator is 3. Imagine the decimal point is at the end of 3 ($3.$). Move the decimal point 1 place to the left.

So, $\frac{3}{10} = 0.3$.

Example: Convert $\frac{45}{100}$ to a decimal.

The denominator is 100, which has two zeros. The numerator is 45. Imagine the decimal point at the end of 45 ($45.$). Move the decimal point 2 places to the left.

So, $\frac{45}{100} = 0.45$.

Example: Convert $\frac{123}{1000}$ to a decimal.

The denominator is 1000 (three zeros). The numerator is 123. Move the decimal point 3 places to the left from $123.$.

So, $\frac{123}{1000} = 0.123$.

Example: Convert $\frac{7}{100}$ to a decimal.

The denominator is 100 (two zeros). The numerator is 7. We need to move the decimal point 2 places to the left. Since there is only one digit in the numerator, we add a zero to the left to create enough places: $07$. Move the decimal point 2 places left from $07.$.

So, $\frac{7}{100} = 0.07$.

Case 2: The denominator is not a power of 10, but can be easily made a power of 10.

Sometimes the denominator is a factor of a power of 10 (like 2 is a factor of 10, 4 is a factor of 100, 8 is a factor of 1000, 20 is a factor of 100, 25 is a factor of 100, 50 is a factor of 100). In such cases, we can convert the fraction into an equivalent fraction with a denominator that is a power of 10. Then, follow the steps from Case 1.

To do this, multiply both the numerator and the denominator by the same number that makes the denominator a power of 10.

Example: Convert $\frac{1}{2}$ to a decimal.

The denominator is 2. We can make it 10 by multiplying by 5. Multiply both numerator and denominator by 5.

Now, convert $\frac{5}{10}$ to a decimal (using Case 1 rule - one zero, move decimal 1 place left).

So, $\frac{1}{2} = 0.5$.

Example: Convert $\frac{3}{4}$ to a decimal.

The denominator is 4. We can make it 100 by multiplying by 25. Multiply both numerator and denominator by 25.

Now, convert $\frac{75}{100}$ to a decimal (using Case 1 rule - two zeros, move decimal 2 places left).

So, $\frac{3}{4} = 0.75$.

Example: Convert $\frac{1}{8}$ to a decimal.

The denominator is 8. We can make it 1000 by multiplying by 125. Multiply both numerator and denominator by 125.

Now, convert $\frac{125}{1000}$ to a decimal (three zeros, move decimal 3 places left).

So, $\frac{1}{8} = 0.125$.

Case 3: The denominator cannot easily be made a power of 10 by multiplication.

If the denominator is not a factor of a power of 10 (e.g., 3, 6, 7, 9, 11, etc.), or if you cannot quickly figure out the multiplier, you can always convert a fraction to a decimal by performing division.

Rule: Divide the numerator by the denominator using long division. Add a decimal point and zeros to the numerator as needed.

Example: Convert $\frac{1}{3}$ to a decimal.

Divide 1 by 3. Since 1 is smaller than 3, we start with a quotient of 0, place a decimal point, and add zeros to 1.

$\begin{array}{r} 0.333... \\ 3{\overline{\smash{\big)}\,1.000\phantom{)}}} \\ \underline{-~\phantom{(}0}\downarrow\phantom{00)} \\ 10\phantom{00)} \\ \underline{-~\phantom{()}9}\downarrow\phantom{0)} \\ 10\phantom{0)} \\ \underline{-~\phantom{()}9}\downarrow \\ 10 \\ \underline{-~\phantom{()}9} \\ 1 \end{array}$

The remainder is always 1, and the digit 3 in the quotient keeps repeating. This is called a repeating decimal.

So, $\frac{1}{3} = 0.333...$. We write this as $0.\overline{3}$, where the bar above the digit indicates that it repeats infinitely.

Example: Convert $\frac{3}{8}$ to a decimal using long division.

Divide 3 by 8. Add a decimal point and zeros to 3.

The remainder is 0. This is called a terminating decimal.

So, $\frac{3}{8} = 0.375$. (Notice we also found this using Case 2 by converting $\frac{3}{8}$ to $\frac{375}{1000}$).

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Converting Decimals to Fractions

Converting a decimal number back into a fraction is also a systematic process. We use the place value of the last digit in the decimal part to determine the denominator.

Steps:

1. Write the decimal number without the decimal point. This will be the numerator of the fraction.
2. Determine the denominator based on the number of digits after the decimal point:
   • If there is 1 digit after the decimal point (tenths), the denominator is 10.
   • If there are 2 digits after the decimal point (hundredths), the denominator is 100.
   • If there are 3 digits after the decimal point (thousandths), the denominator is 1000, and so on.
   • In general, the denominator is $1$ followed by as many zeros as there are digits after the decimal point.
3. Form the fraction $\frac{\text{Numerator from step 1}}{\text{Denominator from step 2}}$.
4. Simplify the resulting fraction to its lowest terms by dividing the numerator and denominator by their HCF.

Example: Convert 0.6 to a fraction.

Step 1: Write the number without the decimal point. The number is $6$. This is the numerator.

Step 2: Count the digits after the decimal point in $0.6$. There is 1 digit (the digit 6) after the decimal point. So, the denominator is 1 followed by 1 zero, which is 10.

Step 3: Form the fraction: $\frac{6}{10}$.

Step 4: Simplify the fraction $\frac{6}{10}$ to its lowest terms. The common factors of 6 and 10 are 1 and 2. The HCF is 2.

So, 0.6 converted to a fraction in lowest terms is $\frac{3}{5}$.

Example: Convert 2.75 to a fraction.

Step 1: Write the number without the decimal point. The number is $275$. This is the numerator.

Step 2: Count the digits after the decimal point in $2.75$. There are 2 digits (7 and 5) after the decimal point. So, the denominator is 1 followed by 2 zeros, which is 100.

Step 3: Form the fraction: $\frac{275}{100}$.

Step 4: Simplify the fraction $\frac{275}{100}$ to its lowest terms. Both 275 and 100 are divisible by 25 (HCF of 275 and 100 is 25).

The result is the improper fraction $\frac{11}{4}$. This can also be written as a mixed number $2\frac{3}{4}$.

So, 2.75 as a fraction is $\frac{11}{4}$ or $2\frac{3}{4}$.

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Like and Unlike Decimal Numbers

When working with decimal numbers, especially when comparing them or performing addition and subtraction, it's helpful to categorise them based on the number of digits they have after the decimal point. This classification leads to the concepts of like and unlike decimal numbers.

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Like Decimal Numbers

Like decimal numbers are decimal numbers that have the exact same number of digits in their decimal part (the digits to the right of the decimal point).

Think of the number of decimal places as the 'length' of the decimal part. Like decimals have the same 'length' after the decimal point.

Example: $0.5$ and $2.3$ are like decimal numbers. Both have exactly one digit after the decimal point (tenths place).

Example: $1.23$ and $45.67$ are like decimal numbers. Both have exactly two digits after the decimal point (tenths and hundredths places).

Example: $12.345$, $6.789$, and $100.015$ are like decimal numbers. All three have exactly three digits after the decimal point (tenths, hundredths, and thousandths places).

Even whole numbers can be considered as like decimals if we add a decimal point and the same number of zeros. For example, $5.00$ and $12.00$ are like decimals, both having two decimal places.

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Unlike Decimal Numbers

Unlike decimal numbers are decimal numbers that have a different number of digits in their decimal part (the digits to the right of the decimal point).

Unlike decimals have different 'lengths' after the decimal point.

Example: $0.5$ (1 decimal place) and $1.23$ (2 decimal places) are unlike decimal numbers.

Example: $12.3$ (1 decimal place) and $4.567$ (3 decimal places) are unlike decimal numbers.

Example: $8.9$ (1 decimal place), $10.11$ (2 decimal places), and $0.555$ (3 decimal places) are all unlike decimal numbers because the number of decimal places is different for each.

A whole number and a decimal number with a non-zero decimal part are also unlike decimals unless we add zeros to the whole number. For example, $7$ and $7.5$ are unlike decimals (0 decimal places vs 1 decimal place).

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Converting Unlike Decimals to Like Decimals

Converting unlike decimal numbers into like decimal numbers is a very useful step, especially when you need to compare them or add/subtract them. The conversion process involves adding one or more zeros to the right end of the decimal part of the number(s) with fewer decimal places until all numbers have the same number of decimal places as the number with the maximum decimal places.

Adding zeros to the extreme right of the decimal part of a number does not change its value. This is because, for example, $0.5$ means $5$ tenths, which is equivalent to $50$ hundredths ($0.50$), which is equivalent to $500$ thousandths ($0.500$), and so on. $\frac{5}{10} = \frac{50}{100} = \frac{500}{1000}$.

Rule: To convert unlike decimals to like decimals, find the maximum number of decimal places among all the numbers. Then, add trailing zeros to the decimal part of each number that has fewer decimal places until all numbers have this maximum number of decimal places.

Example: Convert $0.5, 1.23,$ and $4.567$ into like decimals.

The given decimal numbers are:

• $0.5$ (1 decimal place)
• $1.23$ (2 decimal places)
• $4.567$ (3 decimal places)

The maximum number of decimal places among these numbers is $3$ (from $4.567$). So, we will convert all numbers to have $3$ decimal places.

• For $0.5$: It has 1 decimal place. We need $3 - 1 = 2$ more decimal places. Add two zeros to the right: $0.500$.
• For $1.23$: It has 2 decimal places. We need $3 - 2 = 1$ more decimal place. Add one zero to the right: $1.230$.
• For $4.567$: It already has 3 decimal places. No zeros are needed: $4.567$.

The unlike decimal numbers $0.5, 1.23,$ and $4.567$ have been converted into the like decimal numbers $0.500, 1.230,$ and $4.567$. These new numbers have the same value as the original ones but make comparison and alignment for addition/subtraction much easier.

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Comparison of Decimal Numbers

In earlier sections, we learned about decimal numbers and their place values. Just as we compare whole numbers and fractions to see which is greater or smaller, we also need to compare decimal numbers. For example, if two runners finish a race in 12.5 seconds and 12.3 seconds, we need to compare these decimals to know who finished faster. Comparing decimal numbers helps us to order them and understand their relative values.

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Steps for Comparing Decimal Numbers

To compare two decimal numbers, we compare the digits in each place value position, starting from the leftmost digit (the highest place value) and moving towards the right (towards the smaller place values) until we find a difference.

1. Compare the Whole Number Parts: Start by comparing the digits in the whole number part of the decimals (to the left of the decimal point). Compare the digits in the greatest place value position first (e.g., Hundreds, Tens, Ones). The decimal number with the larger whole number part is the greater number.

   Example: Compare $15.7$ and $12.9$.

   The whole number part of $15.7$ is $15$.

   The whole number part of $12.9$ is $12$.

   Comparing the whole numbers, we have $15$ and $12$. Since $15 > 12$, the decimal number $15.7$ is greater than $12.9$.

   $15.7 > 12.9$

2. If the Whole Number Parts are Equal: If the whole number parts of the two decimals are the same, move to the right of the decimal point and compare the digits in the tenths place (the first digit after the decimal point). The decimal number with the larger digit in the tenths place is greater.

   Example: Compare $5.3$ and $5.8$.

   The whole number parts are equal (both are $5$).

   Compare the digits in the tenths place: $3$ in $5.3$ and $8$ in $5.8$.

   Since $8 > 3$, the decimal number $5.8$ is greater than $5.3$.

   $5.8 > 5.3$

3. If the Whole Number Parts and Tenths Digits are Equal: If the whole number parts and the digits in the tenths place are the same, move further right and compare the digits in the hundredths place (the second digit after the decimal point). The decimal number with the larger digit in the hundredths place is greater.

   Example: Compare $7.45$ and $7.41$.

   Whole number parts are equal (both are $7$).

   Digits in the tenths place are equal (both are $4$).

   Compare the digits in the hundredths place: $5$ in $7.45$ and $1$ in $7.41$.

   Since $5 > 1$, the decimal number $7.45$ is greater than $7.41$.

   $7.45 > 7.41$

4. Continue the Process: If the digits in the hundredths place are also equal, continue comparing the digits in the next place value position to the right (thousandths, ten-thousandths, and so on) until you find the first place where the digits are different. The number with the larger digit in that position is the greater number.

   Example: Compare $10.123$ and $10.125$.

   • Whole number parts are equal (10 vs 10).
   • Tenths digits are equal (1 vs 1).
   • Hundredths digits are equal (2 vs 2).
   • Compare Thousandths digits: $3$ in $10.123$ and $5$ in $10.125$. Since $5 > 3$, $10.125 > 10.123$.

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Using Like Decimals for Comparison

Comparing unlike decimal numbers (those with a different number of decimal places) can sometimes be confusing. Converting them into like decimal numbers by adding trailing zeros simplifies the comparison process, as it aligns the decimal places clearly.

Remember that adding zeros to the extreme right of the decimal part does not change the value of the decimal number (e.g., $0.5 = 0.50 = 0.500$).

To use this method:

1. Determine the maximum number of decimal places among the numbers you want to compare.
2. Convert all the decimal numbers into like decimals by adding trailing zeros so that they all have this maximum number of decimal places.
3. Now, compare these like decimal numbers using the left-to-right digit comparison method described above.

Example: Compare $1.6$ and $1.58$.

$1.6$ has 1 decimal place. $1.58$ has 2 decimal places.

The maximum number of decimal places is 2. Convert $1.6$ to a like decimal with 2 places by adding one zero: $1.60$.

Now compare $1.60$ and $1.58$.

• Whole parts: $1$ and $1$ (Equal).
• Tenths digits: $6$ in $1.60$ and $5$ in $1.58$. Since $6 > 5$, $1.60 > 1.58$.

Therefore, $1.6 > 1.58$.

Example: Compare $0.9$ and $0.900$.

$0.9$ has 1 decimal place. $0.900$ has 3 decimal places.

The maximum is 3. Convert $0.9$ to a like decimal: $0.900$.

Now compare $0.900$ and $0.900$.

Comparing digits from left to right: Whole parts (0 vs 0), Tenths (9 vs 9), Hundredths (0 vs 0), Thousandths (0 vs 0). All digits are equal.

So, $0.9 = 0.900$. This confirms that adding trailing zeros does not change the value of a decimal.

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Algebraic Operations on Decimal Numbers

We have learned what decimal numbers are, their place values, how to represent them on a number line, compare them, and convert them from and to fractions. Now, let's see how to perform the basic arithmetic operations – addition, subtraction, multiplication, and division – using decimal numbers. Working with decimals is very similar to working with whole numbers, with specific rules for handling the decimal point.

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Addition of Decimal Numbers

Adding decimal numbers is straightforward, provided you align the decimal points correctly. Think of it like adding whole numbers, but with a column reserved for the decimal point.

Steps for Adding Decimal Numbers:

1. Write the decimal numbers one below the other such that their decimal points are vertically aligned in a straight column. This ensures that digits corresponding to the same place value are in the same column (ones under ones, tenths under tenths, hundredths under hundredths, etc.).
2. It is often helpful to convert unlike decimal numbers to like decimal numbers before adding. Do this by adding zeros to the right of the last digit in the decimal part of the numbers with fewer decimal places until all numbers have the same number of decimal places. This step helps maintain correct column alignment and prevents errors, although mathematically, it doesn't change the value of the numbers.
3. Start adding the digits from the rightmost column (the smallest place value), just like you would add whole numbers. Carry over any tens to the next column on the left.
4. Place the decimal point in the sum (the result) directly in the same vertical line as the decimal points in the numbers being added.

Example: Add $12.34 + 5.6 + 0.789$.

The given numbers are $12.34$ (2 decimal places), $5.6$ (1 decimal place), and $0.789$ (3 decimal places).

The maximum number of decimal places is 3. Let's write them as like decimals by adding trailing zeros:

• $12.34$ becomes $12.340$
• $5.6$ becomes $5.600$
• $0.789$ remains $0.789$

Now, arrange the numbers vertically, aligning the decimal points, and perform the addition:

Explanation of addition:

• Thousandths column (rightmost): $0 + 0 + 9 = 9$. Write down 9.
• Hundredths column: $4 + 0 + 8 = 12$. Write down 2, carry over 1 to the tenths column.
• Tenths column: $3 + 6 + 7 + 1 (\text{carry}) = 17$. Write down 7, carry over 1 to the ones column.
• Place the decimal point directly below the others.
• Ones column: $2 + 5 + 0 + 1 (\text{carry}) = 8$. Write down 8.
• Tens column: $1 + 0 + 0 = 1$. Write down 1.

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Subtraction of Decimal Numbers

Subtracting decimal numbers is done using the same process as subtracting whole numbers, with the critical step being the alignment of decimal points.

Steps for Subtracting Decimal Numbers:

1. Write the number being subtracted (subtrahend) vertically below the number it is subtracted from (minuend). Align the decimal points in a straight vertical column.
2. Add zeros to the right end of the decimal part of the number(s) with fewer decimal places (usually the minuend if it has fewer decimal places than the subtrahend) to make them like decimals. This is crucial in subtraction to ensure that there is a digit in every column of the minuend corresponding to a digit in the subtrahend, allowing for borrowing when necessary.
3. Subtract the digits column by column, starting from the rightmost digit, just as you do with whole numbers. Borrow from the column to the left when the digit in the minuend is smaller than the digit in the subtrahend.
4. Place the decimal point in the final difference (the result) directly in the same vertical line as the decimal points in the numbers.

Example: Subtract $8.5 - 3.25$.

The numbers are $8.5$ (1 decimal place) and $3.25$ (2 decimal places). The maximum number of decimal places is 2. Convert $8.5$ to a like decimal: $8.50$.

Write the numbers vertically, aligning decimal points, and subtract:

Explanation of subtraction:

• Hundredths column (rightmost): We need to subtract 5 from 0. Borrow from the tenths place. The 5 in the tenths place becomes 4, and the 0 in the hundredths place becomes 10. $10 - 5 = 5$. Write down 5.
• Tenths column: We now have 4 in the tenths place. Subtract 2 from 4. $4 - 2 = 2$. Write down 2.
• Place the decimal point directly below the others.
• Ones column: Subtract 3 from 8. $8 - 3 = 5$. Write down 5.

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Multiplication of Decimal Numbers

Multiplying decimal numbers is a two-step process: first multiply ignoring the decimal points, and then correctly place the decimal point in the final product.

Steps for Multiplying Decimal Numbers:

1. Perform the multiplication as if the decimal numbers were whole numbers. Ignore the decimal points during this multiplication step.
2. Count the total number of digits in the decimal parts of both numbers that were multiplied. This is the sum of the number of decimal places in the first number and the number of decimal places in the second number.
3. In the product obtained in Step 1 (the whole number result), place the decimal point starting from the rightmost digit. Count leftwards by the total number of decimal places calculated in Step 2. If the product has fewer digits than the required number of decimal places, add zeros to the left of the product before placing the decimal point.

Example: Multiply $2.3 \times 1.5$.

1. Multiply the numbers as whole numbers: $23 \times 15$.
   $ \begin{array}{cc} & 23 \\ \times & 15 \\ \hline & 115 \\ + & 23\phantom{0} \\ \hline & 345 \\ \hline \end{array} $

   The product as whole numbers is $345$.

2. Count the total number of decimal places:
   • $2.3$ has 1 decimal place.
   • $1.5$ has 1 decimal place.
   • Total number of decimal places in the product should be $1 + 1 = 2$.
3. Place the decimal point in $345$. Starting from the right, move 2 places to the left. $345. \to 34.5 \to 3.45$.

Example: Multiply $0.12 \times 0.4$.

1. Multiply as whole numbers: $12 \times 4$.
   $ \begin{array}{cc} & 12 \\ \times & 4 \\ \hline & 48 \\ \hline \end{array} $

   The product as whole numbers is $48$.

2. Count decimal places:
   • $0.12$ has 2 decimal places.
   • $0.4$ has 1 decimal place.
   • Total decimal places = $2 + 1 = 3$.
3. Place the decimal point in the product $48$. We need 3 decimal places. Starting from the right, count 3 places left. $48. \to 4.8 \to .48$. We need a zero before the decimal point and also between the decimal and the first digit if needed to make 3 places. $048. \to 04.8 \to 0.48$. Need one more zero. $\to 0.048$.

Multiplying a Decimal by 10, 100, 1000, etc.:

Multiplying a decimal number by a power of 10 is a shortcut that involves shifting the decimal point.

Rule: To multiply a decimal number by 10, 100, 1000, or any higher power of 10, shift the decimal point to the right. The number of places to shift is equal to the number of zeros in the power of 10.

Add trailing zeros to the decimal number if necessary to allow for the required shift.

Example: $3.45 \times 10$. 10 has one zero. Shift decimal 1 place right. Result: $34.5$.

Example: $1.234 \times 100$. 100 has two zeros. Shift decimal 2 places right. Result: $123.4$.

Example: $0.5 \times 1000$. 1000 has three zeros. Shift decimal 3 places right. $0.5$ can be thought of as $0.500$. Shift decimal: $0.500 \to 5.00 \to 50.0 \to 500.0$. Result: $500$.

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Division of Decimal Numbers

Division of decimal numbers is performed using long division, but there are specific steps depending on whether the divisor is a whole number or a decimal.

Case 1: Dividing a Decimal by a Whole Number:

Steps:

1. Write the long division problem with the decimal number as the dividend and the whole number as the divisor.
2. Divide the whole number part of the dividend by the divisor.
3. Place the decimal point in the quotient directly above the decimal point in the dividend as soon as you start dividing the digits after the decimal point in the dividend.
4. Continue the division process with the digits in the decimal part of the dividend. If there is a remainder after using all the digits of the dividend, add trailing zeros to the decimal part of the dividend and continue dividing until the remainder is 0 or you have reached the desired number of decimal places for the quotient.

Example: Divide $6.4 \div 2$.

Divide 6 by 2 (result is 3, write above 6). Place the decimal point in the quotient. Bring down 4. Divide 4 by 2 (result is 2, write above 4).

Example: Divide $1.25 \div 5$.

Divide 1 by 5 (result 0, write above 1). Place decimal point. Consider 12. Divide 12 by 5 (result 2, remainder 2). Write 2 above 2. Bring down 5. Consider 25. Divide 25 by 5 (result 5, remainder 0). Write 5 above 5.

Case 2: Dividing a Decimal by a Decimal:

When the divisor is a decimal number, the first step is to convert the division problem into an equivalent problem where the divisor is a whole number. This is done by shifting the decimal points in both the divisor and the dividend.

Steps:

1. Look at the divisor (the number you are dividing by). Count how many places you need to shift the decimal point to the right to make it a whole number.
2. Shift the decimal point in the dividend (the number being divided) to the right by the exact same number of places you shifted the decimal point in the divisor. If the dividend does not have enough decimal places, add trailing zeros to the right of its decimal part to make the shift possible.
3. Now you have a new division problem where you are dividing a decimal (the new dividend) by a whole number (the new divisor).
4. Perform the long division using the steps described for Case 1 (dividing a decimal by a whole number).
5. Place the decimal point in the quotient directly above the *new* position of the decimal point in the dividend.

Example: Divide $4.5 \div 0.5$.

1. Divisor is $0.5$. To make it a whole number, shift the decimal point 1 place to the right. $0.5 \to 5$.
2. Dividend is $4.5$. Shift the decimal point 1 place to the right (the same number of places as the divisor). $4.5 \to 45$.
3. The new division problem is $45 \div 5$.
4. Perform the long division:
   $ \begin{array}{r} 9. \\ 5{\overline{\smash{\big)}\,45.}} \\ \underline{-~\phantom{()}45} \\ 0 \end{array} $
5. Place the decimal point in the quotient above the decimal point in the new dividend (45.).

Example: Divide $1.44 \div 1.2$.

1. Divisor is $1.2$. Shift decimal point 1 place right $\to 12$.
2. Dividend is $1.44$. Shift decimal point 1 place right $\to 14.4$.
3. The new division problem is $14.4 \div 12$.
4. Perform long division (dividing a decimal by a whole number):
   $ \begin{array}{r} 1.2 \\ 12{\overline{\smash{\big)}\,14.4}} \\ \underline{-~\phantom{(}12}\downarrow \\ 24 \\ \underline{-~\phantom{()}24} \\ 0 \end{array} $
5. Place the decimal point in the quotient above the decimal point in the new dividend (14.4).

Dividing a Decimal by 10, 100, 1000, etc.:

Dividing a decimal number by a power of 10 is a shortcut that involves shifting the decimal point.

Rule: To divide a decimal number by 10, 100, 1000, or any higher power of 10, shift the decimal point to the left. The number of places to shift is equal to the number of zeros in the power of 10.

Add leading zeros to the decimal number if necessary to allow for the required shift (e.g., to divide 1.2 by 1000, you might need to add zeros like $001.2$ before shifting).

• Divide by 10 (1 zero): Shift decimal point 1 place to the left.
• Divide by 100 (2 zeros): Shift decimal point 2 places to the left.
• Divide by 1000 (3 zeros): Shift decimal point 3 places to the left, and so on.

Example: $234.5 \div 10$. 10 has one zero. Shift decimal 1 place left. Result: $23.45$.

Example: $56.7 \div 100$. 100 has two zeros. Shift decimal 2 places left. $56.7 \to 5.67 \to .567 \to 0.567$. Result: $0.567$.

Example: $1.2 \div 1000$. 1000 has three zeros. Shift decimal 3 places left. Think of $1.2$ as $001.2$. Shift 3 places left: $0.0012$. Result: $0.0012$.

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Uses of Decimal Numbers

Decimal numbers are extremely common in our daily lives and are used across various fields of study and professions. This is because they provide a convenient and standardised way to represent parts of whole numbers, especially when dealing with measurements and quantities that are based on units divided into tenths, hundredths, thousandths, and so on.

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Common Applications of Decimal Numbers

Here are some of the many ways decimal numbers are used in the real world:

• Money (Currency): Money is perhaps the most common example of decimals in everyday life. Currency units are often divided into subunits based on powers of 10. In India, $1$ Rupee is equal to $100$ paise. Decimals are used to show Rupees and paise together. For example:

  $\textsf{₹}1 = 100 \text{ paise}$

  $1 \text{ paise} = \frac{1}{100} \text{ Rupee} = \textsf{₹}0.01$

  So, $\textsf{₹}25.50$ is read as "twenty-five Rupees and fifty paise". The digits after the decimal point represent the paise part of the money. $\textsf{₹}50.75$ is "Fifty Rupees and seventy-five paise".

• Measurement: Decimal numbers are extensively used in various types of measurements:

  • Length: Measuring length in units like metres (m) and centimetres (cm) or kilometres (km) often involves decimals. For example, a height of $1.75$ metres, a width of $10.5$ centimetres, or a distance of $12.3$ kilometres. Since $1 \text{ cm} = 0.01 \text{ m}$ and $1 \text{ metre} = 0.001 \text{ km}$, conversions between metric units easily result in decimals.
  • Weight: Measuring weight in kilograms (kg) and grams (g) or grams and milligrams (mg) uses decimals. For example, $2.5$ kilograms of rice, or $500$ grams written as $0.5$ kg (since $1 \text{ kg} = 1000 \text{ g}$).
  • Volume: Measuring liquids in litres (L) and millilitres (mL) or just in litres using decimals. For example, a bottle containing $0.75$ litres of cold drink, or $250$ millilitres written as $0.25$ litres (since $1 \text{ litre} = 1000 \text{ mL}$).

• Temperature: Temperature readings from thermometers are frequently given as decimal numbers, especially in scientific contexts or weather reports (e.g., the temperature is $32.4^\circ\text{C}$ or $98.6^\circ\text{F}$).

• Speed and Distance: Measuring speed (e.g., a car travelling at $65.8 \text{ km/h}$) and recording distances covered (e.g., running $5.2$ kilometres) involve decimals.

• Sports: Many sports use decimals for precise measurements of performance, such as race timings (e.g., finishing a sprint in $11.35$ seconds) or distances in jumping events (e.g., a long jump of $7.82$ metres).

• Reading Meters: Utility meters like electricity meters, water meters, and gas meters, as well as fuel pumps at petrol stations, display readings that typically include decimal numbers to show partial units consumed or dispensed.

• Calculating Averages: When you calculate the average of a set of numbers, the result might not be a whole number and is often expressed as a decimal (e.g., the average height of students in a class is $1.45$ metres, the average score in a test was $78.5$).

The decimal system provides a consistent way to represent and perform calculations with these quantities, making it an indispensable tool in various practical situations.

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