When we look at a number like 457.368, each digit holds a specific value based on its position relative to the decimal point. Breaking this down is called the Expanded Form.
How to read it:
Instead of just saying “four five seven point three six eight,” the mathematically accurate way to read this is:
“Four hundred fifty-seven and three tenths, six hundredths, and eight thousandths.”
The value of a digit depends entirely on where it sits. In the example 457.368:
The 4 is in the hundreds place, so its value is 400.
The 3 is in the first decimal place (tenths), so its value is $\frac{3}{10}$.
The 8 is in the third decimal place (thousandths), so its value is $\frac{8}{1000}$.
There are two primary ways to convert a rational number (like a fraction) into a decimal. Let’s use the example $\frac{12}{5}$.
If you can multiply the denominator to reach 10, 100, or 1000, this method is much faster.
Multiply both the top and bottom by a number that turns the denominator into 10.
Since it is divided by 10, move the decimal one place to the left: 2.4
If the denominator doesn’t easily convert to 10 or 100, use long division:
Divide the numerator (12) by the denominator (5).
5 goes into 12 twice ($5 \times 2 = 10$), leaving a remainder of 2.
Add a decimal point and a zero to the remainder to make it 20.
5 goes into 20 exactly 4 times.
Result: 2.4
| Digit | Position | Value |
| 4 | Hundreds | 400 |
| 5 | Tens | 50 |
| 7 | Ones | 7 |
| 3 | Tenths | 0.3 or $\frac{3}{10}$ |
| 6 | Hundredths | 0.06 or $\frac{6}{100}$ |
| 8 | Thousandths | 0.008 or $\frac{8}{1000}$ |
When we divide the numerator by the denominator, we use long division. For the fraction $\frac{1}{3}$, we are dividing $1$ by $3$.
The Steps:
Step 1: Since $3$ doesn’t go into $1$, we add a decimal point and a zero to make it $10$.
Step 2: $3$ goes into $10$ three times ($3 \times 3 = 9$).
Step 3: Subtract $9$ from $10$ to get a remainder of $1$.
Step 4: Bring down another $0$ and repeat.
The Result:
As you can see in the division, the remainder $1$ keeps appearing. This means the number $3$ will repeat infinitely in the quotient.
This is known as a non-terminating recurring decimal. The bar over the $3$ indicates that it repeats forever.
You don’t always have to do long division to know what kind of decimal you’ll get. There is a simple rule:
The Note: If a fraction is in its simplest form and the prime factors of its denominator are only multiples of 2 and 5, it will be a terminating decimal.
If the denominator has any prime factors other than $2$ or $5$ (like $3$, $7$, or $11$), it will be a non-terminating recurring decimal.
Let’s apply the “Pro Rule” to the fraction $\frac{21}{15}$.
Method 1: Simplification (The Fast Way)
Before dividing, always simplify your fraction!
Analyze the Denominator: The denominator is $5$. Since it only contains the prime factor $5$, our rule tells us this must be a terminating decimal.
Method 2: Long Division
If we use long division ($21 \div 15$):
$15$ goes into $21$ once ($15 \times 1 = 15$).
Remainder is $6$. Add a decimal and a zero to make it $60$.
$15$ goes into $60$ exactly four times ($15 \times 4 = 60$).
Remainder is $0$.
The Result: The decimal “stops” at $1.4$. This is a terminating decimal.
Sometimes, a recurring decimal has a long string of numbers before it starts to repeat. Let’s look at the division of $22$ by $7$:
The Division Process:
$22 \div 7 = 3$ with a remainder of $1$.
Bring down a $0$ to make it $10$; $7$ goes into $10$ once ($10 – 7 = 3$).
Bring down a $0$ to make it $30$; $7$ goes into $30$ four times ($30 – 28 = 2$).
Bring down a $0$ to make it $20$; $7$ goes into $20$ twice ($20 – 14 = 6$).
Bring down a $0$ to make it $60$; $7$ goes into $60$ eight times ($60 – 56 = 4$).
Bring down a $0$ to make it $40$; $7$ goes into $40$ five times ($40 – 35 = 5$).
Bring down a $0$ to make it $50$; $7$ goes into $50$ seven times ($50 – 49 = 1$).
The Pattern:
Notice that after the seventh step, we are back to a remainder of $1$ (starting at $10$ again). This means the entire sequence $142857$ will repeat infinitely.
Important Rule: A number is considered a rational number if its decimal expansion is either terminating (ends) or non-terminating recurring (repeats a pattern).
Converting a terminating decimal back into a fraction (rational number) is a simple process of using place values.
| Decimal | Rational Form (Simplest) | Type |
| $1.2$ | $\frac{6}{5}$ | Terminating |
| $1.25$ | $\frac{5}{4}$ | Terminating |
| $3.\overline{142857}$ | $\frac{22}{7}$ | Non-terminating Recurring |
When a decimal repeats forever, we can use a simple algebraic trick to find its fractional (rational) form.
To solve this, we create an equation to “cancel out” the infinite tail:
Step 1: Let $x = 0.\bar{3}$ (which is $0.3333…$).
Step 2: Multiply by $10$ to shift the decimal: $10x = 3.3333…$.
Step 4: Solve for $x$: $x = \frac{3}{9}$ which simplifies to $\frac{1}{3}$.
When only part of the decimal repeats, we adjust our multiplication:
Step 1: Let $x = 0.2\overline{35}$ ($0.2353535…$).
Step 2: Multiply by $1000$ to get $1000x = 235.353535…$.
Step 3: Multiply by $10$ to get $10x = 2.353535…$.
Step 4: Subtract: $990x = 233$.
Step 5: Final fraction: $x = \frac{233}{990}$.
Understanding decimals is essential for converting units we use every day.
Since $1000 \text{ grams} = 1 \text{ kg}$, we can convert any weight by dividing by $1000$:
$1 \text{ g} = \frac{1}{1000} \text{ kg} = 0.001 \text{ kg}$
$85 \text{ g} = \frac{85}{1000} \text{ kg} = 0.085 \text{ kg}$
$5678 \text{ g} = \frac{5678}{1000} \text{ kg} = 5.678 \text{ kg}$
Since $100 \text{ paise} = \text{Rs } 1$, we divide by $100$ to find the rupee value:
$1 \text{ paise} = \text{Rs } \frac{1}{100} = \text{Rs } 0.01$
$50 \text{ paise} = \text{Rs } \frac{50}{100} = \text{Rs } 0.50$
$225 \text{ paise} = \text{Rs } \frac{225}{100} = \text{Rs } 2.25$
A key takeaway from these exercises is the definition of a Rational Number. A number is rational if its decimal expansion is either:
Terminating (like $1.2$ or $1.25$).
Non-terminating recurring (like $3.\overline{142857}$ or $0.\bar{3}$).
If a number follows either of these patterns, it can always be written as a fraction!
To convert grams into kilograms, you first need to know the conversion factor: 1 kilogram (kg) = 1,000 grams (g).
Example: Write 2 kg 348 g in kilograms using decimals.
Step-by-Step Solution:
Separate the units: Think of the measurement as two parts added together.
$2 \text{ kg} + 348 \text{ g}$
Convert grams to kilograms: Since there are 1,000 grams in a kilogram, divide the gram portion by 1,000.
$2 \text{ kg} + \frac{348}{1000} \text{ kg}$
Turn the fraction into a decimal: Dividing by 1,000 moves the decimal point three places to the left.
$2 \text{ kg} + 0.348 \text{ kg}$
Add them up: Combine the whole kilograms and the decimal.
Final Answer: $2.348 \text{ kg}$
When dealing with Indian currency, the conversion factor is: 1 Rupee (Rs) = 100 Paise.
Example: Write Rs 24 and 50 paise in rupees using decimals.
Step-by-Step Solution:
Separate the units:
$Rs\ 24 + 50 \text{ paise}$
Convert paise to rupees: Since there are 100 paise in one rupee, divide the paise portion by 100.
$Rs\ 24 + Rs\ \frac{50}{100}$
Turn the fraction into a decimal: Dividing by 100 moves the decimal point two places to the left.
$Rs\ 24 + Rs\ 0.50$
Add them up:
$Rs\ 24.50$
Simplify (Optional): In decimal math, a zero at the very end of a decimal doesn’t change the value.
Final Answer: $Rs\ 24.5$