What is a Decimal Number?

The decimal number might sound like a somewhat challenging concept to you, but believe me, after reading this article, you will not fear encountering it on the exam, you will even be glad to see it.
Shall we start?

A decimal number uses a decimal point to represent any real number in our base-10 number system ,using a decimal point to separate the whole part from the fractional part.

In everyday life, we often come across the decimal number and we don't even realize it!
For example, when checking a fever, body temperature readings on the thermometer can be a number like $37.5$ or $36.4$.
When we weigh ourselves, we step on the scale and, also in this case, a decimal number appears! The weight is shown with the decimal point and expresses, in a clear and simple way, a number that is not whole.

The decimal point divides the number into two parts: everything to the left represents whole units, while everything to the right represents parts smaller than one (tenths, hundredths, etc.).

A decimal number is simply another way to write fractions and mixed numbers. Instead of writing $3\frac{1}{2}$, we can write $3.5$. Instead of writing $7\over10$, we can write $0.7$. This system makes calculations much easier and is universally understood.
As we have mentioned, in the decimal number there is a decimal point (or comma).
This point separates the whole part and the decimal part.
Everything that appears to the left of the point or comma is called the whole part (divided into hundreds, tens, and units)
Everything that appears to the right is called the decimal part.
The decimal part is divided into tenths, hundredths, and thousandths.
Let's see this division so we understand it better:

Understanding Decimal Place Values

The decimal place value system follows the same logical pattern as whole numbers. Just as we move from ones to tens to hundreds going left, we move from tenths to hundredths to thousandths going right:

...hundreds | tens | ones | . | tenths | hundredths | thousandths...

Each position represents exactly $1\over10$ of the position to its left. The position of each digit after the decimal point determines its value - the first position is tenths, the second is hundredths, the third is thousandths, and so on. This consistent pattern makes decimal numbers easy to understand once you see the logic behind it.

In the number $4.586$ what does the digit $5$ represent?
The digit $5$ represents the tenths.

In the number $8.701$
What does the digit $1$ represent?
The digit $1$ represents the thousandths.

In the number $45.765$
What does the digit $4$ represent?
The digit $4$ represents the tens. Pay attention - it represents tens of a whole number and not tenths.

There are 2 ways to express a decimal number in words:
1. The first is, simply, to read the numbers as they appear and add the point where it is found.

For example:
How do you read the decimal number $3.56$?
In the following way - "Three point five, six".
How do you read the decimal number $76.304$?

In the following way - "seventy-six point three, zero, four".

2.The second way is to remember that the name derives from the last digit.
First, the whole part should be mentioned, then say "and", we will ask ourselves: What does the last digit represent?
If, for example, the last digit represents hundredths we will pronounce the decimal part as it appears and then the word "hundredths".
Are you confused by this? Come and see how simple it is.

How do you read the decimal number $16.56$?
Sixteen and $56$ hundredths.
First, we have named the whole numbers as is proper, added "and" and asked ourselves What is the last digit $6$ representing?
The hundredths. So we will add $56$ hundredths.

How do you read the decimal number $3.765$?
$3$ and $765$ thousandths.
Explanation: $3$ whole numbers, $5$ last digit is in thousandths place.

How do you read the decimal number $0.8$?
$0$ and $8$ tenths.

When to use each method:

• Method 1 is simpler and commonly used in casual speech
• Method 2 is more formal and preferred in mathematical contexts

Adding Trailing Zeros

An important property of decimal numbers is that adding zeros to the right end of a decimal does not change its value. If we add the figure $0$ to the end of the decimal number (to the right of the point) the value of the number does not change!

For example:
$45.877=45.87700000$
$0.5=0.50$[object Object]
$12.6 = 12.60 = 12.600$

Why this works: Adding trailing zeros is like adding empty place holders in positions that represent smaller and smaller fractions. Since these positions contain zero, they add nothing to the total value.

Practical applications:

• Money: $5.20 and $5.2000 represent the same amount
• Measurements: $3.5$ meters = $3.500$ meters
• This property is useful when aligning decimals for addition and subtraction

Important note: This only works for zeros added to the RIGHT of the decimal point, not zeros inserted elsewhere in the number.

Practice on Decimal Point Placement:

Given the number $76593$ place the decimal point so that the figure $9$ represents tenths.
Solution: We know that for the figure $9$ to represent tenths, it must appear immediately after the decimal point, therefore:
$765.93$

Additional Practice:

1. Given $84672$, place the decimal point so that $6$ represents hundredths.
   Answer: $846.72$ ($6$ is in the hundredths position)
2. Given $52418$, place the decimal point so that $4$ represents tenths.
   Answer: $524.18$ ($4$ is in the tenths position)
3. Given $39175$, place the decimal point so that $7$ represents thousandths.
   Answer: $391.75$ ($7$ is in the thousandths position)

Converting Between Decimals and Fractions

Understanding the connection between decimals and fractions is fundamental to mathematical literacy. These are simply two different ways to express the same value - like saying "half" versus "50 percent" versus "0.5."

Converting Decimals to Fractions

Step-by-Step Process:

1. Identify the place value of the last digit
2. Write the decimal as a fraction using the place value as the denominator
3. Simplify the fraction by finding the greatest common factor

Examples:

Converting $0.75$:

• Last digit ($5$) is in the hundredths place
• Write as $75\over100$
• Simplify: $75 ÷ 25 = 3$, $100 ÷ 25 = 4$
• Final answer: $3\over4$

Converting $0.6$:

• Last digit ($6$) is in the tenths place
• Write as $6\over10$
• Simplify: $6 ÷ 2 = 3$, $10 ÷ 2 = 5$
• Final answer: $3\over5$

Converting $1.25$:

• This is a mixed number: $1$ and $0.25$
• Convert $0.25$: last digit in hundredths = $\frac{25}{100} = \frac{1}{4}$
• Final answer: 1\frac{1}{4}

Converting $0.125$:

• Last digit ($5$) is in the thousandths place
• Write as $125\over1000$
• Simplify: $125 ÷ 125 = 1$, $1000 ÷ 125 = 8$
• Final answer: $1\over8$

Converting Fractions to Decimals

Method 1: Division Divide the numerator by the denominator:

• $\frac{1}{4} = 1 ÷ 4 = 0.25$
• $\frac{3}{8} = 3 ÷ 8 = 0.375$
• $\frac{5}{6} = 5 ÷ 6 = 0.833...$ (repeating)

Method 2: Equivalent Fractions
Convert to a fraction with a denominator of 10, 100, or 1000:

• $\frac{1}{2} = \frac{5}{10} = 0.5$
• $\frac{3}{4} = \frac{75}{100} = 0.75$
• $\frac{7}{20} = \frac{35}{100} = 0.35$

Types of Decimal Numbers

Terminating Decimals: End after a finite number of digits

• $0.5$, $0.25$, $0.125$, $0.875$
• Occur when the denominator has only factors of $2$ and $5$

Repeating Decimals: Have digits that repeat infinitely

• $\frac{1}{3} = 0.333...$ (written as $0.3̄$)
• $\frac{2}{9} = 0.222...$ (written as $0.2̄$)
• $\frac{1}{6} = 0.1666...$ (written as $0.16̄$)

Converting Repeating Decimals to Fractions:

For $0.333...:$

• Let $x = 0.333...$
• Multiply by $10$: $10x = 3.333...$
• Subtract: $10x - x = 3.333... - 0.333...$
• Simplify: $9x = 3$, so $x = \frac{3}{9} = \frac{1}{3}$

Comparing Decimal Numbers

Comparing decimal numbers is a fundamental skill that requires systematic thinking. Many students make mistakes by simply comparing the numbers after the decimal point as if they were whole numbers, but this approach leads to errors.

Step-by-Step Comparison Process:

1. Compare the whole number parts first
2. If whole parts are equal, compare the tenths place
3. If tenths are equal, compare the hundredths place
4. Continue comparing place by place until you find a difference
5. The number with the larger digit in the first differing place is larger

Strategy: Align and Fill When comparing decimals with different numbers of decimal places, align them by the decimal point and add zeros to make them the same length:

Example 1: Basic Comparison Compare $3.47$ and $3.5$

$3.47$
$3.50$ (add zero for easier comparison)

• Whole parts: $3 = 3$ ✓
• Tenths place: $4 < 5$
• Therefore: $3.47 < 3.5$

Example 2: Different Whole Numbers Compare $12.9$ and $8.99$

$12.9$
$8.99$

• Whole parts:$12 > 8$
• Therefore: $12.9 > 8.99$
• (No need to compare decimal parts when whole parts differ)

Example 3: Multiple Decimal Places Compare $0.125$ and $0.13$

$0.125$
$0.130$ (add zero)

• Whole parts: $0 = 0$ ✓
• Tenths: $1 = 1$ ✓
• Hundredths: $2 < 3$
• Therefore: $0.125 < 0.13$

Example 4: Tricky Comparison Compare $0.8$ and $0.799$

$0.800$ (add zeros)
$0.799$

• Whole parts: $0 = 0$ ✓
• Tenths: $8 > 7$
• Therefore: $0.8 > 0.799$

This surprises many students! Even though $0.799$ has more digits, $0.8$ is actually larger.

Common Mistakes and How to Avoid Them:

Mistake 1: "More digits means bigger"

• Wrong thinking: $0.799 > 0.8$ because $799 > 8$
• Correct approach: Compare place by place, not the digits as whole numbers

Mistake 2: "Longer decimal is smaller"

• Wrong thinking: $0.125 < 0.13$ because $125 < 13$
• Correct approach: Align decimals and compare systematically

Mistake 3: "Ignoring leading zeros"

• Wrong thinking: $0.07$ vs $0.7$ → comparing $7$ vs $7$
• Correct approach: $0.07$ vs $0.70$ → comparing $0$ vs $7$ in tenths place