Decimal Notation: Definition, Method, and Examples

What is decimal notation?

Decimal notation is a numerical format that represents parts of a whole using a base-10 number system.

In decimal notation, every place value is a power of ten. Instead of writing a fraction with a numerator and a denominator, the value is written linearly.

Before moving forward, recall that decimal numbers are values that include both a whole quantity and a fractional quantity written using this system.

The standard format ensures that anyone reading the number understands exactly which portion is complete and which portion represents a fraction.

Locate the decimal point

The decimal point is a small dot placed immediately after the ones place, serving as the anchor for the entire number.

It separates the whole number part on its left from the fractional part on its right. The decimal point itself dictates where the ones place sits, establishing the value of every other digit in the sequence.

When reading a decimal number aloud, the decimal point is pronounced as the word "and" to emphasize this separation between complete wholes and fractional pieces.

Name the places around the point

As you move outward from the decimal point, each position has a specific value based on powers of $10$.

To the left, the places represent whole quantities: ones, tens, hundreds, and so on. To the right, the places represent fractional quantities: tenths, hundredths, and thousandths.

If you want to master decimal place value, it helps to see that each step to the right makes the digit's value ten times smaller. This underlying structure is also exactly how we expand and calculate decimals in expanded form.

Write decimals from models

Visual models, such as grids divided into $10$ or $100$ equal parts, show how decimal notation works in practice.

A fully shaded $10 \times 10$ grid represents one whole ($1.0$). If only a portion of the grid is shaded, counting the squares reveals the decimal value.

When reading and writing decimals, visualizing the value prevents confusion. For instance, $45$ out of $100$ small squares represents forty-five hundredths. In decimal notation, this is explicitly written as $0.45$.

Use zeros as placeholders

Placing extra zeros at the far right of a decimal number does not change its mathematical value, but it does change its written precision.

For example, $4.5$, $4.50$, and $4.500$ all represent the exact same quantity. The digit $4$ remains anchored in the ones place, and the digit $5$ remains anchored in the tenths place.

The appended trailing zeros simply communicate that there are zero hundredths and zero thousandths present, which is often useful when lining up numbers for addition or subtraction.

Worked examples

Reviewing structured steps reveals how standard definitions operate when applied to a problem.

Example 1: Writing decimal notation from a fraction

Question: Write the mixed number $34$ and $\dfrac{7}{100}$ in decimal notation.

Method:

1. Identify the whole-number part. The whole number is $34$.
2. Place the decimal point immediately after the ones place.
3. Identify the fractional part. The fraction $\dfrac{7}{100}$ means there are $7$ hundredths and $0$ tenths.
4. Write $0$ in the tenths place and $7$ in the hundredths place.

Answer: The decimal notation is $34.07$.

Check: Read the decimal aloud. The number $34.07$ is read as "thirty-four and seven hundredths," which accurately reflects the original mixed fraction.

Example 2: Expanding decimal notation

Question: Write the decimal number $6.19$ in expanded form using fractions.

Method:

1. Identify the value of each digit based on its position relative to the decimal point.
2. The digit $6$ is in the ones place. Its value is $6$.
3. The digit $1$ is in the tenths place. Its value is $\dfrac{1}{10}$.
4. The digit $9$ is in the hundredths place. Its value is $\dfrac{9}{100}$.
5. Add the individual values together.

Answer: The expanded form is $6 + \dfrac{1}{10} + \dfrac{9}{100}$.

Check: Convert the fractions back into decimal notation. Since $\dfrac{1}{10} = 0.1$ and $\dfrac{9}{100} = 0.09$, combining $6 + 0.1 + 0.09$ yields $6.19$.

Example 3: Verifying equivalent values

Question: Do $8.2$ and $8.20$ represent the same mathematical value?

Method:

1. Identify the whole-number part of both numbers. Both have $8$ ones.
2. Identify the tenths place. Both have $2$ tenths.
3. Identify the hundredths place. The first number has no visible digit in the hundredths place, meaning $0$ hundredths. The second number explicitly shows a $0$ in the hundredths place.

Answer: Yes, they represent the exact same value.

Check: When comparing decimals, align the decimal points vertically. Because $8.2$ and $8.20$ have identical digits in every place value column, they are strictly equal.

Common mistakes

One common mistake is misreading the fractional part as a whole number.

A learner might mistakenly think $0.45$ is larger than $0.8$ because the integer $45$ is greater than the integer $8$. However, $0.8$ represents $8$ tenths, while $0.45$ represents $4$ tenths and $5$ hundredths.

Because $8$ tenths is more than $4$ tenths, $0.8$ is the larger value. Another frequent error is skipping placeholder zeros immediately after the decimal point. If a number has $3$ ones and $7$ hundredths, writing $3.7$ is incorrect because it mistakenly places the $7$ in the tenths position. The accurate decimal notation is $3.07$.

Frequently asked questions

What is the decimal notation of 1?

The decimal notation of $1$ is $1.0$. While it is most commonly written simply as the whole number $1$, adding a decimal point and a trailing zero emphasizes that there are exactly zero fractional parts.

What is the decimal notation of a fraction?

The decimal notation of a fraction expresses the same value using a decimal point rather than a numerator and denominator. For example, the fraction $\dfrac{1}{2}$ is equivalent to $\dfrac{5}{10}$, which is written as $0.5$ in decimal notation.

What is the decimal notation of a percentage?

To write a percentage in decimal notation, divide the percentage value by $100$ and remove the percent sign. This effectively moves the decimal point two places to the left. Following this rule, $79\%$ is written as $0.79$.