Decimal notation

Decimals are another way of writing numbers that are not whole. Decimal notation is the system we use to write these numbers, based on the idea of place value extended to the right of the decimal point.

In this chapter we focus on how decimal numbers are written, read, and understood. Conversions to and from fractions, and percentage work, are handled in later chapters.

Structure of a decimal number

A decimal number has two main parts, separated by a decimal point:

• the part to the left of the decimal point (the whole-number part)
• the part to the right of the decimal point (the fractional part)

For example, in the decimal number
$$
47.263,
$$

• $47$ is the whole-number part,
• $.$ is the decimal point,
• $263$ is the fractional part.

If there is no whole-number part, we usually write $0$:

• $0.5$ instead of $.5$ (especially in formal writing).

Place value to the right of the decimal point

You may already know the place values to the left of the decimal point:

• ones, tens, hundreds, thousands, and so on.

To the right of the decimal point, the places represent fractions with denominators that are powers of $10$.

From left to right:

• first place: tenths
• second place: hundredths
• third place: thousandths
• fourth place: ten-thousandths
• and so on.

In a table:

• $0.1$ is “one tenth” $= \frac{1}{10}$
• $0.01$ is “one hundredth” $= \frac{1}{100}$
• $0.001$ is “one thousandth” $= \frac{1}{1000}$

So for $47.263$:

• $2$ is in the tenths place,
• $6$ is in the hundredths place,
• $3$ is in the thousandths place.

We can write:
$$
47.263 = 4 \times 10 + 7 \times 1 + 2 \times \frac{1}{10} + 6 \times \frac{1}{100} + 3 \times \frac{1}{1000}.
$$

This is the key idea of decimal notation: each digit shows how many of a particular place value you have.

Reading decimal numbers

To read a decimal number in words:

1. Read the whole-number part as you normally would.
2. Say “point” for the decimal point in informal contexts, or say “and” in some traditional English usages (note: in school math, your teacher’s convention may vary).
3. Read each digit to the right of the decimal point separately, or group them and use the place name.

Two common styles:

• Digit-by-digit after the point:
• $3.14$: “three point one four”
• $0.06$: “zero point zero six”
• Using place names (more formal for exact values):
• $3.14$: “three and fourteen hundredths”
• $0.06$: “six hundredths”
• $2.305$: “two and three hundred five thousandths”

Both ways refer to the same number; you choose based on context.

Writing decimals with and without trailing zeros

A trailing zero is a zero at the end of the fractional part.

Examples:

• $0.5$
• $0.50$
• $0.500$

These three numbers are all equal in value:
$$
0.5 = 0.50 = 0.500.
$$

The extra zeros do not change the quantity; they only show the number of decimal places you want to emphasize. For example:

• In money, we often write $5.00$ dollars instead of $5$ to show “zero cents”.
• In a measurement like $3.20$ m, the zero can suggest the measurement was made to the nearest hundredth.

Similarly, you do not usually write unnecessary zeros before the whole-number part:

• we write $7.2$, not $07.2$ in most everyday uses.

However, placing a zero before the decimal point in numbers less than $1$ is important for clarity:

• write $0.7$, not $.7$.

Comparing and ordering decimals

Decimal notation lets you compare sizes of non-whole numbers by using place value.

To compare two decimal numbers:

1. Compare the whole-number parts.
2. If they are equal, compare digits from left to right after the decimal point (tenths, then hundredths, then thousandths, and so on).
3. If one number “runs out” of digits, you can imagine trailing zeros to help comparison.

Examples:

• Compare $3.48$ and $3.5$.
• Whole-number parts: both $3$.
• Tenths: $4$ vs $5$. Since $4 < 5$, we have $3.48 < 3.5$.
• Compare $0.7$ and $0.65$.
• Whole-number parts: both $0$.
• Tenths: $7$ vs $6$. Since $7 > 6$, $0.7 > 0.65$.
• You can think of $0.7$ as $0.70$ to make it easier to see.
• Compare $2.305$ and $2.35$.
• Whole-number parts: both $2$.
• Tenths: $3$ vs $3$ (same).
• Hundredths: $0$ vs $5$. Since $0 < 5$, $2.305 < 2.35$.

The idea of “imagining” trailing zeros (like $0.7 = 0.70$) is just making use of the fact that those zeros do not change the value.

Rounding decimals (idea and notation)

A rounded decimal is an approximate value written with fewer decimal places.

When you round a number to a certain decimal place, you decide which digit to keep at that place and whether to increase it by $1$ based on the next digit to the right.

The notation often makes this clear:

• “Rounded to the nearest tenth”: $3.27 \approx 3.3$
• “Rounded to two decimal places”: $4.678 \approx 4.68$

The symbol $\approx$ is used to show that the rounded value is approximate, not exact.

Specific methods and practice with rounding are part of working with decimals in more detail, but the important notation point is:

• An exact decimal uses $=$.
• A rounded or approximate decimal uses $\approx$.

Terminating and repeating decimals (notation only)

Decimals can behave in two main ways:

• Terminating decimals: the fractional part ends (has a last digit).
• Examples: $0.5$, $3.14$, $2.75$, $10.03$.
• Repeating (recurring) decimals: one or more digits repeat forever.
• Common notation uses a bar (vinculum) over the repeating block:
• $0.\overline{3}$ means $0.3333\ldots$
• $0.1\overline{6}$ means $0.1666\ldots$
• $2.\overline{47}$ means $2.474747\ldots$

The dots “$\ldots$” in $0.333\ldots$ indicate that the pattern continues without end. The bar notation is a compact way to show that.

Later, when we connect decimals to fractions, you will see why some decimals terminate and others repeat.

Zero and negative decimals

Zero and negative numbers fit naturally into decimal notation.

• Zero itself can be written as $0$, $0.0$, $0.00$, and so on. These are all the same number.
• A negative decimal uses a minus sign:
• $-0.5$, $-3.2$, $-7.04$.

The minus sign applies to the entire number, not just part of it:

• $-(3.5)$ is the same as $-3.5$.

When writing negative decimals with no whole-number part, it is still best to include the zero:

• write $-0.3$, not $-.3$.

Common conventions and formatting

A few practical notes about how decimals appear in writing:

• In many English-speaking countries, a dot is used as the decimal separator:
• $1.25$ means “one and twenty-five hundredths”.
• In many other countries, a comma is used as the decimal separator:
• $1,25$ may mean the same as $1.25$.

In this course we consistently use the dot as the decimal point.

• Large whole numbers often use commas (or spaces) to group digits in threes for readability:
• $12{,}345.67$ is “twelve thousand, three hundred forty-five and sixty-seven hundredths”.
• In some regions this could be written as $12\,345.67$ with a space.

These grouping symbols do not change the value; they just help you read the number more easily.

Summary of key ideas

• Decimal notation extends place value to the right of the decimal point using powers of $10$ (tenths, hundredths, thousandths, …).
• A decimal number has a whole-number part and a fractional part, separated by a decimal point.
• Trailing zeros to the right of the decimal point do not change the value: $0.5 = 0.50$.
• To compare decimals, compare whole-number parts, then move right through the decimal places; you can imagine extra zeros at the end when needed.
• The symbol $=$ is for exact decimals; $\approx$ is for rounded decimals.
• Repeating decimals are written with a bar over the repeating digits, like $0.\overline{3}$.
• Zero and negative numbers follow the same decimal notation rules, with a minus sign for negative values.