Math V: Understanding Decimals

 

1. What is a Decimal?

 

You have already learned about fractions — numbers that show parts of a whole. A decimal is simply a fraction written in a special form. For example, instead of writing one-half as ½, you can write it as the decimal 0.5. In 0.5, the zero is in the ones place, and the five is in the tenths place.

 

The word decimal comes from the Latin word decimus, meaning “tenth,” from the root word decem, which means “ten.” This is because decimals are based on powers of 10.

 

Decimals are numbers that have two parts:

 

• A whole number part.

 

• A fractional part.

 

These parts are separated by a decimal point — the dot that you see between them. For example, in 34.5:

 

34 is the whole number part.

 

5 (in the tenths place) is the fractional part.

 

 

2. The History of Decimals

 

The decimal system you use today did not appear overnight. It evolved over many years.

 

In the 1440s, an Italian mathematician named Giovanni Bianchini used decimal fractions and the decimal point in his astronomical and measurement calculations.

 

Later, in the late 1500s, Christopher Clavius popularized the decimal point and influenced John Napier, the inventor of logarithms, to use it widely in mathematics.

 

But even before that, the decimal system had older roots in ancient India and the Arab world. The work of Bianchini, Clavius, and Napier was especially important in shaping the decimal point and making it common around the world.

 

3. Parts of a Decimal Number

 

Think of a decimal as having three pieces:

 

The whole number piece (to the left of the decimal point).

 

The fractional piece (to the right of the decimal point).

 

The decimal point (the dot between them).

 

The fractional part always shows a number less than 1, and it is written to the right of the decimal point. Even a whole number can be written in decimal form by adding a .0 after it. For example, 34 can be written as 34.0.

 

4. How Decimals Relate to Fractions

 

Decimals are another way to write fractions that have denominators that are powers of 10 — like 10, 100, 1000, and so on.

 

0.5 = 5/10 = one-half

 

0.25 = 25/100 = one-quarter

 

This makes decimals very useful for expressing parts of a whole in a shorter and easier form.

 

5. The Decimal Point and Place Value

 

Decimal Point – This separates the whole number part from the fractional part.

 

Place Value – Digits to the left of the decimal point are whole numbers (ones, tens, hundreds, etc.). Digits to the right represent parts of a whole:

 

1st place → tenths

 

2nd place → hundredths

 

3rd place → thousandths

 

Example:

In 12.35:

 

12 is the whole number part.

 

3 is in the tenths place (3/10).

 

5 is in the hundredths place (5/100).

You read this as "twelve and thirty-five hundredths".

 

6. Types of Decimals

 

Decimals come in different forms:

 

Terminating Decimals – These have a fixed number of digits after the decimal point and then stop.

Examples: 0.25, 1.75, 3.0

 

Non-Terminating Recurring Decimals – These go on forever, but have a repeating pattern.

Examples: 0.3333… (3 repeats), 0.142857142857… (sequence repeats)

 

Non-Terminating Non-Recurring Decimals – These also go on forever, but without any repeating pattern.

Example: π (pi) ≈ 3.1415926535…

 

Decimal Fractions – Fractions where the denominator is a power of 10, written as decimals.

Examples: 1/2 = 0.5, 1/4 = 0.25

 

7. How to Read a Decimal Number

 

Follow these steps:

 

• Say the whole number part as you normally would.

 

• Say “and” for the decimal point.

 

• Say the fractional part as a whole number, then name its place value.

 

Example:

 

45.6 → “forty-five and six tenths”

 

7.09 → “seven and nine hundredths”

 

8. Rules When Working with Decimals

 

When you work with decimals in operations like division, remember these rules:

 

• A number divided by one is that same number.

 

• A number cannot be divided by zero.

 

• If zero is divided by a number, the answer is zero.

 

• If you change the divisor from a decimal to a whole number, you must also make the same change to the dividend.

 

 

9. Why Do We Use Decimals?

 

Decimals are everywhere in daily life, especially when precision is important:

 

Money: ₱45.75 is more exact than ₱46.

 

Weight: You might weigh 35.8 kg, not just 36 kg.

 

Length: A rope might be 2.35 meters long.

 

Whole numbers cannot always give you the exact measurement you need, so decimals help show exact values.

 

10. Quick Summary

 

• Decimals are another way to write fractions, especially those with denominators that are powers of 10.

 

• They have a whole number part, a fractional part, and a decimal point in between.

 

• Place value tells you how big or small each digit is.

 

• Decimals can be terminating, non-terminating recurring, or non-terminating non-recurring.

 

• We use decimals in money, measurement, and many everyday activities where accuracy matters.

 

 

Place Value

1. The Idea of Place Value

 

Every digit in a number has a value depending on its place. The same digit can mean different amounts depending on where it is.

 

In 4,532, the 4 means four thousand because it is in the thousands place.

 

In 543.2, the 4 means four tens because it is in the tens place.

 

2. Whole Number Place Values

 

From right to left, the place values are:

 

Ones (1) – single units

 

Tens (10) – groups of ten

 

Hundreds (100) – groups of one hundred

 

Thousands (1,000) – groups of one thousand

 

Ten thousands (10,000)

 

Hundred thousands (100,000)

 

Millions (1,000,000) and so on.

 

3. Decimal Place Values

 

Decimals add places to the right of the decimal point. From left to right after the decimal:

 

Tenths (1/10)

 

Hundredths (1/100)

 

Thousandths (1/1,000)

 

Ten-thousandths (1/10,000) and so on.

 

 

 

Whole Number Place Value

Decimal Place Value