What are Decimals?
Decimals are a way of writing numbers that are not whole. They help us represent parts of a whole, just like fractions do. A decimal number is written with a dot called the decimal point. The numbers to the left of the decimal point represent whole numbers, and the numbers to the right represent the fractional part of a whole.
Example: In the number 3.45:
- The 3 is a whole number.
- The .45 is the fractional part.
Reading a Decimal
To read a decimal, say the whole number part first, then read the decimal point as “point” and name the digits to the right. The numbers after the decimal point should be read separately.
Example: 3.45 is read as “three point four five.” 6.72 should be read as “six point seven two” and not six point seventy two.
Place Value in Decimals
The place value system for decimals is the same as whole numbers, with each place ten times larger than the one to its right. The decimal point is the divider between whole numbers (on the left) and fractional parts (on the right). Hence, the place value of the digit after the decimal point to the right of ones place is ten times smaller than 1 and is represented as tenths (1/10). To the right of tenths, we have hundredths (1/100) and then, thousandths (1/1000) and so on. Thus, each position after the decimal point has a special place value:
- The first place is tenths (1/10).
- The second place is hundredths (1/100).
- The third place is thousandths (1/1000), and so on.
Example: In 5.673, the 5 is in the ones place, the 6 is in the tenths place, the 7 is in the hundredths place, and the 3 is in the thousandths place.
Key Components of Decimals
- Whole Number: The part of the number to the left of the decimal point represents the whole number. For example, in the decimal 35.762, “35” is the whole number.
- Decimal Point: The dot that separates the whole number from the fractional part.
- Tenths: The first digit to the right of the decimal point represents tenths. For example, in 5.762, “7” represents seven tenths.
- Hundredths: The second digit to the right of the decimal point represents hundredths. In 5.762, “6” represents six hundredths.
- Thousandths: The third digit to the right of the decimal point represents thousandths. For example, in 5.672, “2” represents two thousandths.
Example: 4.78 consists of 4 wholes, 7 tenths and 8 hundredths.
Decimals in Expanded Form
Decimals in expanded form show the value of each digit based on its place value. It helps break down a decimal number into its individual parts. For example, the decimal 5.43 in expanded form is written as 5 + 0.4 + 0.03. Here, the digit 5 represents five whole units, 0.4 represents four-tenths, and 0.03 represents three-hundredths. By expressing decimals in expanded form, we can each digit contributes to the overall number.
Example: 38.46 = 30+8+0.4+0.06
It can also be written in the following ways:
38.46 = 30 + 8 + $\frac{4}{10}$+$\frac{6}{100}$
38.46 = 3 tens + 8 ones + 4 tenths + 6 hundredths
Decimals on the Number Line
A number line is a great way to visualize and understand decimals. Just like whole numbers, decimals can be plotted on a number line between two whole numbers. We can divide the spaces between whole numbers into ten parts to show tenths.
To place 2.7 on a number line:
- Identify that it falls between 2 and 3.
- Divide the space between 2 and 3 into 10 equal parts (each representing 0.1).
- Count 7 parts to place 2.7
The space between any two whole numbers can be divided into 100 equal parts to show hundredths.
To place 11.85 on a number line:
- Identify that it falls between 11 and 12.
- Divide the space between 11 and 12 into 100 equal parts (each representing 0.01).
- The 85th part represents 11.85.
Alternatively, the space between two tenths is be divided into 10 equal parts to show hundredths.
To place 4.53 on a number line:
- Identify that it falls between 4 and 5.
- Divide the space between 4 and 5 into 10 equal parts (each representing 0.1).
- Count 5 parts to reach 4.5.
- Divide the space between 4.5 and 4.6 into 10 equal parts (each representing 0.1).
- Count 3 parts to reach 4.53.
Rounding Decimals
Rounding decimals is a way to simplify a number by adjusting it to the nearest value based on a specific place value, making it easier to work with. To round a decimal, follow these steps:
Identify the place value to which you want to round (e.g., nearest whole number, nearest tenth, etc.).
Look at the digit to the right of that place:
- If the digit is 5 or greater, round up by adding 1 to the digit you’re rounding to.
- If the digit is less than 5, round down and leave the digit you’re rounding to unchanged.
Rounding to the nearest whole number
Round 6.84 to the nearest whole number.
- The digit in the ones place is 6.
- Look at the tenths place: 8 (greater than 5), so round up.
- The number becomes 7.
Rounding to the nearest tenth
Round 3.47 to the nearest tenth.
- Look at the digit in the tenths place: 4.
- The digit in the hundredths place is 7 (which is greater than 5), so round up.
- The number becomes 3.5.
Rounding to the nearest hundredth
Round 10.973 to the nearest hundredth.
- Look at the digit in the hundredths place: 4.
- The digit in the thousandths place is 3 (which is less than 5), so round down.
- The number becomes 10.97.
Comparing Decimals
To compare decimals, follow these steps:
Compare the whole number parts first. The number with the larger whole number is greater.
Example: Compare 3.25 and 4.12. Since 4 is greater than 3, 4.12 is the larger number.
If the whole numbers are the same, move to the tenths place and compare the digits.
Example: Compare 5.48 and 5.63. The whole numbers are the same (5), so compare the tenths place. Since 6 is greater than 4, 5.63 is larger.
If the tenths place is the same, move to the hundredths place, and continue comparing digits until you find a difference.
Example: Compare 2.456 and 2.453. The whole number and tenths (2 and 4) are the same. The difference is in the hundredths place: 6 is greater than 3, so 2.456 is larger.
For decimals with different lengths, you can add extra zeros to the shorter decimal to help compare. Extra zeros don’t change the value but make it easier to compare.
Example: Compare 7.4 and 7.38. You can rewrite 7.4 as 7.40. Now, compare 7.40 and 7.38. Since 40 is greater than 38, 7.4 is larger.
Decimals in Real Life
Decimals are everywhere in daily life, helping us represent parts of whole numbers accurately. Whether you’re shopping, measuring, or handling money, decimals play an important role.
Money
Perhaps the most common use of decimals is with money. Most currencies are divided into smaller units, like dollars and cents or rupees and paise. For example, ₹12.50 represents 12 rupees and 50 paise, with 50 paise being half of a rupee (0.50).
Measurements
In measuring lengths, weights, and volumes, decimals provide precision. A carpenter might measure a piece of wood as 4.75 meters, meaning 4 meters and 75 centimeters, while a chef might measure 1.5 liters of water for a recipe.
Shopping Discounts
When calculating discounts, decimals help express percentages. For example, a 25% discount on a ₹200 item would be 0.25 × 200 = ₹50 off, making the final price ₹150.
Sports
In sports, decimals are used to record times in races and competitions. A sprinter’s time might be 10.25 seconds, showing exactly how long they took, down to the hundredth of a second.
Science and Medicine
In science, decimals help in precise measurements and calculations, such as tracking temperature changes or measuring doses in medication. A doctor might prescribe 0.75 ml of a liquid medicine for accurate dosing.
Decimals allow us to handle everyday situations with precision and clarity, making them an essential part of various real-life tasks.