Decimals
Category : 4th Class
LEARNING OBJECTIVES
This lesson will help you to:—
Real Life Examples
One of the most instances will be money! Whenever we have some numbers of cents that do not add up to a full dollar, we express the amount as a decimal. For example $3.75, $12.69, and even $100 are all examples of decimals.
QUICK CONCEPT REVIEW
What are decimals?
Examples:
decimal 
whole – number part 
fractional part 
3.25 4.172 0.168 
3 4 0 
25 172 168 
PLACE VALUE AND DECIMALS

millions 

hundred thousands 

ten thousands 

thousands 

hundreds 
5 
tens 
7 
ones 
. 
and 
4 
tenths 
9 
hundredths 

thousandths 

tenthousandths 

hundredthousandths 

millionths 
\[345.65\]
The leading zeros
Let's look at a normal whole number; 345
Hundreds 
Tens 
Units (ones) 
3 
4 
5 
We can break the number up to see how the number 345 is constructed.
The construction of a number 345 actually means
3 of 100s + 4 of 10s + 5 of ones.
Now imagine extending this number 345 to show some hidden numbers. These numbers have been taken away because they have no real value at all
Thousands 
Hundreds 
Tens 
Unit (ones) 
0 
3 
4 
5 
Amazing Fact
When we add two decimal numbers, the answers will have the same number of decimal digits ads the given numbers.
Historical Preview
According to Joseph Needham, decimal fractions were first developed and used by the Chinese in the 1^{st} century BC, and then spread to the Middle East and from there to Europe.
The Jewish mathematician Immanuel Bonfils invented decimal fractions around 1350.
Similarly the construction of the number 0345 is 0 of 1000s + 3 of 100s + 4 of 10s + 5 of ones.
We can see that 0 of 1000s means zero. So we do not count the number of 0s leading a number.
The trailing zeros after the decimal part of a decimal number.
Let's look at this number 0.650
Decimal point 
Tenths / 10th 
Hundredth / 100th 
Thousandths / 1000th 
0 
3 
4 
5 
The construction of this decimal part of a decimal number means
6/10 + 5/100 + 0/1000.
We can see that 0 out of 1000 is nothing. So we can ignore this 0. What it means is that 0.65 is the same as 0.650.
Similarly 0.6500 is the same as 0.65 because it means
6/10 + 5/100 + 0/1000 + 0/10000.
ADDITION AND SUBTRACTION OF DECIMALS
To add decimal numbers
Let's look at an example:
123 + 0.0079 + 43.5 =
To add these numbers, first arrange the terms vertically, aligning the decimal points in each term. Don't forget, for a whole number like the first term, the decimal point lies just to the right of the ones column. You can add zeroes to the right of the decimal point to make it easier to align the columns. Then add the columns working from the right to the left, positioning the decimal point in the answer directly under the decimal points in the terms.
123.0000
0.0079
+43.5000
166.5079
To subtract decimal numbers:
Here's a subtraction example:
27.583 – 0.2 =
To subtract these numbers, first arrange the terms vertically, aligning the decimal points in each term. You can add zeroes to the right of the decimal point, to make it easier to align the columns. Then subtract the columns working from the right to the left, putting the decimal point in the answer directly underneath the decimal points in the terms. Check your answer by adding it to the second term and making sure it equals the first.
\begin{array}{*{35}{l}}
\,\,\,27.583 \\
\underline{\text{ }0.200} \\
\,\,27.383 \\
\end{array}
Note: To add (or subtract) decimals, always fill empty place values with zeros so that all of the numbers have the same number of decimal places.
CONVERTING FRACTIONS INTO DECIMALS
Fractions and decimals are two different ways to show the same values; parts of wholes.
Step 1: Find a number you can multiply by the bottom of the fraction to make it 10, or 100, or 1000, or any 1 followed by 0s.
Step 2: Multiply both top and bottom by that number.
Step 3: Then write down just the top number, putting the decimal point in the correct spot. (one space from the right hand side for every zero in the bottom number.)
Misconcept/Concept
Misconcept: Longer decimal numeral is larger.
Concept: This misconception is not true. Example: Let’s take two numbers 2.54869 and 3.01. As we can see, the first number is longer than the second number, but it is not larger than the second numeral. 3.01>2.4869
CONVERTING DECIMALS INTO FRACTIONS
Step 1: Write down the decimal divided by 1, like this: decimal/1.
Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there are two numbers after the decimal point, then use 100, if there are three then use 1000, etc.)
Step 3: Simplify (or reduce) the fraction.
ROUNDING OF DECIMALS
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