General rule for representing decimals on a number line:
Step 1: Let us first find on which two integers where the decimal number lie.
Step 2: Divide the length between those two integers into ten equal parts.
Step 3: Move as many steps you want from the preceding integer to the right.
Step 4: You will reach the decimal number on the number line.
Locate \(0.5\) on the number line.
Step 1: The decimal number \(0.5\) lies between \(0\) to \(1\).
Step 2: Divide the length between \(0\) to \(1\) into ten equal parts.
Step 3: Move five-step to the right from \(0\).
Step 4: The reached number in the number line is \(0.5\).
General rule for comparing decimals:
Step 1: First compare the highest place values of whole parts of two decimal numbers.
Step 2: If the highest place values of two decimals are the same, then compare the second-highest place of digits.
Step 3: If the whole part is the same, then compare the decimal part of tenth place.
Step 4: If it is also same, then compare the decimal part of the hundredth place. The same procedure can be extended to any number of decimal digits.
Step 2: If the highest place values of two decimals are the same, then compare the second-highest place of digits.
Step 3: If the whole part is the same, then compare the decimal part of tenth place.
Step 4: If it is also same, then compare the decimal part of the hundredth place. The same procedure can be extended to any number of decimal digits.
Example:
1. Compare \(67.62\) and \(33.41\).
Solution:
Here, the highest place value is tens place, which is different in both the numbers.
\(6\) \(>\) \(3\).
Therefore, \(67.62\) \(>\) \(33.41\).
Adding zeros at the right end of decimal digits do not change the value of that decimal number.
1. \(25.1 = 25.10\)
2. \(621.035 = 621.0350\)
Addition and Subtraction on decimal numbers:
General rules to add decimal numbers:
Step 1: Line up the decimal numbers one by one.
Step 2: To equalize the number of decimal places by adding zeros at the rightmost side of the decimal number.
Step 3: Start adding from the rightmost digit of the decimal number as the usual addition.
Step 4: Finally, put a decimal point in the answer in the same place as the numbers above it.
Example:
Add \(456.23\) and \(56.026\).
Solution:
Here, to equalize the decimal places, add zero at the end of \(456.23\).
Therefore, the addition of \(456.23\) and \(56.026\) is \(512.256\).
General rules to add decimal numbers:
Step 1: Line up the decimal numbers one by one.
Step 2: To equalize the number of decimal places by adding zeros at the rightmost side of the decimal number.
Step 3: Start subtracting from the rightmost digit of the decimal number as the normal addition.
Step 4: Finally, put a decimal point in the answer in the same place as the numbers above it.
Example:
Subtract \(94.56\) from \(156.6\).
Solution:
Here, to equalize the decimal places, add zero at the end of \(156.6\).
Therefore, the answer is \(62.04\).
Estimating sums and differences:
Let's see how to estimate the sums and differences of decimal numbers.
If we add two decimal numbers, their total will always be:
- More than \((\uparrow)\) the sum of just the whole numbers.
- Less than \((\downarrow)\) the sum of the whole numbers plus \(2\) more.
Now, let's test how it works through example.
Example:
Estimate the sum of \(14.246\) and \(9.832\).
Solution:
Add whole numbers only: \(14 + 9 = 23\)
Now, add \(2\) more with whole numbers: \(14 + 9 + 2 = 25\)
Let's add the actual decimals: \(14.246 + 9.832 = 24.078\)
Here, the sum will be more than \(23\) and less than \(25\).
Therefore, the actual sum will always land between the sum of the whole parts, and the whole parts plus \(2\).
Decimal Sequence:
A decimal sequence is a collection of decimal numbers that follow a particular pattern or rule.
Similar to whole number sequences, decimal sequences can increase or decrease by the same amount or by a pattern.
Types of Decimal Sequences
1. Increasing Sequence
Each term is greater than the previous term.
Example: \(7.6\), \(7.9\), \(8.2\), ...
2. Decreasing Sequence
Each term is smaller than the previous term.
Example: \(8.4\), \(8.2\), \(8.0\), ...
3. Alternating Sequence
A sequence where the numbers increase and decrease in a regular pattern.
Example: \(8.4\), \(8.2\), \(8.0\), ...
How to find the next set of numbers in decimal sequences?
- Observe the changes in the numbers.
- Evaluate differences between the terms.
- Find the next numbers using the same rule.