Comparing and Ordering

Comparing numbers is used to determine the relationship between two or more numbers to check whether the numbers are greater, smaller, or equal to each other. To compare numbers, analyze the place values, align decimal points, or convert fractions to a common denominator are required. This concept is essential in everyday activities like ranking scores, measuring quantities, and managing finances.
 

There are symbols we use for comparing numbers. Some of these symbols are mentioned below:

• Greater Than ( > )
  Indicates that the number on the left is larger than the number on the right.
  Example: 8 > 5 (8 is greater than 5).

• Less Than ( < )
  Indicates that the number on the left is smaller than the number on the right.
  Example: 3 < 7 (3 is less than 7).

• Equal To ( = )
  Indicates that two values are the same.
  Example: 4 + 2 = 6 can be written as 6 = 6 (both sides are equal).

• Greater Than or Equal To ( ≥ )
  Indicates that the number on the left is either greater than or equal to the number on the right.
  Example: x ≥ 10 (x can be 10 or more).

• Less Than or Equal To ( ≤ )
  Indicates that the number on the left is either less than or equal to the number on the right.
  Example: y ≤ 5 (y can be 5 or less).

Comparing numbers means checking the number whether it is bigger or smaller or equal to each other. To compare numbers, steps need to be followed are - 

Step 1: Compare the Number of Digits:
Firstly, check which number has the most digits.

Step 2: Compare Place Values:
Then, compare the place values, starting from the leftmost digit to the rightmost digit of the numbers.

Step 3: Double-check and Verify:
After ordering, recheck and verify.

The above steps are used to compare whole numbers. Let us see what steps are used in comparing fractions and decimals.

Comparing Fractions:

There are two types of fractions: like fractions and unlike fractions. 
Let us see how to compare like fractions:

Step 1: Check if the denominators are the same.

Step 2: Compare the numerators directly. The fraction with the larger numerator is greater.

Now, let us understand how to compare unlike fractions:

Step 1: Find the LCD or Lowest Common Denominator of both fractions. 

Step 2: Convert each fraction to an equivalent fraction with the LCD

Step 3: Compare the new numerators. The fraction with the larger numerator is greater.

 Comparing Decimals:

These are the steps we use to compare decimals:

Step 1: Align the decimal points.

Step 2: Add zeros to make the same number of decimal places.

Step 3: Compare the digits from left to right

Step 4: Repeat the same process for more than two decimals.
 

Arranging the number from least to greatest is called ordering a list of numbers. There are two ways to order numbers:

• Ascending order

• Descending order

Let us see what they mean:

Ascending order:

Ascending order means to arrange numbers in increasing order, from smallest to largest. For example,
2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
The symbol used to show ascending order is “<”.

Descending order: 

Descending order means to arrange numbers in decreasing order, from largest to smallest. For example,
30, 27, 24, 21, 18, 15, 12, 9, 6, 3.
The symbol used to show descending order is “>”.
 

Comparing and ordering have numerous applications across various fields. We will now explore how comparing and ordering are used in different areas:

• Money Management and Budgeting: 
  Comparing and ordering numbers is essential in personal finance. When budgeting, people compare expenses, order bills by priority, and determine which purchases fit within their budget. 

• Shopping and Price Comparisons:
  Shopping for products, people compare prices to get the best deals. Online shopping platforms use price comparisons, discounts, or filter prices allowing customers to order products from the cheapest to the most expensive.

• Ranking and Grading in Education: 
  Teachers compare students’ test scores to rank them and determine performance levels. Universities also rank applicants based on grades and test scores for admissions.