Ordering

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Understanding Inequality Symbols

There are six important symbols you need to know and use when comparing numbers:

= (equals)

• This means two values are exactly the same
• Example: 5 = 5, or 0.5 = ½

≠ (does not equal)

• This means two values are different
• Example: 7 ≠ 8, or 0.3 ≠ ⅓

> (greater than)

• This means the first number is larger than the second number
• The symbol points to the smaller number
• Example: 10 > 3 (read as "10 is greater than 3")

< (less than)

• This means the first number is smaller than the second number
• The symbol points to the smaller number
• Example: 4 < 9 (read as "4 is less than 9")

⩾ (greater than or equal to)

• This means the first number is either larger than or equal to the second number
• Example: 5 ⩾ 5 (true, because 5 equals 5) or 7 ⩾ 5 (true, because 7 is greater than 5)

⩽ (less than or equal to)

• This means the first number is either smaller than or equal to the second number
• Example: 3 ⩽ 3 (true, because 3 equals 3) or 2 ⩽ 3 (true, because 2 is less than 3)

Helpful Tip: Think of the inequality symbols < and > as a mouth that always wants to eat the bigger number. The open end (the mouth) always faces the larger value.

Comparing and Ordering Whole Numbers

When comparing whole numbers, look at the number of digits first:

• A number with more digits is always larger (unless it's negative)
• Example: 543 > 98 because 543 has three digits while 98 has only two digits

If two numbers have the same number of digits, compare them digit by digit from left to right:

• Start with the leftmost digit (the highest place value)
• Move right until you find a difference

Example: Compare 4,567 and 4,521

• Both start with 4, so look at the next digit
• The second digit is 5 in both, so look at the next digit
• The third digit is 6 in the first number and 2 in the second
• Since 6 > 2, we know 4,567 > 4,521

Comparing and Ordering Decimals

When comparing decimals, line up the decimal points and compare digit by digit from left to right.

Important: Numbers further to the right of the decimal point are worth less, not more!

• Example: 14 is more than 8, but 0.14 is less than 0.8

To make comparison easier:

• Add zeros to make the numbers have the same number of decimal places
• Example: To compare 3.7 and 3.65, rewrite as 3.70 and 3.65
• Now it's clear that 3.70 > 3.65

Step-by-step method:

1. Line up the decimal points, writing numbers underneath each other
2. Add zeros if needed so all numbers have the same number of decimal places
3. Compare the whole number part first (to the left of the decimal point)
4. If the whole numbers are equal, compare the decimal parts digit by digit from left to right

Example: Order these from smallest to largest: 2.5, 2.18, 2.499

Rewrite with same decimal places:

• 2.500
• 2.180
• 2.499

Compare: All have 2 as the whole number. Look at the first decimal place: 5, 1, and 4. Since 1 < 4 < 5, the order is: 2.18, 2.499, 2.5

Ordering Mixed Types (Fractions, Decimals, Percentages, Negatives)

When you need to order a mixture of fractions, decimals, percentages, and negative numbers:

Step 1: Convert everything to decimals (this is usually the easiest common form)

• For fractions: divide the top by the bottom
• For percentages: divide by 100
• For whole numbers and negatives: write as decimals by adding .0

Step 2: If there are negative numbers, separate them from the positive numbers first and order each group separately

Step 3: Write all decimals with the same number of decimal places (add zeros if needed)

Step 4: Compare using place value, digit by digit from left to right

Step 5: List the numbers in order, starting with the most negative (smallest) and ending with the most positive (largest)

Example: Order these from smallest to largest: 0.45, ⅖, 48%, -0.3

Convert to decimals:

• 0.45 stays as 0.45
• ⅖ = 2 ÷ 5 = 0.4
• 48% = 48 ÷ 100 = 0.48
• -0.3 stays as -0.3

Rewrite with same decimal places: -0.30, 0.40, 0.45, 0.48

Order: -0.3, ⅖, 0.45, 48%