Comparing Numbers

Comparing numbers means finding which number is greater, which is smaller, or whether both numbers are equal.

We compare numbers in daily life also, for example:

Your marks = 87
Your friend’s marks = 91

Here, 91 is greater than 87, so your friend scored more.

Symbols Used in Comparing Numbers

Symbol	Meaning	Example	Read As
>	Greater than	45>32	45 is greater than 32
<	Less than	18<25	18 is less than 25
=	Equal to	50=50	50 is equal to 50

Comparing 1, 2, 3 digit numbers

One-digit numbers are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Example:

7 > 4
2 < 9
5 = 5

The number that comes later while counting is greater.

Two-digit numbers have tens and ones.

Example: 46 and 52

Number	Tens	Ones
46	4	6
52	5	2

First compare the tens digit. 5 tens is greater than 4 tens. So, 52 > 46

Three-digit numbers have: Hundreds, Tens, Ones

Example: 375 and 348

Number	Hundreds	Tens	Ones
375	3	7	5
348	3	4	8

Step 1: Compare hundreds
Both have 3 hundreds.

Step 2: Compare tens
7 tens is greater than 4 tens.

So, 375 > 348

Comparing Large Numbers

For large numbers, follow these rules:

Rule 1: Count the number of digits

The number with more digits is greater.

Example:

8,765 and 76,543

8,765 has 4 digits.
76,543 has 5 digits.

So,

76,543 > 8,765

Rule 2: If digits are equal, compare from left to right

Example: 45,678 and 45,879

Both numbers have 5 digits. Compare digit by digit:

Place	45,678	45,879
Ten-thousands	4	4
Thousands	5	5
Hundreds	6	8

At hundreds place, 8 is greater than 6.

So, 45,879 > 45,678

Place Value and Comparing Numbers

Place value helps us understand the value of each digit in a number.

Example: In 6,482:

Digit	Place	Place Value
6	Thousands	6000
4	Hundreds	400
8	Tens	80
2	Ones	2

When comparing numbers, start from the highest place value.

Example: 6,482 and 6,428

Both have 6 thousands and 4 hundreds. Now compare tens:

8 tens > 2 tens

So, 6,482 > 6,428

Comparing Numbers Using Number Line

A number line helps us compare numbers visually.

Numbers on the right side are greater.
Numbers on the left side are smaller

Forming the Greatest and Smallest Number

A) To form the greatest number from given digits, arrange the digits in descending order.

Example: Digits: 5, 2, 9, 1

Greatest number: 9521

Because 9 > 5 > 2 > 1

B) To form the smallest number from given digits, arrange the digits in ascending order.

Example: Digits: 5, 2, 9, 1

Smallest number: 1259

Because 1 < 2 < 5 < 9

C) Forming the Smallest Number When Zero Is Given

Zero cannot be placed at the beginning of a number.

Example:

Digits: 0, 4, 7, 2

Wrong smallest number: 0247
Because 0247 is actually 247.

Correct method:

Put the smallest non-zero digit first, then zero, then remaining digits.

Digits in order: 2, 0, 4, 7

Smallest number: 2047

Greatest number: 7420

Comparing Decimal Numbers

A) Decimal numbers have a whole number part and a decimal part. Example: 4.5 and 4.8

Whole number part is same: 4

Now compare decimal part:

5 tenths < 8 tenths

So, 4.5 < 4.8

B) Comparing Decimals with Different Digits

Example: 6.25 and 6.3

Write 6.3 as 6.30

Now compare: 6.25 and 6.30

25 hundredths < 30 hundredths

So, 6.25 < 6.3

Important Rule : Adding zeros after decimal does not change the value.

Example:

5.6 = 5.60 = 5.600

Comparing Fractions

Fractions are usually written as:$$\frac{a}{b}$$

where a and b are whole numbers, and $b \neq 0$

In school level, fractions are mostly treated as positive parts of a whole.

Fractions are numbers like: 1/2, 3/4, 5/8

Case 1: Same Denominator

If denominators are same, compare numerators.

Example:

3/7 and 5/7

5 > 3

So,

5/7 > 3/7

Case 2: Same Numerator

If numerators are same, the fraction with the smaller denominator is greater.

Example:

1/3 and 1/5

Imagine one pizza divided into 3 parts and another into 5 parts. One part out of 3 is bigger than one part out of 5.

So, 1/3 > 1/5

Case 3: Different Numerators and Denominators

Use cross multiplication.

Example: 3/4 and 5/6

Cross multiply:

3 × 6 = 18
5 × 4 = 20

Since 20 > 18,

5/6 > 3/4

Comparing Rational Numbers

Rational numbers are numbers that can be written in the form:$$\frac{p}{q}$$​

where $p$ and $q$ are integers, and $q \neq 0$

Examples:$$\frac{2}{3}, -\frac{5}{7}, \frac{0}{4}, \frac{9}{1}, 2.5, -3.75$$

A rational number can be positive, negative, or zero. Therefore, while comparing rational numbers, we must carefully observe the sign, numerator, and denominator.

Basic Rules for Comparing Rational Numbers

Rule 1: Positive rational number is always greater than negative rational number

Example:$$\frac{3}{5} > \frac{-7}{5}$$

because every positive number is greater than every negative number.

Rule 2: Zero is greater than every negative rational number

Example:$$0 > \frac{-5}{8}$$

Rule 3: Zero is smaller than every positive rational number

Example:$$0 < \frac{4}{9}$$

Rule 4: Among negative rational numbers, the number closer to zero is greater

Example:$$\frac{-2}{7} > \frac{-5}{7}$$

because $\frac{-2}{7}$ is closer to zero than $\frac{-5}{7}$

Methods of Comparing Rational Numbers

There are different methods to compare rational numbers.

Method 1: Comparing Rational Numbers with Same Denominator

When denominators are same, compare the numerators. The rational number with the greater numerator is greater.

Example 1

Compare:$$\frac{4}{9} \text{ and } \frac{7}{9}$$

Both have the same denominator 9.

Now compare numerators:$$4 < 7$$

Therefore,$$\frac{4}{9} < \frac{7}{9}$$

Example 2

Compare:$$\frac{-3}{11} \text{ and } \frac{-8}{11}$$

Both have the same denominator 11.

Now compare numerators:$$-3 > -8$$

Therefore,$$\frac{-3}{11} > \frac{-8}{11}$$

Method 2: Comparing Rational Numbers with Same Numerator

When two positive rational numbers have the same numerator, the number with the smaller denominator is greater.

Example

Compare:$$\frac{5}{6} \text{ and } \frac{5}{9}$$

Both have numerator 5.

Since:$$6 < 9$$

Therefore,$$\frac{5}{6} > \frac{5}{9}$$

Reason

If a whole is divided into fewer parts, each part is bigger.

Negative Case

Compare:$$\frac{-5}{6} \text{ and } \frac{-5}{9}$$

First compare the positive values:$$\frac{5}{6} > \frac{5}{9}$$

But both numbers are negative. So, the number with greater numerical value becomes smaller.

Therefore,$$\frac{-5}{6} < \frac{-5}{9}$$

Method 3: Comparing by Making Denominators Same

When denominators are different, we can make them same by using LCM.

Steps

1. Find the LCM of the denominators.
2. Convert both rational numbers into equivalent rational numbers with the same denominator.
3. Compare the numerators.

Example 1

Compare:$$\frac{2}{3} \text{ and } \frac{3}{5}$$

LCM of 3 and 5 is:$$15$$

Denominators are 3 and 5.

Now convert both rational numbers:

$$\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$$

$$\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$$

Now compare:$$\frac{10}{15} > \frac{9}{15}$$

Therefore,$$\frac{2}{3} > \frac{3}{5}$$

Example 2

Compare:$$\frac{-4}{7} \text{ and } \frac{-2}{3}$$

Denominators are 7 and 3.

LCM of 7 and 3 is:$$21$$

Now convert:

$$\frac{-4}{7} = \frac{-4 \times 3}{7 \times 3} = \frac{-12}{21}$$

$$\frac{-2}{3} = \frac{-2 \times 7}{3 \times 7} = \frac{-14}{21}$$

Now compare:$$-12 > -14$$

Therefore,$$\frac{-4}{7} > \frac{-2}{3}$$

Method 4: Comparing by Cross Multiplication

Cross multiplication is a quick method to compare rational numbers.

For two rational numbers:$$\frac{a}{b} \text{ and } \frac{c}{d}$$

where $b$ and $d$ are positive denominators, compare:$$a \times d \text{ and } c \times b$$

Example 1

Compare:$$\frac{5}{6} \text{ and } \frac{7}{9}$$

Cross multiply:

$$7 \times 6 = 42$$

$$5 \times 9 = 45$$

Since:$$45 > 42$$

Therefore,$$\frac{5}{6} > \frac{7}{9}$$

Example 2

Compare:$$\frac{-3}{4} \text{ and } \frac{-5}{6}$$

Cross multiply:

$$-5 \times 4 = -20$$

$$-3 \times 6 = -18$$

Since:$$-18 > -20$$

Therefore,$$\frac{-3}{4} > \frac{-5}{6}$$