Numbers are the foundation of mathematics and our daily life. When we count objects, measure distances, read a calendar, or check the population of a city, we use numbers. The numbers that help us count things are called natural numbers. When we add zero to natural numbers, we get whole numbers. In this chapter, we learn how to read and write very large numbers, compare them, arrange them, estimate them, and round them off. These skills are used in everyday situations such as reading the price of a car (which may be in lakhs) or the population of a country (which may be in crores).
The ancient Indian numeral system developed the place value idea, where the position of a digit in a number decides its value. The same digit 5 means five, fifty, or five thousand depending on where it is placed. This idea of place value, along with the decimal system that uses ten symbols (0 to 9), allows us to express numbers as large as we want. In this chapter, we also study Indian and International systems of numeration, comparison of numbers using greater than and less than signs, and the concepts of ascending and descending order.
Estimation and approximation are also important in this chapter. We cannot always measure or count exactly, so we round numbers to the nearest ten, hundred, or thousand to make quick calculations. The chapter also introduces the use of brackets (BODMAS rule) for simplifying expressions, Roman numerals, and the process of finding the largest and smallest numbers under given conditions.
When we compare two numbers, we first look at the number of digits. The number with more digits is always the greater number. For example, 4,521 has four digits while 876 has only three digits, so 4,521 is greater than 876.
If two numbers have the same number of digits, we compare the digits from the leftmost position. The number having the larger digit at the leftmost (highest) place is greater. If those digits are equal, we move to the next digit to the right, and so on. For example, 8,459 and 8,562 both have four digits. Compare the thousands digit: both are 8. Compare the hundreds digit: 5 and 5 again equal. Compare the tens digit: 4 and 6. Since 6 is greater than 4, we conclude that 8,562 is greater than 8,459.
The symbols used for comparison are the greater than sign (>) and the less than sign (<). For example, 23,456 > 23,455 and 987 < 1,000.
Arranging numbers from the smallest to the largest is called ascending order. Arranging numbers from the largest to the smallest is called descending order.
For example, the numbers 5, 2, 9, 1 arranged in ascending order are 1, 2, 5, 9 and in descending order are 9, 5, 2, 1. To arrange numbers, first compare them using the rules above, then place them in order.
In the Indian system, large numbers are grouped as ones, thousands, lakhs and crores. Starting from the right, the groups are: ones (3 digits), thousands (2 digits), lakhs (2 digits), crores (2 digits). For example, the number 4,56,72,913 has 4 crores, 56 lakhs, 72 thousands and 913 ones. It is read as "four crore fifty-six lakh seventy-two thousand nine hundred thirteen".
In the International system, numbers are grouped in threes: ones, thousands, millions and billions. The same number 45,672,913 is read as "forty-five million six hundred seventy-two thousand nine hundred thirteen". The digit 9,99,99,999 is nine crore ninety-nine lakh ninety-nine thousand nine hundred ninety-nine in the Indian system.
| Crores | Lakhs | Thousands | Ones |
|---|---|---|---|
| TC | C | TL | L |
| 10 Crore | Crore | Ten Lakh | Lakh |
The place value of a digit = digit value multiplied by the place it occupies. For example, in 4,56,72,913 the digit 5 is in the ten lakh place, so its place value is 5 x 10,00,000 = 50,00,000.
Real-life situations involve very large numbers. The population of India is counted in crores. Distances in the solar system are measured in lakhs of kilometres. The price of a house may be a few crores of rupees. When we read such numbers, we use commas correctly: in the Indian system a comma is placed after every two digits after the first three digits from the right. For example, 5,08,015 is read as five lakh eight thousand fifteen.
The number just after a given number is called its successor. The successor of a number is obtained by adding 1. For example, the successor of 99,999 is 1,00,000. The number just before a given number is called its predecessor. The predecessor is obtained by subtracting 1. The predecessor of 10,000 is 9,999.
Often we do not need exact numbers, only a good approximation. Estimation means finding a value close to the exact value.
To estimate a sum, we first round the numbers and then add. For example, 4,890 + 2,110 estimated to the nearest hundred gives 4,900 + 2,100 = 7,000. To estimate a product, we round to the highest place value. For example, 28 x 412 is estimated as 30 x 400 = 12,000.
Brackets are used to give priority to certain operations in an expression. For example, in the expression 6 + 3 x 2, if we add first we get 18, but if we multiply first we get 12. To remove this confusion we use brackets: (6 + 3) x 2 = 18.
The order of simplification is given by the BODMAS rule: - B stands for Brackets - O stands for Of - D stands for Division - M stands for Multiplication - A stands for Addition - S stands for Subtraction
First solve the brackets, then of, then division and multiplication (in order from left to right), and finally addition and subtraction (in order from left to right).
The Romans used seven basic symbols: I (1), V (5), X (10), L (50), C (100), D (500) and M (1000). Rules to write Roman numerals are: 1. If a smaller symbol appears after a larger symbol, it is added: VI = 6, XI = 11, LX = 60. 2. If a smaller symbol appears before a larger symbol, it is subtracted: IV = 4, IX = 9, XL = 40. 3. The same symbol cannot be repeated more than three times in a row. So 4 is IV, not IIII. 4. V, L and D are never repeated.
For example, the number 24 is written as XXIV = 10 + 10 + (5 - 1) = 24.
To form the largest number using given digits, write the digits in descending order. To form the smallest number, write them in ascending order. For example, using the digits 4, 1, 7, 9 the largest number is 9,741 and the smallest is 1,479. If zero is one of the digits, zero cannot be placed at the leftmost position in the smallest number. Using digits 4, 0, 7, 9 the smallest number is 4,079.
| Term | Meaning | Example |
|---|---|---|
| Natural numbers | Numbers used for counting (1, 2, 3, ...) | 1, 2, 3, 100 |
| Whole numbers | Natural numbers plus zero | 0, 1, 2, 3 |
| Successor | Number obtained by adding 1 | Successor of 999 is 1000 |
| Predecessor | Number obtained by subtracting 1 | Predecessor of 1000 is 999 |
| Place value | Value of a digit based on its position | In 5,321 the digit 3 has value 300 |
| Ascending order | Smallest to largest | 3, 8, 12, 25 |
| Descending order | Largest to smallest | 25, 12, 8, 3 |
| Rounding off to | Look at digit | Example |
|---|---|---|
| Nearest ten | Ones digit | 74 becomes 70 |
| Nearest hundred | Tens digit | 354 becomes 400 |
| Nearest thousand | Hundreds digit | 8,450 becomes 8,000 |
Worked examples help us apply all the rules of the chapter in a single problem. Consider this question: arrange the numbers 24,765, 24,756, 23,999, and 25,001 in ascending order, and then write the successor and predecessor of the largest number. To begin, we compare the numbers using the digit-count rule; all four have five digits, so we move to the leftmost digit. The first digit of each number is 2, so we compare the second digit. Here 23,999 has a 3 in the ten-thousands position, while 24,765 and 24,756 both have a 4, so 23,999 is the smallest. Comparing 24,765 and 24,756, the first four digits are identical, but at the tens place we have 6 and 5, so 24,756 is smaller than 24,765. Hence the ascending order is 23,999, 24,756, 24,765, 25,001. The largest number is 25,001, whose successor is 25,002 and whose predecessor is 25,000.
A second common type of problem asks us to round numbers before estimating. For example, estimate the sum of 6,849 and 3,126 by rounding each to the nearest hundred. The hundreds digit of 6,849 is 8 and the tens digit is 4, so 6,849 rounds down to 6,800. The tens digit of 3,126 is 2, so it rounds down to 3,100. The estimated sum is 6,800 + 3,100 = 9,900. Notice that the estimate is close to but not exactly equal to the true sum, which is what makes estimation useful when we need a quick answer. Estimation never replaces exact calculation; it simply helps us check whether a computed answer is reasonable and gives us a way to judge quantity quickly in everyday life.
A third important skill is using brackets and the BODMAS rule to simplify a single expression. Consider the expression (18 + 6) divided by (12 - 9). First we solve the brackets: 18 + 6 = 24 and 12 - 9 = 3. Then we divide: 24 divided by 3 is 8. If we had ignored the brackets and worked from left to right, we would have obtained a very different and incorrect result. This example shows why the order of operations must always be followed: brackets first, then of, division and multiplication, and finally addition and subtraction. Students who practise such mixed problems with brackets build the careful habits needed for all future arithmetic.
Knowing Our Numbers teaches us the language of large numbers that appear everywhere around us, from populations to distances and prices. We learnt to compare, order, estimate, round off and write numbers in Indian and International systems. The skills of using brackets, following the BODMAS rule and writing Roman numerals prepare us for more complex number work. Practising these concepts builds a strong foundation for all future chapters in mathematics.