0Introduction to Counting Numbers
Counting is required to represent a collection of objects with an exact numeral quantity.It helps to identify the larger collection in a group of collections.
Comparing numbers
Numbers are compared to check which one is higher/smaller than the others. Following things are checked to know that a number is greater or smaller:
If the number of digits in the numbers are different. The number having more digits is greater and the other is smaller.
- For example, among the two numbers 324 and 22, 324 is higher as it has more number of digits. 22 has lower number of digits, hence it is smaller.
- For example, among 221, 34, 1356, 222, 45225, 45225 is the highest and 34 is the lowest.
If the number of digits is equal then the digit at the highestplace is compared.
- If the digits at the highest place are different, the higher value is larger number and the lower value is the smaller number.
- For example, among 235 and 643, the number of digits are same but digit at highest (here hundreds) place, is 2 and 6. Since 6 is higher than 2, hence 643 is higher and 235 is smaller.
- If the digits at the highest place are equal, then the next higher place is compared and so on.
- For example, among 235 and 245, the number of digits and digit at highest place are same so digit at 2nd highest (tens) place is compared. Since 4 is higher than 3, hence 245 is higher and 235 is smaller.
- For example, among 267542, 267894 and 267843, the number of digits and digits at 4 highest places are same (for 2nd and 3rd number) so digit at 5th highest place is compared. Since 9 is higher than 4, hence 267894 is higher than 267843 and 267542.
Problem: Find the greatest and smallest numbers.
Solution: The green and yellow marked digits are qualifying digits for greatest and smallest numbers respectively. When number of digits are equal, then digits at ten thousands, thousands etc are compared until we get a highest and lowest digit. When number of digits is unequal, the number having more digits than all others is greatest and the number having lowest number of digits than all others is smallest.
| Values | Number of digits | Number at Ten Thousands place | Number at Thousands Place | Number at Hundreds place | Number at Tens place | Number at Units Place | Greatest and Smallest Number |
| (a) 4536, 4892, 4370, 4452 | All have 4 digits | No | 4,4,4,4 | 5,8,3,4 | Not Reqd | Not Reqd | 4892- G 4370 - S |
| (b) 15623, 15073, 15189, 15800 | All have 5 digits | 1,1,1,1 | 5,5,5,5 | 6,0,1,8 | Not Reqd | Not Reqd | 15800- G 15073 -S |
| (c) 25286, 25245, 25270, 25210 | All have 5 digits | 2,2,2,2 | 5,5,5,5 | 2,2,2,2 | 8,4,7,1 | Not Reqd | 25286 -G 25210 -S |
| (d) 6895, 23787, 24569, 24659 | 3 have 5 digits and 1 has 4 digits | 0,2,2,2 | 3,4,4 (comparing only last 3 values) | 5,6 (comparing only last 2 values) | Not Reqd | Not Reqd | 24659 -G 6895 -S |
Making numbers from individual digits
When we have a few single digits, a variety of numbers can be formed by arranging the digits in different orders.To make a new number from existing, shift places of digits.
Eg. 1357 can be made as 5731, 7351, 5317, 1735 etc. by shifting the digits.
To make largest number from a given number of digits,
- Keep the largest digit at the highest place.
- Keep the second largest digit at the second highest place and so on.
To make smallest number from a given number of digits,
- Keep the smallest digit at the highest place.
- Keep the second smallest digit at the second highest place and so on.
Problem: Use given digits without repetition and make smallest and greatest 4-digit numbers.
Solution: Since a 4 digit number is to be made, 0 cannot be put at the highest place as it will make the number a 3 digit number. So, if there is a 0 in 4 digits, put the third largest number at the highest place to make the smallest 4 digit number.
| Values | Largest Digit | Second Largest Digit | Third Largest Digit | Fourth Largest Digit | Smallest Number | Greatest Number |
| (a) 2,8,7,4 | 8 | 7 | 4 | 2 | 2478 | 8742 |
| (b) 9,7,4,1 | 9 | 7 | 4 | 1 | 1479 | 9741 |
| (c) 4,7,5,0 | 7 | 5 | 4 | 0 | 4057(Since 0457 is a 3 digit number) | 7540 |
| (d) 1,7,6,2 | 7 | 6 | 2 | 1 | 1267 | 7621 |
| (e) 5,4,0,3 | 5 | 4 | 3 | 0 | 3045 (Since 0345 is a 3 digit number) | 5430 |
Ordering the numbers
Random Numbers can be arranged in two orders:
- Ascending: Here numbers are arranged in smallest to largest order.
- Descending:Here numbers are arranged in largest to smallest order.
Problem: Arrange the following numbers in ascending and descending order.
Solution:
| Values | Ascending | Descending |
| (a) 847, 9754, 8320, 571 | 571, 847, 8320, 9754 | 9754, 8320, 847, 571 |
| (b) 9801, 25751, 36501, 38802 | 9801, 25751, 36501, 38802 | 38802, 36501, 25751, 9801 |
| (c) 5000, 7500, 85400, 7861 | 5000, 7500, 7861, 85400 | 85400, 7861, 7500, 5000 |
| (d) 1971, 45321, 88715, 92547 | 1971, 45321, 88715, 92547 | 92547, 88715, 45321, 1971 |
Increase in number of digits by adding 1
When 1 is added to the highest number of a n-digits, the result will be lowest number of n+1 digits. For eg,
| Number of Digits | Highest Number | After adding 1 | New number of digits | Terminology or Number name |
| 1 | 9 | 10 | 2 | Ten or 10x1 |
| 2 | 99 | 100 | 3 | Hundred or 10x10 |
| 3 | 999 | 1000 | 4 | Thousand or 10x100 |
| 4 | 9999 | 10000 | 5 | Ten Thousand or 10x1000 |
| 5 | 99999 | 100000 | 6 | Lakh or 10 x 10000 |
| 6 | 999999 | 1000000 | 7 | Ten Lakh or 10 x 100000 and so on |
| 7 | 9999999 | 10000000 | 8 | Crore or 10 x 1000000 |
Expanding numbers and place values
More than 1 digit numbers can be expanded by multiplying the individual digits with multiples of 10. The multiplication factor of 10 represents the digit’s place in the number. Foreg.
- 56 can be written as 50 + 6 = 5 x 10 + 6
- 8324 can be expanded as 8000 + 300 + 20 + 4 = 8 x 1000 + 3 x 100 + 2 x 10 + 4
- 36135 can be expanded as 30000 + 6000 + 100 + 30 + 5 = 3 x 10000 + 6 x 1000 + 1 x 100 + 3 x 10 + 5. Since the number 3 is multiplied by 10000 (Ten Thousand), it is said to be at Ten Thousands place. Similarly other places are shown below.
- 243677 can be expanded as 200000 + 40000 + 3000 + 600 + 70 + 7 = 2 X 100000 + 4 x 10000 + 3 x 1000 + 6 x 100 + 7 x 10 + 7
- 35585004 can be expanded as 30000000 + 5000000 + 500000 + 80000 + 5000 + 4 = 3 x 10000000 + 5 x 1000000 + 5 x 100000 + 8 x 10000 + 5 x 1000 + 4
Using Commas
Commas are used while reading and writing large numbers. In Indian System of Numeration, Commas are used to mark thousands, lakhs and crores.
- The first comma comes after hundreds place (three digits from the right) and marks thousands.
- The second comma comes two digits later (five digits from the right). It comes after ten thousands place and marks lakh.
- The third comma comes after another two digits (seven digits from the right). It comes after ten lakh place and marks crore.
For eg. 68537954 can be written as 6,85,37,954.
| Problem: Read the numbers. Write them using placement boxes and put commas according to Indian and International System of Numeration. | |||
| Number | Indian Numeration | International Numeration | Number Name |
| 527864 | 5,27,864 | 527,864 | INDIAN: Five Lakh Twenty Seven Thousand Eight Hundred Sixty Four INTERNATIONAL: Five Hundred Twenty Seven Thousand Eight Hundred Sixty Four |
| 95432 | 95,432 | 95,432 | Ninety Five Thousand Four Hundred Thirty two (same in both numeration systems) |
| 18950049 | 1,89,50,049 | 18,950,049 | INDIAN: One Crore Eighty Nine Lakh Fifty Thousand Forty Nine INTERNATIONAL: Eighteen Million Nine Hundred Fifty Thousand Forty Nine |
| 70002509 | 7,00,02,509 | 70,002,509 | INDIAN: Seven Crore Two Thousand Five Hundred Nine INTERNATIONAL: Seventy Million Two Thousand Five Hundred Nine |
Measuring Large Numbers or Quantities
We use units to measure large quantities.
| Type | Quantity | Conversion | Used to Measure |
| Length | Millimeter (mm) | N/A | eg. thickness of pencil |
| Length | Centimeter (cm) | 1 cm = 10 mm | eg. length of a pencil |
| Length | Meter (m) | 1 m = 100 cm | eg. length of a classroom |
| Length | KiloMeter (km) | 1 km = 1000 m | eg. distance between cities |
| Weight | Gram (g) | N/A | eg. weight of a mobile phone |
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