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Unit: Number System
Chapter: Playing with Digits
Reference: – Numbers in General Form, Games with Numbers, Reversing Two-Digit Numbers, Reversing Three-Digit Numbers, Forming Three-Digit Numbers with Given Digits, Letters for Digits, Tests of Divisibility (by 2, 3, 4, 5, 6, 8, 9, 10, 11), Common Factors and Common Multiples, Prime Factorization, HCF and LCM
After studying this chapter, you should be able to understand:
- Numbers in General Form
- Games with Numbers & Digit Reversal
- Letters for Digits (Cryptarithmetic)
- Tests of Divisibility
- Common Factors & Common Multiples
Knowing Our Numbers
Definitions
- Natural numbers are simply counting numbers 1, 2, 3, 4, …… etc with the smallest number starting from 1 and there is no largest number.
- Numbers are formed using the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These symbols are called digits or figures.
- Face value of a digit in a number is the digit itself.
- Place value of a digit in a number depends upon the place it occupies in the place value chart.
Place value of a digit in a number
= Face Value of the Digit × Value of the place it occupies in the place value chart
Comparison of numbers
- When two numbers have different number of digits:
The number with more number of digits is greater than the other one.
Example: Compare 4,98,726 and 57,46,225
So, 4,98,726 < 57,46,225.
- When two numbers have same number of digits
For numbers with same digits, we start comparing the digits with leftmost place in both numbers. The number with greater digit in the same place is greater.
Example:
Compare 52,62,781 and 52,26,781
Both are 7-digit numbers. We compare both the numbers starting from leftmost place and compare the digits of the two numbers. We see
6 Ten Thousands > 2 Ten Thousands
So, 52,62,781 > 52,26,781.
Greatest and Smallest Numbers
- The greatest single-digit number is 9. When 1 is added to the greatest single-digit number, we get 10, which is the smallest two-digit number.
∴ the greatest single-digit number + 1 = the smallest two-digit number.
- The greatest two digit-number is 99. When 1 is added to the greatest two-digit number, we get 100, which is the smallest three-digit number.
∴ the greatest two-digit number + 1 = the smallest three-digit number.
- The greatest three-digit number is 999. When 1 is added to the greatest three-digit number, we get 1000, which is the smallest four-digit number. ∴ the greatest three-digit number + 1 = the smallest four-digit number.
- The greatest four-digit number is 9999. When 1 is added to the greatest four-digit number, we get 10,000, which is the smallest five-digit number.
∴ the greatest four-digit number + 1 = the smallest five-digit number.
- The greatest five-digit number is 99999. When 1 is added to the greatest five-digit number, we get 1,00,000, which is the smallest six digit number.
∴ the greatest five-digit number + 1 = the smallest six-digit number.
This smallest six-digit number is called one-lakh.
- The greatest six-digit number is 999999. When 1 is added to the greatest six-digit number, we get 10,00,000, which is the smallest seven-digit number.
∴ the greatest six-digit number + 1 = the smallest seven-digit number.
This smallest six-digit number is called ten-lakh.
- The greatest seven-digit number is 9999999. When we add 1 to this seven-digit number, we get 10000000, which is the smallest eight-digit number.
∴ the greatest seven-digit number + 1 = the smallest eight-digit number.
This smallest six-digit number is called one-crore.
Indian System of Numeration
In this system, we start grouping the number from right in group of 3 and further in group of 2. Starting from the right, the first period is ones, consisting of three place values i.e. ones, tens and hundreds. The next period is thousands, consisting of two places i.e. thousands and ten-thousands. The third period is lakhs consisting of two places i.e. lakhs and ten-lakhs and then crores and so on.
Source: Kapdec.com
- Commas in Indian system of numeration
Commas are placed between the numbers so that we can read and write large numbers easily. As per the Indian system of numeration, the first comma is placed after the hundreds place. Further commas are placed after every two digits.
Examples-
- 98,75,747
This number can be read as ninety-eight lakhs seventy-five thousand seven hundred and forty-seven.
- 4,67,79,056
The number can be read as four crores sixty-seven lakh seventy-nine thousand and fifty-six.
The International Place Value System
In this system, we start grouping the number from right in group of 3 called period. Starting from the right, the first period is ones, consisting of three place values i.e. ones, tens and hundreds. The next period is thousands, consisting of three places i.e. thousands, ten-thousands and hundred-thousands. The third period is millions consisting of three places i.e. millions, ten-millions and hundred-millions and then billions, trillions and so on.
Source: Kapdec.com
- Commas in International system of Numeration
As per International numeration system, the first comma is placed after the hundreds place. Commas are then placed after every three digits.
Examples-
- 9,875,747
The number can be read as nine million eight hundred seventy-five thousand seven hundred and forty-seven.
- 46,779,056
The number can be read as forty-six million seven hundred seventy-nine thousand and fifty-six.
Comparison between Indian & International Place Value System
The major difference between two systems of breaking up numbers is the position of placing commas for grouping numbers into periods.
- In Indian number system, we start grouping the number from right in group of 3 and further in group of 2 but in International system, we start grouping the number from right in group of 3 called period.
- Also, in the Indian number system we use terms like ones, tens, hundreds, thousands, lakhs and crores whereas in the International number system we use terms like ones, hundreds, thousands, millions and so on.
Let us look at the table below to compare the two systems:
Source: Kapdec.com
We can observe from the table that:
- 10 lakhs = 1 million
- 1 crore = 10 millions
- 10 Crores = 100 millions
Comparing Numbers
Ascending order
The arrangement of numbers from the smallest to the greatest is called ascending order.
Example-
3459 < 3769 < 4897 < 6663 < 7683
Descending order
The arrangement of numbers from the greatest to the smallest is called descending order.
Example-
7683 > 6663 > 4897 > 3769 > 3459
Formation of Numbers
When there are few single digits, we can form a variety of numbers by arranging the digits in different orders. Different numbers can be formed using the digits with or without repetition of digits.
Formation of Numbers without Repetition of Digits
Suppose we have four digits 6, 8, 9, 5. We want to form different
4-digit numbers using these digits such that there is no repetition of digits.
# Which is the greatest and the smallest number that we can form using these digits without repetition?
To form the greatest number from a given set of digits without repetition of digits-
- The largest digit is kept at highest place i.e. the first place starting left.
- The second largest digit is kept at the second highest place i.e. the second place from left and so on.
In the above given set of digits, the largest digit is 9 followed by 8, 6 and 5 respectively. So, the greatest number that can be formed using these digits without repetition is 9865.
To form the smallest number from a given set of digits without repetition of digits-
- The smallest digit is kept at highest place i.e. the first place starting left.
- The second smallest digit is kept at the second highest place i.e. the second place from left and so on.
In the above given set of digits, the smallest digit is 5 followed by 6, 8 and 9 respectively. So, the smallest number that can be formed using these digits without repetition is 5689.
# Do Remember
- If one of the digits is ‘0’, we keep 0 in the second highest place and the second smallest digit in the highest or the first place starting left.
- For example, suppose we have 4 digits 9, 0, 3 and 7. The smallest digit here is 0 followed by 3, 7 and 9 respectively. The smallest 4 digit number that can be formed using these digits would be 3079 and not 0379 as 0379 would become a 3 digit number.
Formation of Numbers with Repetition of Digits
Suppose we have three digits 3, 7, 0. We want to form different 4-digit numbers using these digits such that there is repetition of digits.
Which is the greatest and the smallest number that we can form using any one of these digits twice along with the remaining digits?
To form the greatest number from a given set of digits with repetition of digits–
- In the above given set of digits, the largest digit is 7 followed by 3 and 0 respectively. Now, since we need to form greatest number by repeating any one of the digit, we will repeat 7 since it is the largest of all. So, the greatest number that can be formed using these digits with repetition is 7730.
To form the smallest number from a given set of digits without repetition of digits–
- In the above given set of digits, the smallest digit is 0 followed by 3 and 7 respectively. Now, since we need to form smallest number by repeating any one of the digit, we will repeat 0 since it is the smallest of all. So, the smallest number that can be formed using these digits with repetition is 3007.
Shifting the Digits
- What would happen if we shift the digits in a number from one place to another? A largest number could become the smallest one given the set of digits.
- Suppose we interchange the digits at thousands place and units place in the number 6009. Is the new number greater, smaller or equal to the former number?
- The new number would be 9006 which would be greater than the original number.
Fun Fact
- The name of the popular search engine ‘Google’ came from a misspelling of the word ‘googol’, which is a very large number (the number one followed by one hundred zeros to be exact)
- Even more zeroes- A ‘googolplex’ is the number 1, followed by a googol zeros, a number so ridiculously big that it can’t be written because there literally isn’t enough room in the entire universe to fit it in!
