Introduction to Numbers
Numbers are fundamental to our understanding of the world and are used in various ways in everyday life and academic subjects like mathematics. Let's have a breakdown of the key aspects you might find interesting:
Comparing numbers
Numbers are compared to check which one is higher/smaller than the others. The following things are checked to know that a number is greater or smaller
Different Number of Digits
- 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.
Equal No. of Digits
- If the number of digits is equal, then the digit at the highest place is compared.
- If the digits at the highest place are different, the higher value is a 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.
(i) For example, among 235 and 245, the number of digits and digit at highest place are same so digit at 2 highest (tens) place is compared. Since 4 is higher than 3, hence 245 is higher and 235 is smaller.
(ii) For example, among 267542, 267894 and 267843, the number of digits and digits at 4 highest places are same (for 2 and 3 number) so digit at 5 highest place is compared. Since 9 is higher than 4, hence 267894 is higher than 267843 and 267542.
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.
Making Largest Digit
To make the 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.
Making Smallest Digit
To make the
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.
Example: Use given digits without repetition and make smallest and greatest 4-digit numbers.
Sol: 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.
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.
Example: Arrange the following numbers in ascending and descending order.
Sol:
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: 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. For example: (i) 56 can be written as 50 + 6 = 5 x 10 + 6.(ii) 8324 can be expanded as 8000 + 300 + 20 + 4 = 8 x 1000 + 3 x 100 + 2 x 10 + 4.(iii) 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. Using Commas
Commas are used while reading and writing large numbers. (I) Indian System of NumerationIn this, 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.
(II) International System of Numeration
- The International System of Numeration organizes numbers into specific places, each with its own value.
- These places range from ones all the way up to millions, billions, and beyond.
- Each place, like ones, tens, hundreds, etc., represents a different quantity.
- For instance, 1 million equals 1000 thousands, and 1 billion equals 1000 millions.
Estimation of Numbers by Rounding off
- When dealing with very large numbers, we often approximate them to the closest reasonable value. This process is called estimation.
- Rounding off means changing the most insignificant value to its nearest zero figure. This makes the number more readable and easy for estimation
- Estimation involves quickly and roughly calculating the result of number operations by rounding off the numbers involved.
Rules of Estimation
- To round to the nearest tens, we consider numbers 1 to 4 as 0 and numbers 6 to 9 as 10.
- For rounding to the nearest hundred, numbers 1 to 49 become 0, while numbers 51 to 99 become 100.
Rounding to the nearest thousands involves considering numbers 1 to 499 as 0 and 501 to 999 as 1000
Estimating Sum/Difference
(i) Estimate 7890 + 437.
Here 7890 > 437.
Therefore, round off to hundreds.
7890 is rounded off to 7900
437 is rounded off to + 400
Estimated Sum = 8300
Actual Sum = 8327
(ii) Estimate 5678 – 1090.
Here 5678 > 1090.
Therefore, round off to thousands.
5678 is rounded off to 6000
1090 is rounded off to – 1000
Estimated Difference = 5000
Actual Difference = 4588
Estimating multiplication
Example: Estimating the product of 199 and 31:
Here, 199 is rounded off to 200
=31 is rounded off to 30
Estimated Product = 200 × 30 = 6000
Actual Result = 199 × 31 = 6169