Class 6 Maths Chapter 3 Number Play Notes
Class 6 Maths Chapter 3 Notes – Class 6 Number Play Notes
Numbers
Numbers are symbols used to represent quantities or values. They help us count, measure, and describe things in the world around us. With the help of the numbers, we all can add, subtract, divide, and multiply.
Numbers can Tell us Things
- Numbers are arithmetic values.
- Numbers are used to convey the magnitude of everything around us.
e.g. Let us suppose some children are standing in a park. Each one says a number in the sequence 0, 1, 1, 2, 1, 0, 2, 0.
- A child says 1 if there is only one taller child standing next to them.
- A child says 2 if both the children standing next to them are taller.
- A child says 0 if neither of the children standing next to them is taller.
- Thus, we conclude that each person says several taller neighbors they have.
Supercell
A cell is called a supercell if the number contained in it is larger than its adjacent cells.
e.g. Observe the numbers given in the table below.
Here, a cell is circled if the number in it is larger than its adjacent cells.
i.e. 646 is circled as it is larger than 566 and 355 where as 202 is not circled as it is smaller than 566.
The number 195 is circled as it has only one adjacent cell with 108 in it, and 195 is larger than 108.
Example 1.
Mark the supercells in the table below.
Solution:
Patterns of Numbers on the Number Line
Number Line:
A number line is the pictorial representation of numbers on a straight line. The pattern of numbers on a number line is a repeating pattern with equal spacing between the marks. Any number can be place their appropriate position on the line.
Example 2.
Mark the numbers, 1180, 1754, 1400, 2600, 8950, 2060, and 5050 on the number line.
Solution:
Digits
We start writing numbers from 1, 2, 3,…… and so on.
There are nine 1-digit numbers i.e. 1, 2, 3, 4, 5, 6, 7, 8, 9.
1-Digit Numbers:
1- digit numbers are integers from 0 to 9.
e.g. 2, 3, 4, 5, 9
2-Digit Numbers:
2-digit numbers are the numbers 10 to 99, which have digits in the tens and ones places.
e.g. 32, 65, 98
3-Digit Numbers:
3-digit numbers are the numbers 100 to 999, which have digits in the hundreds, tens, and ones places.
e.g. 541, 873, 929
4-Digit Numbers:
4-digit numbers are the numbers 1000 to 9999, which have digits in the thousands, hundreds, tens, and ones places.
e.g. 4321, 8754, 9024
5-Digit Numbers:
5-digit numbers are the numbers 10000 to 99999, which have digits in the ten thousand, hundreds, tens, and ones places.
Example 3.
Write the number of numbers having 1-digit, 2-digit, 3-digit, 4-digit, and 5-digit.
Solution:
1-digits – 9
2-digits – 90
3-digits – 900
4-digits – 9000
5-digits – 90000
Digit Sums of Numbers
There exist certain numbers such that when we add up the digits of these numbers, the sum is the same.
e.g. adding the digits of 45 and 117 or 324 will give the same sum 9.
Example 4.
The sum of the digits of the number 63 is 9. Write other numbers whose digits add up to 9.
Solution:
27, 36, 108, 333, 1008, 1233, etc adds up to 9.
Example 5.
How many times does the digit 6 occur among the numbers 1-100?
Solution:
1, 2,… 6, 7, …16,… 26,… 36,… 46…. 56, … 60, 61, 62, 63, 64, 65, 66, 67, 68, 69,…, 76,…86,…96.
Therefore, the digit 6 occurs 20 times among the numbers 1-100.
Palindromic Patterns
Palindromic Number
A number which remains the same when its digits are reversed. In other words, the numbers read the same from left to right and from right to left are called palindromes or palindromic.
e.g. 11, 121, 75257, 16461, 15251, etc are palindrome numbers.
Example 6.
Write some palindromes using digits 1, 4, and 5.
Solution:
The numbers 141, 151, 454, 515, 414, etc. are some palindromes using these digits.
Reverse and Add Palindromes
Steps: Begin with any number. Add the number to its reverse. If the result is a palindrome, stop. If not, repeat the process by reversing the digits of the result and adding.
Example 7.
Write Four 4-digits palindromes.
Solution:
1221, 5445, 6226, 9889
Example 8.
On a 12-hour clock, write the palindromic time between 4 o’clock and 5 o’clock.
Solution:
04:40
Example 9.
Write two dates that have palindrome.
Solution:
12/02/2021 and 02/02/2020
The Magic Number
In 1949, a mathematics teacher D.R. Kaprekar discovered an interesting property of the number 6174, when playing with any 4-digit numbers.
- Step 1: Take a 4-digit number having at least two different digits.
- Step 2: Make the largest number from these digits. Call it A.
- Step 3: Make the smallest number from these digits. Call it B.
- Step 4: Subtract B from A. Call it C.
C = A – B
e.g. Consider the 4-digit number 2368.
A =8632
B = 2368
C = 8632 – 2368 = 6264
A = 6642
B = 2466
C = 6642 – 2466 = 4176
A = 7641
B = 1467
C = 7641 – 1467 = 6174
Thus, we will always reach the magic number 6174.
This number 6174 is called as Kaprekar constant.
Mental Maths
Addition and Subtraction
Addition and Subtraction are arithmetic operations in mathematics that are used to calculate the sum and difference between different operands.
Example 10.
Add the following 5-digit numbers.
(i) 11,460 + 35,434 = ___________
(ii) 20,321 + 12,460 = ___________
Solution:
Example 11.
Subtract the following two 5-digit numbers.
(i) 84592 – 42745
(ii) 54542 – 36895
Solution:
Digits and Operations
Digits are used in various mathematical operations to perform calculations and problems.
Example 12.
Write an example for each of the statements given below.
(i) 5-digit + 5-digit to give a sum more than 80500.
(ii) 5-digit + 5-digit to give a 6-digit sum.
(iii) 3-digit + 3-digit to give a 5-digit sum.
(iv) 5-digit – 5-digit to give a 2-digit difference.
(v) 4-digit – 3-digit to give a 4-digit difference.
Solution:
(i) An example for the statement ‘5-digit + 5-digit to give a sum more than 80500’ is 42500 + 40000 = 82500.
(ii) An example for the statement ‘5-digit + 5-digit to give a 6-digit sum’ is 55000 + 50000 = 105000.
(iii) The maximum sum of two 3-digit numbers (999 + 999) is 1998. Thus, it is impossible to get a 5-digit number by adding two 3-digit numbers.
(iv) An example for the statement ‘5-digit – 5-digit to give a 2-digit difference’ is 95500 – 95450 = 50.
(v) An example for the statement ‘4-digit – 3-digit to give a 4-digit difference’ is 9999 – 999 = 9000.
Number Patterns
A pattern is a repeated arrangement of numbers, shapes, colors, and so on. The pattern can be related to any type of event or object. If the set of numbers are related to each other in a specific rule or manner is called a pattern. Patterns are also known as sequences.
Example 13.
Find out the sum of the numbers in each of the below figures.
Solution:
(i) 50 × 4 + 60 × 10 + 90 × 8 = 200 + 600 + 720 = 1520
or
50 + 50 + 50 + 50 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 60 + 90 + 90 + 90 + 90 + 90 + 90 + 90 + 90 + 90 = 1520
(ii) 23 × 24 + 46 × 8 = 552 + 368 = 920
The Collatz Conjecture
The Collatz Conjecture states that if you take any positive integer n and apply a simple set of rules repeatedly, eventually you will always end up with the number 1. It is one of the most famous unsolved problems in mathematics. It is named after a mathematician Lothar Collatz, who introduced the idea in 1937.
Rule
- If the previous term is even, the next term is one-half of the previous term.
- If the previous term is odd, the next term is 3 times the previous term plus 1.
e.g. start with the number 5.
Since 5 is an odd number, the next term is 3 times 5 plus 1.
i.e. 3 × 5 + 1 = 16
next term is 8. [∵ 16 is even]
Thus, we have 5 → 16 → 8 → 4 → 2 → 1
Example 14.
Write the Collatz sequence, starting with the number 7.
Solution:
7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
Simple Estimation
Estimation of numbers is the process of estimating/approximating or rounding off the numbers in which the value is used for some other purpose to avoid complicated calculations. Estimation is the process of finding the number which is close enough to the exact solution. But it is not an exact answer.
e.g. in class VI, section A has 28 children, section B has 37 children and section C has 31 children. Thus, the number of children in class VI is about 100.
Example 15.
Write the estimated values of the following questions.
(i) Number of words in your maths textbook
(ii) Number of students in your school.
Solution:
(i) Less than 5000
(ii) More than 200
Games and Winning Strategies
Numbers can also be used to play games and develop winning strategies.
e.g. The first player says a number between 1 to 10. Then, the two players take turns multiplying a number between 1 and 10 by the previous number said. The first player to reach 200 wins.
→ Numbers are used in different contexts and in many different ways to organize our lives. We used numbers to count, and applied the basic operations of addition, subtraction, multiplication, and division to them, and to solve problems related to our daily lives.
→ Numbers can be used for many different purposes, including to convey information, make and discover patterns, estimate magnitudes, pose and solve puzzles, and play and win games.
→ A fascinating and magical phenomenon while playing with 4-digit numbers is that with any 4-digit number by making the largest and smallest number using the digits of any 4-digit number and subtracting them and further proceeding in the same manner always reached the Kaprekar constant which is “6174”.
→ Also, if we start a sequence with any number; if the number is even, take half of it; if the number is odd, multiply it by 3 add 1, and repeat. We eventually reached the number 1. This rule is known as the Collatz Conjecture.
→ Two given numbers can be compared by counting their number of digits. The number having more digits is greater than a number having less digits.
→ For comparing two numbers having the same number of digits, we start comparing the digits from the leftmost position. If this digit also happens to be the same, we look at the next digits and so on.
→ The numbers that read the same from left to right and from right to left are called palindromes or palindromic numbers.
→ A cell becomes a supercell, if the number in it is greater than all the numbers in its neighboring cells.
→ As per the Collatz Conjecture rule, any sequence of counting numbers can end at 1 by using it.