SQUARE AND SQUARE ROOT CLASS 8 | NCERT | INTRODUCTION
SQUARE AND SQUARE ROOT CLASS 8 | NCERT | INTRODUCTION | CBSE | INSIGHTFUL MATHS
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00:00Hello everyone, welcome to insightful maths.
00:09In today's session we are going to discuss the topic square and square roots which is
00:13a class 8 topic.
00:15The usage is also there in grade 7.
00:17You must have done Pythagoras theorem wherein to find the hypotenuse you are supposed to
00:23find the square root of the number or at times the squares of the number.
00:28So beginning with square and square root let's understand what are squares, perfect square
00:34and how to find the root of the number.
00:37Before starting the session I request if you have not followed or subscribed to my channel
00:42please do subscribe.
00:44Okay, so square number or a perfect square.
00:48If you have any natural number, we are not starting with the whole numbers because the
00:52smallest whole number is 0 which is not a square number.
00:56So if any natural number, let's say it's p, if it can be written in the form of y square,
01:04how to write that?
01:05If I have a number p which can be written as the product of two similar number y into
01:13y or y square, then this number p, this natural number p is known as the square number or
01:20a perfect square, okay.
01:23Every number is not a perfect square, there are limited numbers which can be represented
01:28in this form.
01:30These are perfect square.
01:31Have a look at this chart, 1 square that is equal to 1, square means the number multiplied
01:38by itself.
01:39If I write 2 square, 2 square means 2 is multiplied by itself and answer is 4.
01:47In the similar manner, if I am finding 3 square, that is 3 into 3, 9.
01:53If I am finding 10 square, that is 10 into 10 which is 100.
01:57Now a very interesting thing you can note here is between 1 to 100.
02:03If I give you a question, between 1 to 100, how many square numbers are there?
02:09So you can see here 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
02:17Only 10 square numbers are there, okay.
02:20Coming to the next series of 11 square till 20 square, what is 20 square?
02:2520 square means 20 multiplied by 20, that is equal to 400.
02:31So now you can notice here beginning from 1 till 400.
02:36So if I ask you between 1 and 400, how many square numbers are there?
02:42There are only 20 square numbers, okay.
02:45Now between 1 and 900, have a look, 30 square is 900.
02:51So between 1 and 900, there are only 30 perfect square number.
02:56I hope this much is understood what is a perfect square number, what is square number.
03:02Any number which can be represented as the product of two similar numbers.
03:08Coming on to what is the square root now.
03:11The reverse of square is known as square root.
03:16Have a look, 1 square is equal to 1.
03:18So if I need to know that this is square number, which two similar numbers got multiplied to
03:25got this number, that is the square root.
03:28Let us understand by this manner, 2 square is equal to 2 multiplied by 2, it is 4.
03:35If I ask you, what is the square root of 4?
03:40Square root means going back to this number.
03:44Square root means I am asking for that number, which multiplied by itself has given you 4.
03:51So have a look, under root of 4 means you are going back to this basic number, under
03:56root of 4 is 2.
03:58So if I ask you what is under root of 9?
04:02Under root of 9 means under root of 3 multiplied by 3.
04:08If two similar numbers are there inside the root, one number gets cancelled, root get
04:13cancelled and another number goes out, answer is 3.
04:18Once again, if I ask you what is under root of 100?
04:23Under root of 100 can be written as under root of 10 multiplied by 10, two similar numbers
04:30inside the under root, one number and root got cancelled, your answer is just 10.
04:37So I advise, you need to remember the square of the numbers from 1 square at least till
04:4320 square, so as to make your calculation easier.
04:48Coming on to the properties of square numbers now.
04:52Square numbers end with these digits only, 0, 1, 4, 5, 6, 9.
04:58I'll explain you how.
05:00Square number never ends with these digits, 2, 3, 7, 8.
05:05It never ends with these digits.
05:08Please have a look at this table, 1 square is 1, 4, 9.
05:13Now have a look at the ones place, 6, 5, 6, 9, 4, 1 or 0.
05:19One more interesting thing is there.
05:21If it is ending with 0, if ones place is 0, 0 will always be coming even number of times,
05:28there cannot be a single 0.
05:31It has to be either two 0s or four 0s or six 0s, it has to be in pair, okay.
05:37So have a look, it is 1, 4, 6, 9, 5, 1, 4, 6, 9, 5, once again, 1, 4, 6, 9, 5 and if
05:470s are there, it is two 0s, again 1, 4, 6, 9, 5 and if 0s are there, two 0s.
05:55So you need to remember the square number always end with 0, 1, 4, 5, 6, 9 or you can
06:02exclude 2, 3, 7, 8.
06:05It will never be ending with 2, 3, 7, 8, the ones digit of a square number, okay.
06:12Let's understand if a number is being given to you, 36 and question says, if I square
06:20this number, what will be the ones digit of your answer?
06:24How do we get the square of this number?
06:26That is 36 multiplied by 36.
06:30You need not to do the complete calculation, please understand how are we getting the ones
06:35digit.
06:366 multiplied by 6 that is 36, 6 here and 3 will go carry, whatever the number is here
06:45hardly makes any difference, we are interested in knowing just the ones digit.
06:50So ones digit of 36 square will be 6.
06:53Once again, if I ask you, what is the ones digit of 104 square, 104 square means 104
07:02multiplied by 104, whatever the answer is, we are looking for just the ones digit, 4
07:09multiplied by 4, it is 16, 6, 1 carry here, I need not to do any other calculation, so
07:17ones place will be 6.
07:19Now if I ask you, what is the ones digit, if I am finding 23 square, please calculate
07:27what will be the ones digit, if you find 23 square, that is 23 multiplied by 23, 3 into
07:363 is 9 and we are not bothered about the number here, answer is 9.
07:41In this manner, you can easily find out what will be the ones digit of the square of a
07:46number without actually finding the square.
07:50Another thing is that I have already told you, only even number of zeros will be there
07:56in a square number.
07:57So if I ask you, is 1000 a square number, the rule says square number do end with 0,
08:05but the zeros should be coming in pair.
08:08We can see here only 2 zeros in pair and 1 is left out, straight away you can say it
08:12is not a square number, okay.
08:15It's not necessary if all zeros are there, 100% it will be a square number, it may or
08:20may not be, but if the zeros are not paired, then definitely it will not be a square number.
08:27Coming to the triangular numbers, if you can arrange any number in the form of a triangle,
08:35if you can arrange any number in the form of a triangle, that number or those numbers
08:40are known as triangular number.
08:43Please have a look, 1 is a triangular number.
08:47If I take 2 more dots below that, it becomes 3, triangular number, abhi humne 2 dot leyan,
08:54now followed by 3 dots, 3 and 3, it is 6, again a triangular number.
09:00After 3, you have to take 4 dots, it is 6 plus 4 is 10.
09:06After 10, you have to take 5 dots, that is 50.
09:09So all these 1, 3, 6, 10, 15, these are the triangular numbers.
09:15If a number can be arranged in the form of a triangular pattern, it is a triangular number.
09:23One more interesting thing is there, if you add 2 consecutive triangular numbers, your
09:29answer is a square number.
09:32Have a look, consecutive means one after the other.
09:35For example, I have numbers 1, 2, 3, 4, 5, 6, so on, this series.
09:40So 2 is coming after 1, 3 is coming after 2, 4 is coming after 3 and so on.
09:47These are consecutive numbers, one after the other.
09:50So have a look, if I add 1 and 3, what we are saying, if we add 2 consecutive triangular
09:58numbers, we get a square number, 1 plus 3.
10:03What is the answer?
10:04It is 4.
10:05Is 4 a square number?
10:07Yes, it is 2 square.
10:09Now 2 consecutive numbers, 3 and 6.
10:13If you take 3 and 6, then you add them up, you will be getting 9.
10:19What is 9?
10:20That is 3 square, it is again a square number.
10:23Take another pair, 10 plus 6, 10 plus 6, what is that?
10:28It is 16 and 16 is what?
10:31It is 4 square.
10:33So we have seen that when we are adding the triangular numbers, 2 consecutive triangular
10:38numbers, your answer is coming as a square number.
10:42Square number means a number which can be represented multiplying 2 similar numbers
10:49together.
10:50Okay?
10:51Let's move to the next topic now.
10:53The numbers between the square number, another very interesting pattern.
10:58If you have 2 square numbers, let's say 1 number is, I will take the smaller number
11:04first so that you understand the topic, 2 square and 3 square.
11:092 square numbers are there, now after 2 square, which is the next square number, it should
11:13be 3 square.
11:14After 3 square, it should be 4 square, okay?
11:17So if I ask you, between 2 square and 3 square, how many non-perfect square numbers are there?
11:252 square is what?
11:26It is 4 and 3 square is what?
11:29It is 9.
11:30Without calculating, without direct calculation, let me explain you first.
11:34It is 5, 6, 7, 8.
11:37So we can say that between 4 and 9, 4 is what?
11:422 square and it is 3 square.
11:44Between these 2, we are having 4 numbers, okay?
11:48Now how do we get this 4?
11:50We cannot calculate writing the numbers every time and then counting up.
11:55There has to be a rule.
11:57We are having 4 numbers.
11:59How do we get this 4?
12:01This smaller number multiplied by 2.
12:04That means if 2 consecutive square numbers are given, just multiply the smaller number
12:10by 2 and that will be your answer.
12:13That means if 1 number is n square, another number has to be n plus 1.
12:20What is the next number of n?
12:22n plus 1 is its square.
12:24So if I ask you between these 2 square numbers, how many non-square numbers are there?
12:30Twice of the smaller number.
12:33That means 2 times 10.
12:36Example, between 3 square and 4 square, how many non-square numbers are there?
12:43Which is smaller out of these 2?
12:453.
12:46So you just need to multiply 2 and 3.
12:49Your answer is 6.
12:51Between 6 square and 7 square, how many non-square numbers are there?
12:57The smaller number 6 and multiplied by 2, answer is 12.
13:02You can always recheck your answer.
13:04You need not to count it every time.
13:06Just smaller number multiplied by 2.
13:10So if 2 consecutive square of the natural numbers are there, n square and n plus 1 square,
13:16smaller number multiplied by 2 will be your answer.
13:19One more thing, if I give you the question in the form of algebra, okay, when the values
13:24are not being given, let me just see if you can guess the answer.
13:29If I have n minus 1 square and n square, which number is smaller?
13:35Of course, these 2 are consecutive square numbers.
13:39Consecutive means one after the other.
13:41So out of n minus 1 and n, which is smaller?
13:45It's a common sense.
13:46If something is subtracted from n, it is a smaller number.
13:50So how many non-square numbers are there in between?
13:53Twice of the smaller number.
13:56And the smaller number is n minus 1 here.
13:58That's your answer.
13:59So I hope it is understood.
14:02Coming to the next thing, when we are adding odd consecutive numbers, 2 things are there.
14:09It has to be odd.
14:12Consecutive means one after the other.
14:14When you are adding odd consecutive numbers, without actually adding the numbers, you can
14:20make out what will be the sum of this series, okay?
14:24So what we are supposed to do, have a look first, 1 and 3, 1 plus 3.
14:32How many consecutive odd numbers I have taken?
14:35Only 2.
14:36So answer is 2 square.
14:38It is 4.
14:39Of course, 1 plus 3 is 4.
14:42Now if you take 1 plus 3 plus 5, again, we have started with the first odd number and
14:49the series continues, 1, 3, 5, 3 numbers, 1, 3, 5, 3 numbers are there.
14:57So answer has to be 3 square, which is equal to 9.
15:01Recheck your answer.
15:025 plus 3 is 8, 8 plus 1 is 9.
15:06That's correct.
15:07So what we are doing, if consecutive, the series of odd consecutive natural numbers
15:13is given, just count how many numbers are there and take the square of that number.
15:191, 2, 3, 4.
15:22Four numbers are taken.
15:23Answer is 4 square, which is 16.
15:25You can recheck.
15:27This one, 1, 2, 3, 4, 5, 6.
15:31Answer has to be 6 square.
15:336 square means 6 into 6, which is 36.
15:36Now check this yourself, how many numbers are there and take the square of that number.
15:42I hope this much is understood.
15:45Coming to the next topic, every square number can be written as the sum of two consecutive
15:53positive natural numbers.
15:56What does it mean?
15:57If you have a square number, you can write that number as a sum of two consecutive positive
16:04natural numbers.
16:06I'll take this using an example.
16:10It is 3 square minus 1 by 2 and 3 square plus 1 by 2.
16:183 square is 9, 9 minus 1 by 2, 3 square is 9, 9 plus 1 by 2.
16:27What is 9 minus 1?
16:29It is 8 by 2.
16:30Your answer is 4, 9 plus 1 is 10 and 10 by 2 is 5.
16:36So how we have done?
16:38This 3 square, 3 square is 9, we wanted to write 9 as the sum of two consecutive natural
16:46numbers.
16:47Which numbers we have got?
16:484 and 5.
16:494 plus 5 is 9 only.
16:52How did we get that?
16:53Whichever square number is given to you, plus 1 by 2 and minus 1 by 2.
16:58I'm reading the statement once again.
17:01Every square number is the sum of two consecutive positive natural numbers.
17:07Okay?
17:08I'll take one more example.
17:14If I have 5 square, it's a square number, 5 square, it has, it is 25.
17:22So it has to be written as the sum of two natural numbers, positive and consecutive.
17:28So what am I going to do?
17:29This number, 5 square plus 1 upon 2, it should be the first number, and 5 square minus 1
17:38by 2, it has to be the second number.
17:405 square, it is 25 plus 1 by 2, another number is 25 minus 1 by 2.
17:48Now 25 plus 1 is 26 by 2, this is 24 by 2.
17:5426 by 2 is 13 and 24 by 2 is 12.
17:59And 12 plus 13, is it 25?
18:02It is coming 25.
18:03So you can write this as the sum of 12 and 13.
18:08Okay?
18:09So we have, just observe one thing, till now I have taken only the odd square numbers.
18:16Let us check if it follows for the even square numbers as well.
18:21If I take the even square number, let's say 6 square.
18:25So as per the rule, it has to be 6 square minus 1 by 2 and 6 square plus 1 by 2.
18:326 square is 36 and 36 minus 1 is 35 by 2, 36 plus 1, it is 37 by 2.
18:41If I divide these numbers, I will be getting the decimal numbers and it is not, decimals
18:48are not included in the natural number.
18:51So what we can conclude, this follows if the square of an odd number is being taken.
18:57Please note, this follows for the square of any odd natural number, not for even, not
19:06for even.
19:08This much is okay?
19:09Now the product of two consecutive even or odd numbers.
19:15If you have two consecutive odd or even numbers, it can be written in this form, a plus 1
19:22and a minus 1 and their answer will be first number square minus 1.
19:29What does it mean?
19:30Let me take two consecutive odd numbers first.
19:34Okay?
19:35Let's say 1 is 3 and 1 is 5.
19:40Okay?
19:41I need to find the product of these two.
19:45Now you will split it in this form, a plus 1, a minus 1.
19:49You have to take two numbers.
19:51If you add and subtract, you should get this one.
19:54Best way out is, start with the bigger number.
19:575 can be written as 4 plus 1 and 3 can be written as 4 minus 1.
20:04We are taking the product.
20:05What did I say?
20:07Bigger number square minus small number square or 1, ek hi baat hai.
20:134 square is 16 minus 1.
20:16Your answer is 15.
20:17Please have a look.
20:183 into 5 is 15 and it's 15 here.
20:22Okay?
20:23Once again, I'm taking two consecutive even numbers are now 2 and 4.
20:284 can be written as 3 plus 1 and 2 can be written as 3 minus 1.
20:37We have split it in a plus 1, a minus 1.
20:40Now, bigger number square minus smaller number square.
20:443 square is 9 minus 1.
20:47Your answer is 8.
20:48It's here.
20:49Okay?
20:50So, in this manner, you can find the square of the two numbers.
20:55Okay?
20:56So, rest, I will be continuing in part 2 of this video.
21:01I hope you are able to understand it here.
21:04If you have not watched, if you have not understood any of the topic, I request, please watch
21:10both of the video once again and follow the next upcoming videos.
21:15You will be able to complete all the chapter easily.
21:18I will be covering all the NCRT questions as well.
21:21Okay?
21:22I hope you are able to understand the topic.
21:24Thank you so much for watching.
21:26Please like and subscribe and follow my channel.
21:28Thank you so much for watching.