Part-2 of Exercise 2.2 - 9th Class Math 9 کاس میتھ مشق نمبر 2۔2
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00:00How? I will write it once and after writing if you see the example, you will understand it well.
00:09For example, how is it written in the book?
00:11It is written in the book that for all real numbers, represent those real numbers with a, b and it belongs to R.
00:21For all real numbers a, b, the trichotomy property says that either a will be greater than b or a will be equal to b or a will be smaller than b.
00:49So, what is the conclusion?
00:51Either the relation between two real numbers will be greater or it will be equal or it will be smaller.
00:59Now, you will understand this thing better from the example.
01:03For example, if I write 5 here and 5 here and 5 here.
01:09Now, there can be three types of relations of 5.
01:12For example, 5 is greater than 2.
01:17It means that 5 is greater than 2.
01:23Or what can be the other relation of 5 in real numbers?
01:26It can be that 5 is equal to 5.
01:30Or what can be another relation that 5 is smaller than 8, 9 or 6.
01:36So, in this way, three relations can be formed between any real numbers.
01:44This tells us the trichotomy property in math.
01:48When you do questions, you solve them by following these rules and properties.
01:53So, I hope you have understood the trichotomy property.
01:56Let's go to the next property.
01:58So, here we have the next property, the transitive property.
02:01And before going to this, you have to tell one more thing that this was written just to explain.
02:06You have to understand the trichotomy property from this only.
02:12This is the original.
02:13So, we were talking about the transitive property.
02:16What is the meaning of transitive property?
02:18Let me write this first.
02:20For all real numbers in this property, three real numbers have been talked about to explain.
02:28So, for all a, b, c belong to R.
02:34That means a, b, c are representing all the real numbers here.
02:39And after representing the real numbers, what does this property tell us?
02:43This property tells us that if a is smaller than b and b is smaller than c.
02:56So, which implies that?
02:58So, what will be the conclusion?
03:00The conclusion will be that a, b will be smaller than c.
03:07One thing you have to learn is that this symbol means which implies that.
03:13And if I say it in Urdu, it will mean that it means.
03:18If a is smaller than b and b is smaller than c, it means that a is also smaller than c.
03:27So, this is the property.
03:29It is an easy property.
03:30Let me explain it with an example.
03:32Let's keep a as a number.
03:35For example, if I keep 4, then 4 is smaller than 5.
03:39This thing is done.
03:41I have done this to understand.
03:44And let's write n here.
03:48b is smaller than c.
03:50So, we have kept b as 5.
03:52So, I keep 5 here.
03:54And this number is smaller than 6, 7, 8.
03:59So, let's write 6 to understand.
04:01Now, this condition has come to us in the form of numbers.
04:05If 4 is smaller than 5 and 5 is smaller than 6.
04:10So, it means that if a is equal to 4, then 4 will be smaller than 6.
04:20Because c is equal to 6 and a is equal to 4.
04:22So, if you look closely, what is wrong in this?
04:27If 4 is smaller than 5 and 5 is smaller than 6, then 4 will be smaller than 6.
04:32You can understand it like this.
04:34Don't go to writing.
04:35Just try to understand that if Ali's height is smaller than Waqas's height.
04:42Ali's height is smaller than Waqas's height.
04:45So, Waqas will come here.
04:49And Waqas's height is smaller than Qasim's height.
04:52So, which implies that the conclusion is that Ali's height will be smaller than Qasim's height.
04:59So, it has been told in this.
05:01So, it is just a general thing that when you are doing these things in math,
05:08then the transitive property will also help you to solve your questions somewhere.
05:13So, I hope you have understood this thing.
05:16So, this property has been described in two ways.
05:18Now, it is understood from the point of view of lesser.
05:20So, we will have to understand it from the point of view of greater.
05:24So, from the point of view of greater, if I give it the name of a.
05:29It is the same in your book.
05:31I gave it the name of a.
05:32And it has been discussed from the point of view of lesser.
05:36So, in the same way, from the point of view of greater, this property also plays the same role.
05:41That if a is big.
05:44I made it small here.
05:45Now, I will make it big in all.
05:47a will be bigger than b.
05:49And b will be bigger than c.
05:54So, this means that a will be bigger than c.
06:00So, I hope I don't need to give the example of this.
06:04But still, I will give it in the form of numbers.
06:07If we keep a equal to 4k.
06:10And if we want to make a bigger than any number, then we keep b equal to 3k.
06:14And here end.
06:16Now, we have kept b equal to 3k.
06:19So, if we increase 3 from 2.
06:26So, which implies that 4 will be bigger than 2.
06:32So, this is the same thing.
06:33a is 4k and c is 2k.
06:35So, I hope I tried my best to make the weak child understandable.
06:41So, up to this point, it was a matter of transitive property.
06:45It will have to be represented in the form of a and b.
06:48So, the property mentioned on the third number is additive property.
06:52In terms of equality, what does additive property say?
06:55It says that for all a, b, c belong to R.
07:01a, b, c means all real numbers.
07:04So, two parts of this property have been made in terms of explaining.
07:08First, it is said that if a is less than b.
07:13That is, if a is less than b, which implies that a is less than b.
07:21That is, if the same real number is added on both sides.
07:27Then b, a will be less than b.
07:31That is, there will be no difference in equality.
07:35Let me explain this with an example.
07:37If I keep a equal to 4 and b equal to 5 to explain to you.
07:42So, 4 is less than 5.
07:47Which implies that 4 will be less than 5 if both sides are equal.
07:58Like if I add 3 here and 3 here.
08:01So, this end will be less than this end.
08:05Like 4 and 3 are 7 and 5 and 3 are 8.
08:11So, you can still see that this end is smaller than this.
08:14So, this thing was just told.
08:16I hope you understood.
08:18At last, I am telling again that by adding the same number on both sides.
08:23There is no difference in equality.
08:27The number that was small will remain small.
08:29The number that was big will remain big.
08:32Now, here in this property, one more thing has also been told to you.
08:36That is, if a is smaller than b.
08:41So, in such a scenario, if you change their position.
08:44That is, if you move a forward.
08:46That is, if you change their location.
08:49So, it does not make a difference.
08:51For example, if I write here.
08:54Like here a plus c is written.
08:57Here I write c plus a.
08:59And here I write c plus b.
09:01So, I wrote c first in both.
09:05So, this symbol of inequality will not make a difference.
09:11This end will remain smaller than this.
09:15This will remain small and this will remain big.
09:18So, I hope you understood this thing too.
09:21Now, how is this thing told in B point?
09:25It is told in B point.
09:27That if a is bigger than b.
09:34So, in such a situation, a will remain bigger than b.
09:40If the same real number is added on both sides.
09:45So, there is no need to give an example of this.
09:47And this thing has also been described in this way.
09:50As above has been described.
09:51That a will remain bigger than b.
09:54If the terms at both ends or the location of their alphabets or numbers.
10:01They are changed.
10:03Like c plus a.
10:05And here I write c plus b.
10:07Still, this symbol of inequality will not make a difference.
10:12So, I hope you understood the additive property too.
10:17So, the fourth property is called multiplicative property.
10:21And in terms of inequality, it has been divided into two parts.
10:25In the first part, it is told.
10:30For all a, b, c belong to R.
10:36If a, b, c belong to R.
10:38In the first part, we will talk about the end.
10:40If c is greater than 0.
10:44That is, if c is greater than 0.
10:48In that scenario, what does this property say?
10:51We will understand this first.
10:52And in the second part, we will understand.
10:55If c is less than 0.
10:57We will understand that.
10:59So, I hope you understood.
11:00Why two parts?
11:02On what basis have two parts of this property?
11:05So, what is said in this multiplicative property?
11:10In this, it is said.
11:12If a is greater than b.
11:19Which implies that.
11:21So, what will happen in multiplicative?
11:23If on both sides, a is greater than b.
11:27On both sides, if I multiply c.
11:31So, there will be no effect on inequality.
11:35a will always be greater than b.
11:38How?
11:39I will give an example of this too.
11:41For example.
11:42Either a is equal to 5.
11:45Or b is equal to 4.
11:47Which implies that.
11:495 to the 10th.
11:514 to the 8th.
11:53C can be any real number.
11:55So, let's take 2.
11:57Multiply 2 on both sides.
11:59So, you can see.
12:015 to the 10th.
12:034 to the 8th.
12:05This side is still greater.
12:07This means.
12:09There was no effect on inequality.
12:11Just like the previous property.
12:13From the point of view of addition.
12:15It is the same.
12:18And the second thing is.
12:20It is said that.
12:22Even if you change their places.
12:24There is no effect.
12:25So, let's write that too.
12:26So, this is the first case.
12:28So, let's write here.
12:29a is greater than b.
12:32Even if you change the location c.
12:35There is no effect on the sign of this inequality.
12:39You can also write by changing their places.
12:42Okay.
12:43So, it came from the aspect of greater.
12:45Now, let's do it from the aspect of lesser.
12:47If I do it from the point of view of lesser.
12:49a is lesser than b.
12:52So, this will also be less.
12:56If I multiply c on both sides from b.
12:59Okay.
13:00So, there is nothing to explain in this.
13:01It is the same.
13:02Earlier, it was from the point of view of bigger.
13:04Now, it is from the point of view of lesser.
13:06And let's do the second line in the same way.
13:08If you change the location of c in the multiplication.
13:11There will be no effect.
13:13So, a is less than b.
13:15Which implies that c a.
13:18You write like this.
13:19And c b.
13:20There will be no effect.
13:22It was given the name of the first.
13:24So, let's give it the name of the second.
13:26Okay.
13:27So, in this way, the first case was completed with the name of a.
13:31Now, let's go to the second case.
13:33So, let's represent the second case with b.
13:35I write here.
13:36I represent it with b.
13:38And the same thing will happen in it.
13:40For all a, b, c belong to R.
13:44It belongs to R.
13:46End.
13:47Now, here c is less than 0.
13:50That is, instead of c, there is a number which is less than 0.
13:54Now, let me tell you one more thing here.
13:58Like, on the number line, there is a positive number on the side of 0.
14:021, 2, 3, 4 and so on.
14:04Here, minus 1, minus 2, minus 3, minus 4 and so on.
14:10And you will also know that 0 is bigger than all the negative numbers.
14:16And similarly, minus 2 and minus 1.
14:19Which is the bigger number in this?
14:21The bigger number in this is minus 1.
14:24Similarly, which number is bigger than minus 3?
14:27Minus 1 and minus 2.
14:28Both are bigger than minus 3.
14:30So, what do you understand?
14:33You understand that in minus, the bigger number becomes smaller.
14:39And the smaller number becomes bigger.
14:41Like here.
14:42That is, 2 is bigger than 1.
14:48Similarly, 3 is bigger than 2.
14:51But here it is the opposite.
14:53That minus 2 is smaller than minus 1.
14:59So, minus 1 is bigger than minus 2.
15:02And minus 2 is bigger than minus 3.
15:05So, I think you must have understood.
15:07I want to explain that in minus, the bigger number is smaller.
15:12Now, you will understand what I will explain.
15:15So, let's divide this into two parts.
15:17First, let's talk in terms of greater.
15:20So, if A is greater than B.
15:24Which implies that A will be smaller than B.
15:30If the same number which is represented by C is multiplied.
15:37Now, you will be a little worried.
15:39Whenever C is smaller than 0.
15:43The symbol changes whenever the same number is multiplied.
15:49I will give you an example.
15:53So, let's do it above.
15:55And I write an example here.
15:57Let's suppose 5 is greater than 4.
16:00Now, you see.
16:02What does A mean?
16:05Here is 5.
16:06And what will come here?
16:09What does B mean?
16:10Here is 4.
16:11So, here 5 is greater than 4.
16:14So, it should be bigger here too.
16:15But here it will be smaller.
16:17How?
16:18If C is smaller than 0.
16:22Here will be a minus number.
16:23So, multiply by any number.
16:27So, if I multiply by minus 2.
16:29If I multiply by minus 2 here too.
16:31So, you see.
16:32This will be minus 10.
16:33And this will be minus 4 to the 8.
16:35So, now you see which number is greater.
16:37So, this number is greater.
16:39And this is smaller.
16:40Because in minus, a greater number is called smaller.
16:43So, this is the same thing.
16:45When numbers are smaller than 0.
16:48So, if you multiply the same number on both sides.
16:51So, the sign in the middle will be opposite of the inequality.
16:55Earlier it was bigger.
16:57Okay.
16:58And then it converted to smaller.
17:01So, if you ask questions, you will understand better.
17:04Let's erase these examples.
17:06And after erasing.
17:08Now, let's change the location of C and write.
17:12So, it will be like this.
17:14If A is greater than B.
17:17So, which implies that.
17:19Despite this.
17:20C A will be less than C B K.
17:25So, what did you know?
17:26That changing the location does not make a difference.
17:30The symbol is reversed if C is smaller than 0.
17:35I have told you again and again.
17:36I hope you have understood this thing too.
17:40So, we have done it from the point of view of bigger.
17:43Now, let's do it from the point of view of smaller.
17:45So, let's write 2 here.
17:47So, if A is less than B.
17:55So, which implies that.
17:57Now, if A is smaller than C.
18:01So, it will be reversed.
18:03And here it will be B C.
18:06So, basically C was smaller than 0.
18:10That's why this symbol is reversed.
18:13If we change the location of C and write.
18:15So, here it will be A is less than B.
18:20And if we change the location.
18:21So, C A will still be bigger than C B.
18:26So, this was the whole thing.
18:28There was only one conclusion.
18:30That inside the minus.
18:32When the sign of the inequality is reversed.
18:36And whenever C is bigger.
18:38So, it will not be reversed.
18:40So, I hope you have understood this property too.
18:42So, the fifth property is called multiplicative inverse property.
18:46So, before discussing this.
18:48You should know what is multiplicative inverse.
18:51So, let me tell you again.
18:52If it is written somewhere 5.
18:54So, what will be its multiplicative inverse?
18:57It will be 1 divided by 5.
18:59So, the multiplicative inverse of 5 is 1 divided by 5.
19:03And the multiplicative inverse of 1 divided by 5 is 5.
19:06And it looks like 5.
19:08But actually there is 1 below it.
19:10Whether you write 1 or not.
19:11It does not matter.
19:12So, whatever real number you see.
19:146, 2, 3.
19:15There is 1 below them.
19:16But we do not write that 1.
19:18But when we use it.
19:19When we have to take LCM etc.
19:20So, we use it at that time.
19:22So, you got to know one thing about multiplicative inverse.
19:25Now it will be easy for you to understand.
19:27So, in this property.
19:28According to inequality.
19:29We have been told that.
19:31For all.
19:32All the real numbers.
19:34Like A, B for example.
19:36Mentioning real numbers.
19:38Or expressing.
19:39So, A, B belong to R.
19:41And what is special here.
19:43A should never be equal to 0.
19:47And B should never be equal to 0.
19:50Because 0 is itself a real number.
19:54So, this multiplicative inverse property.
19:58Is applicable on all real numbers.
20:03Except 0.
20:06That's why it is mentioned separately.
20:08While writing the property.
20:10Or while understanding.
20:11Or while explaining.
20:12So, for all.
20:14A, B belong to R.
20:16Now, let's divide the property into two parts.
20:18First, I will mention less than.
20:21Second, I will mention according to greater than.
20:24So, according to less than.
20:25So, if A is less than B.
20:29Okay.
20:30So, in between.
20:31If and only if symbol comes.
20:34So, what is the reciprocal of A?
20:38The reciprocal of A is 1 divided by A.
20:40And what is the reciprocal of B?
20:421 divided by B.
20:44So, this property says.
20:46That when the multiplicative is taken of two numbers.
20:49Then the symbol of inequality in between.
20:51It goes upside down.
20:53Like it was like this.
20:54So, now it is like this.
20:56So, what is known from this.
20:57That on this side.
20:58This A is smaller than B.
21:01But when it comes in the multiplicative inverse form.
21:04Both numbers.
21:05So, this 1 divided by A is bigger than 1 divided by B.
21:09So, if I try to explain this thing.
21:11So, simple.
21:12You must have seen that 4.
21:13If I talk about a simple number.
21:15Positive number.
21:16So, between 4 and 5.
21:17Comparison is done.
21:18So, 4 will be smaller than 5.
21:21But now.
21:22If I change it in the multiplicative inverse.
21:25So, 1 divided by 4 and 1 divided by 5.
21:28Now, 1 divided by 4 will be bigger.
21:31So, you can see.
21:32In the multiplicative.
21:33The sign.
21:35The sign of inequality.
21:38They change.
21:39This thing.
21:40If I try to explain this thing.
21:43Like there is a number line.
21:44On the number line.
21:45There is 0 in the middle.
21:46Here positive 1.
21:47Positive 2.
21:48Then positive 3.
21:50Positive 4.
21:51And positive 5.
21:52Here then minus 1.
21:53Minus 2.
21:54Minus 3.
21:55Minus 4.
21:56Minus 5.
21:57So, now what happens here.
21:58Let's look at this case.
21:59If you look at the positive side of 0.
22:00On this side.
22:01So, 4 is smaller than 5.
22:02Okay.
22:03It is clear.
22:04But when we come to this side.
22:05So, it won't be like this here.
22:06Here.
22:07Minus 4.
22:08It will be bigger than minus 5.
22:09Because in minus.
22:10It goes upside down.
22:11Small number becomes big.
22:12Big.
22:13Becomes small.
22:14So, this thing.
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40:05Okay?
40:06So, we have a distributive relation now.
40:07Let me write it here.
40:08So, I have written this.
40:09Distributive property.
40:10Let's come next property.
40:11It is easy to understand.
40:12It is additive inverse.
40:13Let me first write it and then tell you.
40:14Why it is additive inverse?
40:15Because, any real number having additive inverse,
40:16including opposite sign.
40:17They are same.
40:18Like 2
40:18answer is always equal to zero. So, you can see that the additive inverse of pi is minus pi and the answer is equal to zero.
40:32So, let's finish this and remove the extra things from here. Let's go to the next property.
40:37So, here is the next property. What is written in this? One real number is this and one real number is this.
40:42If we multiply these two, we will get the real number. Yes, this is the closure property with respect to multiplication.
40:49So, I will write it here. So, this is the closure property and what is written in the closure property?
40:53If you multiply any real number like 3 with any other real number, then the answer will always be the real number.
41:05So, if we multiply two real numbers, then the answer will always be the real number.
41:10So, this is the closure property. I hope you understood this. Let's go to the next property.
41:15So, this is the last property. What is this property? It is the multiplicative inverse.
41:19So, I will write it and tell you how. So, I have written the multiplicative inverse.
41:23Now, the question arises that how is this multiplicative inverse?
41:26So, if I write 5 here and tell you what is the multiplicative inverse of this, then you will write it as 1 divided by 5.
41:35One more thing should be known that where there is nothing, there is 1 below.
41:39So, let's do one more example like this. What is the multiplicative inverse of 4? It is 1 divided by 4.
41:45So, where there is nothing below, there is 1.
41:48So, if you see carefully, what I am doing is that I am reversing in the multiplicative inverse.
41:53If it was 5 divided by 1, then I have made it 1 divided by 5. If it was 4 divided by 1, then I have made it 1 divided by 4.
41:58So, if you see above, then this is what has happened here too.
42:01If it was 5 divided by 8, then I have made it 8 divided by 5.
42:06So, it means that we do this work in the multiplicative inverse.
42:11The second thing is that the result of the multiplicative inverse is always equal to 1.
42:18So, you can see that here too it is equal to 1.
42:21So, that is why it is a multiplicative inverse.
42:23So, I hope you have understood.
42:25So, finally, this exercise is over for you.
42:29I hope you have learned a lot.
42:32These properties are very important.
42:34I have also recorded it.
42:36Maybe it is not so important from the point of view of numbers.
42:38But if you want to understand math well, then this was a very important exercise.
42:44Take good care of yourself.
42:46Forgive me if I have made a mistake unknowingly.
42:49And InshaAllah, see you in the next video.
42:52Take care of yourself. Allah Hafiz.