0:03 hi friends if you watch this whole video
0:06 you'll find the topic real easy the
0:10 topic will seem natural to you so what's
0:14 the topic for today that's right this
0:18 video is about numbers in this video
0:20 we'll explore the different type of real
0:23 numbers such as natural numbers whole
0:27 numbers integers rational and irrational
0:30 numbers and then we'll finish off with
0:33 our top three test oriented questions on
0:36 this topic let's go back to the
0:38 kindergarten days and start with
0:42 counting numbers remember the Rhine one
0:45 two three four five once I caught a fish
0:51 alive well these numbers such as 1 2 3
0:53 and so on are called
0:56 natural numbers and they are denoted by
1:00 the letter M let's draw a circle to
1:04 represent the set of natural numbers now
1:07 whole numbers includes all the natural
1:11 numbers and zero so we can draw the
1:14 whole number circle around the natural
1:19 numbers integers include all the whole
1:22 numbers and negative numbers such as
1:27 minus 1 minus 2 minus 3 and so on so we
1:30 can draw the integer circle around the
1:33 whole numbers one important thing for
1:36 you to keep in mind is that zero is
1:39 considered neither positive nor negative
1:42 let's place the diagram we just did on
1:46 our concept board now the question is
1:49 are there any numbers between any two
1:55 integers let's say between 0 & 1 yes and
1:58 we use them all the time for example
2:03 half one-third point seven point eight
2:07 so between any two integers you have
2:10 these non integer numbers which are
2:13 known as fractions or decimals
2:17 and this entire set of integers and
2:20 fractions and decimals are called
2:23 rational numbers and they are denoted by
2:28 the letter Q any rational number can be
2:32 written as a fraction P by Q where P and
2:37 Q are integers the only restriction is
2:41 that Q should not be 0 since division by
2:45 zero is undefined now is the natural
2:49 number 3 also a rational number what do
2:53 you think so 3 can be written as 3 by 1
2:56 it can be written as a fraction you know
3:01 P by Q form so 3 is also a rational number
3:01 number
3:06 now what about 2.5 so we can write it as
3:10 25 by 10 and if you simplify the
3:15 fraction you get 5 by 2 so 2.5 is also a
3:19 rational number so any rational number
3:24 can be expressed in a P by Q form that
3:27 is in this fraction form the only
3:30 restriction is Q should not be equal to
3:34 0 and typically P and Q don't have any
3:38 common factor except 1 so we write the
3:42 fraction in its simplified form let's
3:45 add the non integers to our set diagram
3:49 the integers and non integers that is
3:51 fractions and decimals together are
3:55 called rational numbers let's talk more
3:57 about rational numbers that are not
4:01 integers as we learned rational numbers
4:05 are fractions represented by the P by Q
4:09 form you can calculate the fraction by
4:12 dividing and we end up with a decimal
4:19 number for example 7 by 10 is 0.75 by 2
4:26 is 2.5 7 by 4 is 1.75 these are all examples
4:27 examples
4:29 apples of terminating decimal numbers
4:33 that is the decimal digits stop or
4:37 terminate after a point because on
4:39 dividing by the denominator the
4:42 remainder is zero but there are other
4:45 fractions which on dividing don't
4:49 terminate for example two by three is
4:53 zero point six six six six and it just
4:56 goes on so we write it as zero point six
5:01 with a dot or a line on top to represent
5:04 that the digit six is recurring which
5:08 means repeating let's take another example
5:08 example
5:12 one by seven so if you look at the
5:14 answer here can you see the repeating
5:19 pattern that's right it's zero point one
5:22 four two eight five seven recurring
5:26 let's take a look at another example one
5:30 by six so that's going to be zero point
5:33 one six six six six and it just keeps
5:36 repeating but notice that we write it
5:39 with a dot or a bar only on top of six
5:44 because only the six is repeating so
5:46 these are called recurring decimal
5:50 numbers well on dividing by the
5:53 denominator the remainder is never zero
5:55 and the digits in the quotient keep
5:59 repeating if you take any fraction how
6:02 do you determine that on division you're
6:03 going to get a terminating decimal
6:06 number or a recurring decimal number of
6:09 course one simple way is to actually do
6:11 the division
6:13 but there's another trick where you
6:16 don't need to divide now what's the
6:18 trick you need to take a look at the
6:22 denominator and if the denominator has
6:26 factors of two and five only then you're
6:28 going to get a terminating decimal
6:31 number otherwise it's going to be a
6:34 recurring decimal number let me pull up
6:37 some examples to illustrate this let's
6:39 start with the example of seven by
6:42 for now what are the factors of the
6:47 denominator for its two into two now our
6:49 goal is to convert the denominator into
6:52 multiples of tens so I'm going to
6:55 multiply 5 into 5 in the numerator and
7:00 in the denominator so I get 175 by 100
7:06 that's 1.75 so this is a terminating
7:10 decimal number but if we take 37 by 150
7:12 so what are the factors of the denominator
7:13 denominator
7:19 they are 2 into 3 into 5 into 5 and if
7:23 you divide we get zero point 2 4 6 6 6
7:26 and so on so this is a recurring decimal
7:30 number and why is that because if you
7:32 carefully look at the factors in the
7:34 denominator you can see that there's a
7:37 three there for it to be a terminating
7:39 decimal number you need factors of two
7:43 and five only now our rational question
7:46 is are there numbers that are not
7:49 rational the answers yes
7:53 these are called irrational numbers for
7:57 example root 2 root 3 cube root of 10
8:01 these are all irrational numbers but we
8:03 look at these in a separate video
8:06 let's try to place the different numbers
8:08 that we've learnt on a number line are
8:10 you familiar with the number line an
8:14 everyday example is a ruler or a
8:17 measuring tape like this with numbers
8:20 marked on it unlike the measuring tape
8:23 the number line has both positive and
8:27 negative numbers so let's draw a number
8:29 line and try to place the different
8:32 numbers on it so here's our number line
8:36 let's mark the natural numbers on it 1 2
8:41 3 and so on let's draw another number
8:45 line and Mark the whole numbers on it so
8:47 it's going to be the natural numbers and
8:52 0 in our third number line let's mark
8:53 the integers
8:55 so it's going to be all the whole
8:59 numbers and the negative numbers minus 1
9:04 minus 2 minus 3 and so on in the next
9:06 number line let's mark the rational
9:09 numbers so it's going to be all the
9:14 integers and decimals and fractions for
9:17 example one point five is here midway
9:20 between one and two two point eight is
9:23 here and it's closer to three and minus
9:28 half is midway between 0 and minus 1 we
9:31 can also visualize the number system by
9:34 starting from the top that is from the
9:37 real numbers let me pull up the concept
9:41 board for you real numbers can be
9:44 divided into rational and irrational
9:48 numbers rational numbers can be divided
9:52 into integers and non integers now
9:55 integers can be divided into negative
9:58 integers and whole numbers whole numbers
10:02 can be split into zero and positive
10:05 integers which are natural numbers
10:08 coming back to the non integers
10:12 fractions and decimals these can be
10:14 divided into terminating decimals and
10:18 non terminating or recurring decimals
10:21 now let's say you want to find rational
10:24 numbers between any two given rational
10:29 numbers for example between 3 & 5 one
10:32 simple answer is 4 we can get it by
10:37 finding the average 3 plus 5 by 2 which
10:41 is equal to 4 if you want to find more
10:44 rational numbers then you can do 3 plus
10:51 4 by 2 that's 3 point 5 & 4 plus 5 by 2
10:55 which is 4 point 5 now let me pull up
10:58 some more interesting examples for you
11:01 if you want to find two rational numbers
11:05 between two fractions for example 1 by 7
11:07 & 4 bytes
11:10 since the denominators here are equal
11:14 and the numerators have a gap we can
11:16 fill in the fractions in the gap so our
11:21 answer is going to be 2 by 7 and 3 by 7
11:24 but what if there's no gap for example
11:27 if we have to find two rational numbers
11:32 between 1 by 7 and 2 by 7 then what do
11:34 you do we can create a gap by
11:36 multiplying the numerator and
11:40 denominator with a number since we need
11:43 two rational numbers we will multiply by
11:48 2 plus 1 which is 3 multiplying the
11:50 numerator and denominator of the two
11:56 fractions by 3 we get 3 by 21 and 6 by
11:59 21 again the denominators are the same
12:02 but can you see that we've got a gap in
12:05 the numerator so the two fractions we
12:10 can insert our 4 by 21 and 5 by 21 now
12:12 let's look at the case where the
12:14 denominators of the two fractions are
12:18 not the same for example if we want to
12:22 insert six rational numbers between 1/2
12:26 and 2/3 as you can see the denominators
12:29 are not equal so the first step is to
12:33 make the denominators equal so our two
12:38 fractions become 3 by 6 and 4 by 6
12:41 similar to our previous case now the two
12:43 denominators are equal but there's no
12:46 gap in the numerators so what should we
12:50 do that's right we need to multiply by a
12:54 number and since we need to insert six
12:56 rational numbers our number is going to
13:00 be 6 plus 1 which is 7 so let's multiply
13:03 the numerator and denominator of the two
13:08 fractions by 7 so here we have 21 by 42
13:12 and 28 by 42 so here are the six
13:15 rational numbers that we can insert
13:18 between these two fractions and on
13:21 simplifying the six rational numbers
13:28 and that's our answer let's talk about
13:31 fractions and decimals how to convert
13:35 fractions to decimals and decimals to
13:38 fractions converting fractions to
13:42 decimals is easy you just need to divide
13:48 for example 3 by 4 is 0.75 a terminating
13:54 decimal number 1 by 3 is 0.33 3
13:59 recurring a recurring decimal number now
14:02 if you want to convert 0.75 to a
14:05 fraction then we can write it as 75 by
14:11 100 simplifying we get 3 by 4 so these
14:13 are simple but a more interesting
14:16 question is how do you convert a
14:19 recurring decimal number to a fraction
14:26 for example 0.333 recurring now we know
14:34 that 0.3 is 3 by 10 0.33 is 33 by 100
14:41 0.333 is 333 by thousand but we are
14:45 looking for zero point 3 3 3 3 recurring
14:48 so let me show you the technique how to
14:50 convert a recurring decimal number to a
14:54 fraction coming up for you right now let
14:57 X be the recurring decimal number which
15:01 is 0.3 recurring in this case so let's
15:05 write it as X equal to 0.33 3 3 and so
15:09 on now we need another number where the
15:13 part after the decimal is the same so
15:17 let's multiply X by 10 and we have 10 X
15:22 equal to 3 point 3 3 3 and so on so as
15:25 you can see here X and 10x have the same
15:28 part after the decimal now let's
15:33 subtract X from 10x so I'm going to copy
15:35 the first line here
15:41 on subtracting we get 9x equal to 3 so X
15:45 is 3 by 9 and if we simplify we get 1 by
15:49 3 so we have converted our recurring
15:52 decimal number 0.3 recurring to a
15:56 fraction 1 by 3 let's look at another
16:01 example again let X be the recurring
16:04 decimal number which is one point two
16:07 seven recurring to find another number
16:10 with the decimal part matching this time
16:13 we can't multiply by 10 we need to
16:17 multiply by 100 and as you can see in
16:21 hundred X the decimal part matches so
16:24 once again subtracting X from 100 X we
16:29 get 99 X equal to 126 and on simplifying
16:33 we get 14 by 11 so we've converted our
16:36 recurring decimal number to a fraction
16:38 now that we are done with the topic of
16:41 rational numbers let's take a look at
16:43 the top three questions on this topic
16:47 coming up for you right now
16:51 I would like you to pause the video
16:53 right here and try solving these
16:56 questions let's make this more
16:59 interactive than just watching the video
17:03 so do post your answers in the comments
17:05 below or if you have any doubts
17:09 questions feel free to write it in the
17:12 comments below and I'm going to make a
17:14 commitment to answer all of them
17:17 promptly so I'm going to move off and
17:25 let you solve these questions thanks for
17:28 watching the whole video I hope the
17:31 number system is really easy for you now
17:34 and the numbers seem more rational and
17:39 natural to you all puns intended in that
17:43 sentence and do remember to Like comment
17:48 and share out this video and go and hit
17:50 the subscribe button for my channel
