 Hello and welcome to the session. In this session we will discuss real numbers and their decimal expansions. We know that all the rational and irrational numbers make up the collection of real numbers. Now we have decimal expansion of a rational number is either terminating non-terminating recurring. Moreover we can also say that a number whose decimal expansion is terminating or non-terminating recurring is a rational number. Then decimal expansion of an irrational number is non-terminating non-recurring. We can also say that a number whose decimal expansion is non-terminating non-recurring is irrational. Like consider the number 5 upon 8 now this is equal to 0.625 that is this is the decimal expansion of this number. Now this decimal expansion is terminating so hence we can say that this number 5 upon 8 is a rational number. Now consider the irrational number root 2 its decimal expansion is given by 1.41421356237 and so on. So as you can see that the decimal expansion of root 2 is non-terminating non-recurring. We can find many irrational numbers between two given numbers suppose the two given numbers are 5 upon 7 and 9 upon 11. Let's try and find out three different irrational numbers between 5 upon 7 and 9 upon 11. Now 5 upon 7 has a decimal expansion 0.714285714285 and so on. Then 9 upon 11 has decimal expansion as 0.8181818182 and so on. Now to find irrational number between these two numbers we find a number which is non-terminating non-recurring lying between these two numbers. We can find infinitely many such numbers like the first irrational number between these two numbers given by 0.750750075000 and so on. Then the second irrational number can be given by 0.76007報76700767000 and so on, and so on. Then the third irrational number is given by 0.8080080008 4 times 0 and so on. So this is how we find the irrational numbers between given two numbers. Now we discuss representing real numbers on the number line. We know that every real number is represented by a unique point on the number line. Further every point on the number line represents one and only one real number. Consider the number 4.26 bar. We need to visualize this number on the number line up to four decimal places. So up to four decimal places this number can be written as 4.2626. Now you can clearly see that this number would lie between the numbers four and five. That is this number lies in this area that is between the numbers four and five. Now we draw another number line to visualize the area between the numbers four and five. Now we divide this region between the numbers four and five into ten equal parts. So this is the middle point of this number line which represents the number 4.5. Now our given number is 4.2626. So thus we can say that this number would lie between the numbers 4.2 and 4.3 that is this region. Now we will draw another number line which represents the region between 4.2 and 4.3 and we visualize this region. Again we will divide this region into ten equal parts. Now this is the middle point 4.25. This is the middle point of this number line. Now we have reached the location 4.26. Now 4.262 would lie in the region 4.26 to 4.27 that is this region. Now we draw another number line to visualize the region between the numbers 4.26 and 4.27. Now the middle point of this number line is given by the number 4.265. Now we have reached the number 4.262. Now 4.2626 would lie in this region that is between the numbers 4.262 and 4.263. Now we will visualize the region between the numbers 4.262 and 4.263 to get the required number on the number line. Now this is the middle point of this number line representing the number 4.2625. This is the required point which represents the number 4.2626. So as you can see we started from this number line where we magnified or visualized the region between the numbers 4 and 5. Then we visualized the region between the numbers 4.2 and 4.3. Then we visualized the region between the numbers 4.26 and 4.27. And after that we visualized the region between the numbers 4.262 and 4.263. And from there we get the required point 4.2626. This completes this session. Hope you have understood real numbers and their decimal expansions and how we represent real numbers on the number line.