Real numbers are numbers that can be found on the number line. This includes both the rational and irrational numbers.
An irrational number is number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. For example:- The number 2=1.41421356..... does not end. The same can be said for any irrational number.
Express rational numbers on number line
Let's see to how express any irrational number on number line For example:- Take 2 as irrational number 1. Draw a number line, mark the origin and other integers. 2. Now, find distance between 0 and 1 using compass. 3. Draw a perpendicular to 1 of the same length as between 0 and 1. 4. Join origin and other end of the new line i.e. perpendicular. You will get a figure as shown. 5. Name origin as A, 1 on the number line as B and end of perpendicular as C. 6. Distance between A and C is 2. 7. Use compass to find distance between A and C. 8. Place compass at A and mark the same length as between A and C on the number line. 9. And you get the point 2 at D. Similarly, you can do it for other irrational numbers like 3,5,..
Define and identify examples of terminating decimals
A terminating decimal is a decimal that ends i.e. it has finite number of digits. For a fraction in decimal form, while performing division after a certain number of steps, we get the remainder zero. The quotient obtained as decimal is called the terminating decimal. For eg:- 83=0.375 has 3 digits after decimal point 45=1.25 has 2 digits after decimal point Hence, 0.375 and 1.25 are terminating decimals.
Non-Terminating Recurring Decimals
While expressing a fraction in the decimal form, when we perform division we get some remainder. If the division process does not end i.e. we do not get the remainder equal to zero; then such decimal is known as non-terminating decimal. In some cases, a digit or a block of digits repeats itself in the decimal part, then the decimal is non-terminating recurring decimal. For eg:- 1.6666....,0.141414...
Non-terminating and non-recurring decimals
While expressing a fraction in the decimal form, when we perform division we get some remainder. If the division process does not end i.e. we do not get the remainder equal to zero; then such decimal is known as non-terminating decimal. And if a digit or a block of digits does not repeats itself in the decimal part, such decimals are called non-terminating and non-recurring decimals. For eg:- 1.41421356.....
Euclid's Division Lemma
Let a and b be any two positive integers. Then there exist unique integers q and r such that a=bq+r, where 0≤r<b If b∣a, then r=0. Otherwise, r satisfies the stronger inequality 0≤r<b
HCF using Euclid's Divison
If a and b are positive integers such that a=bq+r, then every common divisor of a and b is a common divisor of b and r, and vice-versa. Example: Find HCF of 420 and 130. Since 420>130 we apply the division lemma to 420 and 130 to get ,Since 300 , we apply the division lemma to 130 and 30 to get 130=30×4+10 420=130×3+30 Since 100 , we apply the division lemma to 30 and 10 to get 30=10×3+0 The remainder has now become zero, so our procedure stops. Since the divisor at this step is 10, the HCF of 420 and 130 is 10.
HCF using Fundamenal Theorem
Find the HCF of 6 and 20 by the prime factorization method. We have : 6=2×3 20=2×2×5 20=2×2×5. Common factors of 6 and 20 are 21 and 22 So for HCF take the common number with lowest exponent. HCF =2×1=2