Hello, my dear students! Welcome to today's mathematics class. I am so happy to see you all here, ready to learn something new and exciting. Today, we are going to begin a very important chapter - Chapter 1: Number Systems. Now, I know what you might be thinking - "Sir/Ma'am, we already know about numbers! We have been studying numbers since our primary classes!" And you are absolutely right! But trust me, what we are going to learn today will take your understanding of numbers to a completely new level. We will explore the fascinating world of different types of numbers, discover numbers that might surprise you, and understand how all these numbers fit together on the number line. So, let's begin our journey!
In your earlier classes, you have learnt about the number line and how to represent various types of numbers on it. Just imagine you start from zero and go on walking along this number line in the positive direction. As far as your eyes can see, there are numbers, numbers and numbers! It seems like there is no end to them, isn't it?
Now suppose you start walking along the number line, and you want to collect some of the numbers. Imagine you have a bag ready to store them! You might begin with picking up only natural numbers like 1, 2, 3, and so on. You know that this list goes on forever - there is no largest natural number. So, now your bag contains infinitely many natural numbers! We denote this collection by the symbol N. N stands for natural numbers.
Now turn and walk all the way back, pick up zero and put it into the bag. You now have the collection of whole numbers, which is denoted by the symbol W. So, whole numbers include zero and all natural numbers.
Now, stretching in front of you are many, many negative integers. Put all the negative integers into your bag. What is your new collection? It is the collection of all integers, and it is denoted by the symbol Z. You might wonder why we use the letter Z. Well, Z comes from the German word "zahlen", which means "to count". Interesting, isn't it?
Now, are there some numbers still left on the line? Of course there are! There are numbers like 1/2, 3/4, or even -2005/2006. If you put all such numbers also into your bag, it will now be the collection of rational numbers. The collection of rational numbers is denoted by Q. The word 'rational' comes from the word 'ratio', and Q comes from the word 'quotient'. This makes sense because rational numbers are essentially ratios of two integers.
Now, let me give you the formal definition of rational numbers. A number r is called a rational number if it can be written in the form p/q, where p and q are integers and q is not equal to zero. Why do we insist that q is not equal to zero? Well, think about it - if q were zero, we would be dividing by zero, which is not allowed in mathematics! So, we always need q ≠ 0.
Now, students, notice that all the numbers now in the bag can be written in the form p/q, where p and q are integers and q ≠ 0. For example, -25 can be written as -25/1; here p = -25 and q = 1. Therefore, the rational numbers also include the natural numbers, whole numbers, and integers. In fact, every natural number, every whole number, and every integer is also a rational number! This is because we can always write them with denominator 1.
You also know that the rational numbers do not have a unique representation in the form p/q, where p and q are integers and q ≠ 0. For example, 1/2 = 2/4 = 10/20 = 25/50 = 47/94, and so on. These are called equivalent rational numbers or equivalent fractions. However, when we say that p/q is a rational number, or when we represent p/q on the number line, we assume that q ≠ 0 and that p and q have no common factors other than 1 - that is, p and q are co-prime. So, on the number line, among the infinitely many fractions equivalent to 1/2, we will choose 1/2 to represent all of them. This is called the simplest form of the fraction.
Now, let us solve some examples about the different types of numbers, which you have studied in earlier classes.
Example 1: Are the following statements true or false? Give reasons for your answers.
(i) Every whole number is a natural number.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
Let us look at each statement one by one.
For statement (i): Is every whole number a natural number? Well, whole numbers are 0, 1, 2, 3, 4, and so on. Natural numbers are 1, 2, 3, 4, and so on. Notice that 0 is a whole number but NOT a natural number. So, the statement "Every whole number is a natural number" is FALSE.
For statement (ii): Is every integer a rational number? Well, an integer like -5 can be written as -5/1. Here p = -5 and q = 1, which are both integers, and q ≠ 0. So yes, every integer can be written in the form p/q with q ≠ 0. Therefore, every integer is a rational number. So, statement (ii) is TRUE.
For statement (iii): Is every rational number an integer? Well, consider the rational number 3/5. Is 3/5 an integer? No, because 3/5 equals 0.6, which is not an integer. So, statement (iii) is FALSE.
Now students, let me ask you a question - can you find rational numbers between any two given rational numbers? Let us see!
Example 2: Find five rational numbers between 1 and 2.
We can approach this problem in at least two ways.
Solution 1: Recall that to find a rational number between r and s, you can add r and s and divide the sum by 2, that is (r + s)/2 lies between r and s. So, if we take r = 1 and s = 2, then (1 + 2)/2 = 3/2 = 1.5. So, 3/2 is a number between 1 and 2. You can proceed in this manner to find four more rational numbers between 1 and 2. These four numbers are 5/4, 11/8, 13/8 and 7/4.
Solution 2: The other option is to find all the five rational numbers in one step. Since we want five numbers, we write 1 and 2 as rational numbers with denominator 5 + 1, that is 6. So, 1 = 6/6 and 2 = 12/6. Then you can check that 7/6, 8/6, 9/6, 10/6 and 11/6 are all rational numbers between 1 and 2. So, the five numbers are 7/6, 4/3, 3/2, 5/3 and 11/6.
Now students, notice something very important here! In Example 2, you were asked to find five rational numbers between 1 and 2. But, you must have realised that in fact there are infinitely many rational numbers between 1 and 2. In general, there are infinitely many rational numbers between any two given rational numbers. This is a very important property of rational numbers.
Now, let us take another look at the number line. Have we picked up all the numbers? Not yet! The fact is that there are infinitely many more numbers left on the number line! There are gaps in between the places of the numbers we picked up, and not just one or two but infinitely many. The amazing thing is that there are infinitely many numbers lying between any two of these gaps too!
So we are left with some important questions. First, what are the numbers that are left on the number line called? Second, how do we recognise them? That is, how do we distinguish them from the rational numbers? These are very important questions, and we will answer them in the next section.
Now students, let me summarize what we have learned so far. We started with natural numbers, then added zero to get whole numbers, then added negative integers to get integers, and then added fractions to get rational numbers. Each time we added more numbers to our collection, we expanded our understanding of the number system. But as we will see, even rational numbers are not enough to describe all the numbers on the number line!
In our next section, we will discover numbers that are not rational - these are called irrational numbers. And together, rational and irrational numbers form what we call real numbers. So, let's continue our journey!
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Now, let us move on to Section 1.2: Irrational Numbers.
We saw in the previous section that there may be numbers on the number line that are not rationals. In this section, we are going to investigate these numbers. So far, all the numbers you have come across are of the form p/q, where p and q are integers and q ≠ 0. So, you may ask: are there numbers which are not of this form? There are indeed such numbers!
Students, let me tell you an interesting historical story. The Pythagoreans in Greece, followers of the famous mathematician and philosopher Pythagoras, were the first to discover the numbers which were not rationals, around 400 BC. These numbers are called irrational numbers, because they cannot be written in the form of a ratio of integers. There are many myths surrounding the discovery of irrational numbers by the Pythagorean, Hippacus of Croton. In all the myths, Hippacus has an unfortunate end, either for discovering that √2 is irrational or for disclosing the secret about √2 to people outside the secret Pythagorean sect! This shows how shocking this discovery was to the ancient mathematicians!
Now, let us formally define these numbers. A number s is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
You already know that there are infinitely many rationals. It turns out that there are infinitely many irrational numbers too. Some examples are: √2, √3, √15, π, 0.10110111011110...
Now, let me make an important remark. When we use the symbol √, we assume that it is the positive square root of the number. So √4 = 2, though both 2 and -2 are square roots of 4.
Some of the irrational numbers listed above are familiar to you. For example, you have already come across many of the square roots listed above and the number π.
The Pythagoreans proved that √2 is irrational. Later in approximately 425 BC, Theodorus of Cyrene showed that √3, √5, √6, √7, √10, √11, √12, √13, √14, √15 and √17 are also irrationals. Proofs of irrationality of √2, √3, √5, etc., shall be discussed in Class X. As to π, it was known to various cultures for thousands of years, it was proved to be irrational by Lambert and Legendre only in the late 1700s.
Now, let us return to the questions raised at the end of the previous section. Remember the bag of rational numbers? If we now put all irrational numbers into the bag, will there be any number left on the number line? The answer is no! It turns out that the collection of all rational numbers and irrational numbers together make up what we call the collection of real numbers, which is denoted by R. Therefore, a real number is either rational or irrational. So, we can say that every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number. This is why we call the number line the real number line.
In the 1870s, two German mathematicians, Cantor and Dedekind, showed that corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number. This is a fundamental result in mathematics!
Now, let us see how we can locate some of the irrational numbers on the number line.
Example 3: Locate √2 on the number line.
It is easy to see how the Greeks might have discovered √2. Consider a square OABC, with each side 1 unit in length. Then you can see by the Pythagoras theorem that OB = √(1² + 1²) = √2. How do we represent √2 on the number line? This is easy. Transfer this square onto the number line making sure that the vertex O coincides with zero.
We have just seen that OB = √2. Using a compass with centre O and radius OB, draw an arc intersecting the number line at the point P. Then P corresponds to √2 on the number line. This is a beautiful geometric construction!
Example 4: Locate √3 on the number line.
Let us return to the previous figure. Construct BD of unit length perpendicular to OB. Then using the Pythagoras theorem, we see that OD = √((√2)² + 1²) = √3. Using a compass, with centre O and radius OD, draw an arc which intersects the number line at the point Q. Then Q corresponds to √3.
In the same way, you can locate √n for any positive integer n, after √(n-1) has been located. This is amazing! We can locate all square roots on the number line using geometry!
Now students, let me recap what we have learned in this section. We learned that irrational numbers are numbers that cannot be written as a ratio of two integers. We learned about famous irrational numbers like √2, √3, and π. We also learned that when we combine rational and irrational numbers, we get the collection of real numbers, which is denoted by R. And we saw how to locate √2 and √3 on the number line using geometric constructions.
In our next section, we will explore real numbers from a different perspective - through their decimal expansions. This will help us distinguish between rational and irrational numbers even more clearly. So, let's continue!
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Now, let us move on to Section 1.3: Real Numbers and their Decimal Expansions.
In this section, we are going to study rational and irrational numbers from a different point of view. We will look at the decimal expansions of real numbers and see if we can use the expansions to distinguish between rationals and irrationals. We will also explain how to visualise the representation of real numbers on the number line using their decimal expansions. Since rationals are more familiar to us, let us start with them. Let us take three examples: 10/3, 7/8, and 1/7.
Pay special attention to the remainders and see if you can find any pattern.
Example 5: Find the decimal expansions of 10/3, 7/8 and 1/7.
Let us work through each one:
For 10/3, when we divide 10 by 3, we get 3 as the quotient and 1 as the remainder. Then we bring down a 0, making it 10 again, and we get 3 again with remainder 1. This continues forever! So the decimal expansion is 3.3333... The remainders are: 1, 1, 1, 1, 1... and they keep repeating.
For 7/8, when we divide 7 by 8, we get 0 as the quotient and 7 as the remainder. We bring down a 0, making it 70, and we get 8 as the quotient with remainder 6. We bring down a 0, making it 60, and we get 7 as the quotient with remainder 4. We bring down a 0, making it 40, and we get 5 as the quotient with remainder 0. So the decimal expansion is 0.875. The remainders are: 7, 6, 4, 0 - and then it stops!
For 1/7, when we divide 1 by 7, we get 0 as the quotient and 1 as the remainder. We bring down a 0, making it 10, and we get 1 as the quotient with remainder 3. We bring down a 0, making it 30, and we get 4 as the quotient with remainder 2. We bring down a 0, making it 20, and we get 2 as the quotient with remainder 6. We bring down a 0, making it 60, and we get 8 as the quotient with remainder 4. We bring down a 0, making it 40, and we get 5 as the quotient with remainder 5. We bring down a 0, making it 50, and we get 7 as the quotient with remainder 1. And now we are back to remainder 1! So the pattern will repeat. The decimal expansion is 0.142857142857... The remainders are: 1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1... and they repeat in a cycle of 6 digits.
What have you noticed? You should have noticed at least three things:
First, the remainders either become 0 after a certain stage, or start repeating themselves.
Second, the number of entries in the repeating string of remainders is less than the divisor. In 10/3, one number repeats itself and the divisor is 3. In 1/7, there are six entries 326451 in the repeating string of remainders and 7 is the divisor.
Third, if the remainders repeat, then we get a repeating block of digits in the quotient. For 10/3, 3 repeats in the quotient and for 1/7, we get the repeating block 142857 in the quotient.
Although we have noticed this pattern using only the examples above, it is true for all rationals of the form p/q (q ≠ 0). On division of p by q, two main things happen - either the remainder becomes zero or never becomes zero and we get a repeating string of remainders. Let us look at each case separately.
Case (i): The remainder becomes zero.
In the example of 7/8, we found that the remainder becomes zero after some steps and the decimal expansion of 7/8 = 0.875. Other examples are 1/2 = 0.5, 639/250 = 2.556. In all these cases, the decimal expansion terminates or ends after a finite number of steps. We call the decimal expansion of such numbers terminating.
Case (ii): The remainder never becomes zero.
In the examples of 10/3 and 1/7, we notice that the remainders repeat after a certain stage forcing the decimal expansion to go on for ever. In other words, we have a repeating block of digits in the quotient. We say that this expansion is non-terminating recurring. For example, 10/3 = 3.3333... and 1/7 = 0.142857142857142857...
The usual way of showing that 3 repeats in the quotient of 10/3 is to write it as 3.3̄. Similarly, since the block of digits 142857 repeats in the quotient of 1/7, we write 1/7 as 0.1̄4̄2̄8̄5̄7̄, where the bar above the digits indicates the block of digits that repeats. Also 3.57272... can be written as 3.5̄7̄2̄. So, all these examples give us non-terminating recurring decimal expansions.
Thus, we see that the decimal expansion of rational numbers have only two choices: either they are terminating or non-terminating recurring.
Now suppose, on your walk on the number line, you come across a number like 3.142678 whose decimal expansion is terminating, or a number like 1.272727... that is, 1.2̄7̄, whose decimal expansion is non-terminating recurring, can you conclude that it is a rational number? The answer is yes!
We will not prove it but illustrate this fact with a few examples. The terminating cases are easy.
Example 6: Show that 3.142678 is a rational number. In other words, express 3.142678 in the form p/q, where p and q are integers and q ≠ 0.
We have 3.142678 = 3142678/1000000, and hence is a rational number. Notice that we just moved the decimal point 6 places to the right to get the numerator, and used 1 followed by 6 zeros as the denominator.
Now, let us consider the case when the decimal expansion is non-terminating recurring.
Example 7: Show that 0.3333... = 0.3̄ can be expressed in the form p/q, where p and q are integers and q ≠ 0.
Since we do not know what 0.3̄ is, let us call it 'x' and so x = 0.3333...
Now here is where the trick comes in. Look at 10x = 10 × (0.333...) = 3.333...
Now, 3.3333... = 3 + x, since x = 0.3333...
Therefore, 10x = 3 + x
Solving for x, we get 9x = 3, i.e., x = 1/3
So, 0.3̄ = 1/3. Isn't that wonderful? We found the fraction representation of an infinite decimal!
Example 8: Show that 1.272727... = 1.2̄7̄ can be expressed in the form p/q, where p and q are integers and q ≠ 0.
Let x = 1.272727... Since two digits are repeating, we multiply x by 100 to get
100x = 127.2727...
So, 100x = 126 + 1.272727... = 126 + x
Therefore, 100x – x = 126, i.e., 99x = 126
i.e., x = 126/99 = 14/11
You can check the reverse that 14/11 = 1.2̄7̄.
Example 9: Show that 0.2353535... = 0.2̄3̄5̄ can be expressed in the form p/q, where p and q are integers and q ≠ 0.
Let x = 0.2̄3̄5̄. Over here, note that 2 does not repeat, but the block 35 repeats. Since two digits are repeating, we multiply x by 100 to get
100x = 23.53535...
So, 100x = 23.3 + 0.23535... = 23.3 + x
Therefore, 99x = 23.3
i.e., 99x = 233/10, which gives x = 233/990
You can also check the reverse that 233/990 = 0.2̄3̄5̄.
So, every number with a non-terminating recurring decimal expansion can be expressed in the form p/q (q ≠ 0), where p and q are integers. Let us summarise our results in the following form:
The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.
So, now we know what the decimal expansion of a rational number can be. What about the decimal expansion of irrational numbers? Because of the property above, we can conclude that their decimal expansions are non-terminating non-recurring.
So, the property for irrational numbers, similar to the property stated above for rational numbers, is:
The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.
Recall s = 0.10110111011110... from the previous section. Notice that it is non-terminating and non-recurring. Therefore, from the property above, it is irrational. Moreover, notice that you can generate infinitely many irrationals similar to s.
What about the famous irrationals √2 and π? Here are their decimal expansions up to a certain stage.
√2 = 1.4142135623730950488016887242096...
π = 3.14159265358979323846264338327950...
Note that we often take 22/7 as an approximate value for π, but π ≠ 22/7. 22/7 is just an approximation.
Over the years, mathematicians have developed various techniques to produce more and more digits in the decimal expansions of irrational numbers. For example, you might have learnt to find digits in the decimal expansion of √2 by the division method. Interestingly, in the Sulbasutras (rules of chord), a mathematical treatise of the Vedic period (800 BC - 500 BC), you find an approximation of √2 as follows:
√2 = 1 + 1/3 + (1/4 × 1/3) − (1/34 × 1/4 × 1/3) = 1.4142156
Notice that it is the same as the one given above for the first five decimal places. The history of the hunt for digits in the decimal expansion of π is very interesting.
The Greek genius Archimedes was the first to compute digits in the decimal expansion of π. He showed 3.140845 < π < 3.142857. Aryabhatta (476 – 550 C.E.), the great Indian mathematician and astronomer, found the value of π correct to four decimal places (3.1416). Using high speed computers and advanced algorithms, π has been computed to over 1.24 trillion decimal places! Isn't that amazing?
Now, let us see how to obtain irrational numbers.
Example 10: Find an irrational number between 1/7 and 2/7.
We saw that 1/7 = 0.142857. So, you can easily calculate 2/7 = 0.285714.
To find an irrational number between 1/7 and 2/7, we find a number which is non-terminating non-recurring lying between them. Of course, you can find infinitely many such numbers.
An example of such a number is 0.150150015000150000...
Notice that this number has increasing numbers of zeros between the 1's, so it never repeats. It is non-terminating and non-recurring, hence irrational!
Now students, let me recap what we learned in this section. We learned that rational numbers have decimal expansions that either terminate or repeat. We learned how to convert terminating and repeating decimals back to fractions. We also learned that irrational numbers have decimal expansions that are non-terminating and non-recurring. This gives us a powerful way to identify whether a number is rational or irrational - just look at its decimal expansion!
In our next section, we will learn about operations on real numbers. We will see how to add, subtract, multiply, and divide real numbers, and we will also learn about rationalizing denominators. So, let's continue!
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Now, let us move on to Section 1.4: Operations on Real Numbers.
You have learnt in earlier classes that rational numbers satisfy the commutative, associative and distributive laws for addition and multiplication. Moreover, if we add, subtract, multiply or divide (except by zero) two rational numbers, we still get a rational number. That is, rational numbers are 'closed' with respect to addition, subtraction, multiplication and division. It turns out that irrational numbers also satisfy the commutative, associative and distributive laws for addition and multiplication. However, the sum, difference, quotients and products of irrational numbers are not always irrational. For example, (√6) + (−√6) = 0, which is rational. (√2) − (√2) = 0, rational. (√3)·(√3) = 3, rational. And √17/√17 = 1, rational. So, when we combine two irrationals, we might get a rational number!
Now, let us look at what happens when we add and multiply a rational number with an irrational number. For example, √3 is irrational. What about 2 + √3 and 2√3? Since √3 has a non-terminating non-recurring decimal expansion, the same is true for 2 + √3 and 2√3. Therefore, both 2 + √3 and 2√3 are also irrational numbers.
Example 11: Check whether 7√5, 7/√5, √2 + 21, π − 2 are irrational numbers or not.
We know that √5 = 2.236..., √2 = 1.4142..., π = 3.1415...
Then 7√5 = 15.652..., 7/√5 = 7√5/√5√5 = 7√5/5 = 3.1304...
√2 + 21 = 22.4142..., π − 2 = 1.1415...
All these are non-terminating non-recurring decimals. So, all these are irrational numbers.
Now, let us see what generally happens if we add, subtract, multiply, divide, take square roots and even nth roots of these irrational numbers, where n is any natural number. Let us look at some examples.
Example 12: Add 2√2 + 5√3 and √2 − 3√3.
We have (2√2 + 5√3) + (√2 − 3√3) = (2√2 + √2) + (5√3 − 3√3) = (2 + 1)√2 + (5 − 3)√3 = 3√2 + 2√3
Example 13: Multiply 6√5 by 2√5.
6√5 × 2√5 = 6 × 2 × √5 × √5 = 12 × 5 = 60
Example 14: Divide 8√15 by 2√3.
8√15 ÷ 2√3 = (8√3 × √5)/(2√3) = 4√5
These examples may lead you to expect the following facts, which are true:
(i) The sum or difference of a rational number and an irrational number is irrational.
(ii) The product or quotient of a non-zero rational number with an irrational number is irrational.
(iii) If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational.
Now, let us turn our attention to the operation of taking square roots of real numbers. Recall that if a is a natural number, then √a = b means b² = a and b > 0. The same definition can be extended for positive real numbers.
Let a > 0 be a real number. Then √a = b means b² = a and b > 0.
In Section 1.2, we saw how to represent √n for any positive integer n on the number line. We now show how to find √x for any given positive real number x geometrically. For example, let us find it for x = 3.5, that is, we find √3.5 geometrically.
Mark the distance 3.5 units from a fixed point A on a given line to obtain a point B such that AB = 3.5 units. From B, mark a distance of 1 unit and mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semicircle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then, BD = √3.5.
More generally, to find √x for any positive real number x, we mark B so that AB = x units, and mark C so that BC = 1 unit. Then, as we have done for the case x = 3.5, we find BD = √x. We can prove this result using the Pythagoras Theorem.
Notice that in the right-angled triangle OBD, the radius of the circle is (x + 1)/2 units. Therefore, OC = OD = OA = (x + 1)/2 units. Now, OB = x − ((x + 1)/2) = (x − 1)/2. So, by the Pythagoras Theorem, we have BD² = OD² − OB² = ((x + 1)/2)² − ((x − 1)/2)² = 4x/4 = x. This shows that BD = √x.
This construction gives us a visual and geometric way of showing that √x exists for all real numbers x > 0. If you want to know the position of √x on the number line, then let us treat the line BC as the number line, with B as zero, C as 1, and so on. Draw an arc with centre B and radius BD, which intersects the number line in E. Then, E represents √x.
Now, we would like to extend the idea of square roots to cube roots, fourth roots, and in general nth roots, where n is a positive integer. Recall your understanding of square roots and cube roots from earlier classes.
What is ³√8? Well, we know it has to be some positive number whose cube is 8, and you must have guessed ³√8 = 2. Let us try ⁵√243. Do you know some number b such that b⁵ = 243? The answer is 3. Therefore, ⁵√243 = 3.
From these examples, can you define ⁿ√a for a real number a > 0 and a positive integer n?
Let a > 0 be a real number and n be a positive integer. Then ⁿ√a = b, if bⁿ = a and b > 0. Note that the symbol '√' used in √2, ³√8, ⁿ√a, etc. is called the radical sign.
We now list some identities relating to square roots, which are useful in various ways. You are already familiar with some of these from your earlier classes. The remaining ones follow from the distributive law of multiplication over addition of real numbers, and from the identity (x + y)(x − y) = x² − y², for any real numbers x and y.
Let a and b be positive real numbers. Then:
(i) √(ab) = √a √b
(ii) √(a/b) = √a/√b
(iii) (√a + √b)(√a − √b) = a − b
(iv) (a + √b)(a − √b) = a² − b
(v) (√a + √b)(√c + √d) = √(ac) + √(ad) + √(bc) + √(bd)
(vi) (√a + √b)² = a + 2√(ab) + b
Let us look at some particular cases of these identities.
Example 15: Simplify the following expressions:
(i) (5 + √7)(2 + √5)
(ii) (5 + √5)(5 − √5)
(iii) (√3 + √7)²
(iv) (√11 − √7)(√11 + √7)
Solution:
(i) (5 + √7)(2 + √5) = 10 + 5√5 + 2√7 + √35
(ii) (5 + √5)(5 − √5) = 5² − (√5)² = 25 − 5 = 20
(iii) (√3 + √7)² = (√3)² + 2√3√7 + (√7)² = 3 + 2√21 + 7 = 10 + 2√21
(iv) (√11 − √7)(√11 + √7) = (√11)² − (√7)² = 11 − 7 = 4
Remark: Note that 'simplify' in the example above has been used to mean that the expression should be written as the sum of a rational and an irrational number.
Now, we end this section by considering the following problem. Look at 1/√2. Can you tell where it shows up on the number line? You know that it is irrational. Maybe it is easier to handle if the denominator is a rational number. Let us see if we can 'rationalise' the denominator, that is, to make the denominator into a rational number. To do so, we need the identities involving square roots. Let us see how.
Example 16: Rationalise the denominator of 1/√2.
We want to write 1/√2 as an equivalent expression in which the denominator is a rational number. We know that √2 · √2 is rational. We also know that multiplying 1/√2 by √2/√2 will give us an equivalent expression, since √2/√2 = 1. So, we put these two facts together to get:
1/√2 = 1/√2 × √2/√2 = √2/2
In this form, it is easy to locate 1/√2 on the number line. It is halfway between 0 and √2.
Example 17: Rationalise the denominator of 1/(2 + √3).
We use Identity (iv) given earlier. Multiply and divide 1/(2 + √3) by 2 − √3 to get:
1/(2 + √3) × (2 − √3)/(2 − √3) = (2 − √3)/(4 − 3) = 2 − √3
Example 18: Rationalise the denominator of 5/(√3 − √5).
Here we use Identity (iii) given earlier.
So, 5/(√3 − √5) = 5/(√3 − √5) × (√3 + √5)/(√3 + √5) = 5(√3 + √5)/(3 − 5) = (−5/2)(√3 + √5)
Example 19: Rationalise the denominator of 1/(7 + 3√2).
1/(7 + 3√2) = 1/(7 + 3√2) × (7 − 3√2)/(7 − 3√2) = (7 − 3√2)/(49 − 18) = (7 − 3√2)/31
So, when the denominator of an expression contains a term with a square root (or a number under a radical sign), the process of converting it to an equivalent expression whose denominator is a rational number is called rationalising the denominator.
Now students, let me recap what we learned in this section. We learned about operations on real numbers. We saw that rational and irrational numbers both satisfy the commutative, associative, and distributive laws. We learned that the sum or difference of a rational and an irrational is always irrational, and the product or quotient of a non-zero rational with an irrational is always irrational. We also learned about geometric construction for finding square roots on the number line. And most importantly, we learned how to rationalize denominators - a very useful technique in mathematics!
In our next and final section, we will learn about the laws of exponents for real numbers. This will extend what we already know about exponents to rational exponents. So, let's continue!
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Now, let us move on to Section 1.5: Laws of Exponents for Real Numbers.
Do you remember how to simplify the following?
(i) 17² · 17⁵ =
(ii) (5²)⁷ =
(iii) 23¹⁰/23⁷ =
(iv) 7³ · 9³ =
Did you get these answers? They are as follows:
(i) 17² · 17⁵ = 17⁷
(ii) (5²)⁷ = 5¹⁴
(iii) 23¹⁰/23⁷ = 23³
(iv) 7³ · 9³ = 63³
To get these answers, you would have used the following laws of exponents, which you have learnt in your earlier classes. Here a, n and m are natural numbers. Remember, a is called the base and m and n are the exponents.
(i) aᵐ · aⁿ = aᵐ⁺ⁿ
(ii) (aᵐ)ⁿ = aᵐⁿ
(iii) aᵐ/aⁿ = aᵐ⁻ⁿ, m > n
(iv) aᵐbᵐ = (ab)ᵐ
What is a⁰? Yes, it is 1! So you have learnt that a⁰ = 1. So, using law (iii), we can get 1/aⁿ = a⁻ⁿ. We can now extend the laws to negative exponents too.
So, for example:
(i) 17² · 17⁻⁵ = 17⁻³ = 1/17³
(ii) (5²)⁻⁷ = 5⁻¹⁴
(iii) 23⁻¹⁰/23⁷ = 23⁻¹⁷
(iv) (7)⁻³ · (9)⁻³ = (63)⁻³
Now suppose we want to do the following computations:
(i) 2^(2/3) · 2^(1/3)
(ii) (3^(1/5))⁴
(iii) 7^(1/5)/7^(1/3)
(iv) 13^(1/5) · 17^(1/5)
How would we go about it? It turns out that we can extend the laws of exponents that we have studied earlier, even when the base is a positive real number and the exponents are rational numbers. But before we state these laws, and to even make sense of these laws, we need to first understand what, for example, 4^(3/2) is. So, we have some work to do!
We define ⁿ√a for a real number a > 0 as follows:
Let a > 0 be a real number and n a positive integer. Then ⁿ√a = b, if bⁿ = a and b > 0.
In the language of exponents, we define ⁿ√a = a^(1/n). So, in particular, ³√2 = 2^(1/3).
There are now two ways to look at 4^(3/2).
4^(3/2) = (4^(1/2))³ = 2³ = 8
4^(3/2) = (4³)^(1/2) = (64)^(1/2) = 8
Therefore, we have the following definition:
Let a > 0 be a real number. Let m and n be integers such that m and n have no common factors other than 1, and n > 0. Then, a^(m/n) = (ⁿ√a)ᵐ = ⁿ√aᵐ
We now have the following extended laws of exponents:
Let a > 0 be a real number and p and q be rational numbers. Then, we have:
(i) aᵖ · aᵠ = aᵖ⁺ᵠ
(ii) (aᵖ)ᵠ = aᵖᵠ
(iii) aᵖ/aᵠ = aᵖ⁻ᵠ
(iv) aᵖbᵖ = (ab)ᵖ
You can now use these laws to answer the questions asked earlier.
Example 20: Simplify:
(i) 2^(2/3) · 2^(1/3)
(ii) (3^(1/5))⁴
(iii) 7^(1/5)/7^(1/3)
(iv) 13^(1/5) · 17^(1/5)
Solution:
(i) 2^(2/3) · 2^(1/3) = 2^((2/3)+(1/3)) = 2^(3/3) = 2¹ = 2
(ii) (3^(1/5))⁴ = 3^(4/5)
(iii) 7^(1/5)/7^(1/3) = 7^((1/5)-(1/3)) = 7^((3-5)/15) = 7^(-2/15)
(iv) 13^(1/5) · 17^(1/5) = (13 × 17)^(1/5) = 221^(1/5)
Now students, let me recap what we learned in this section. We learned about the laws of exponents for real numbers. We extended our knowledge of exponents from natural numbers to rational numbers. We learned that a^(m/n) means the nth root of a raised to the power m. And we learned the four fundamental laws of exponents that work for rational exponents just as they work for natural number exponents.
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Now, we have come to the end of our lesson. Let me give you a complete summary of everything we have learned in this chapter.
In this chapter, you have studied the following important points:
1. A number r is called a rational number if it can be written in the form p/q, where p and q are integers and q ≠ 0.
2. A number s is called an irrational number if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
3. The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.
4. The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.
5. All the rational and irrational numbers make up the collection of real numbers.
6. If r is rational and s is irrational, then r + s and r − s are irrational numbers, and rs and r/s are irrational numbers, r ≠ 0.
7. For positive real numbers a and b, the following identities hold: - √ab = √a √b - √(a/b) = √a/√b - (√a + √b)(√a − √b) = a − b - (a + √b)(a − √b) = a² − b - (√a + √b)² = a + 2√ab + b
8. To rationalise the denominator of 1/(√a + b), we multiply this by (√a − b)/(√a − b), where a and b are integers.
9. Let a > 0 be a real number and p and q be rational numbers. Then: - aᵖ · aᵠ = aᵖ⁺ᵠ - (aᵖ)ᵠ = aᵖᵠ - aᵖ/aᵠ = aᵖ⁻ᵠ - aᵖbᵖ = (ab)ᵖ
Students, this chapter forms the foundation for your understanding of the number system. You have learned about natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. You have learned how to distinguish between rational and irrational numbers using their decimal expansions. You have learned about operations on real numbers and how to rationalize denominators. And you have learned about laws of exponents for rational exponents.
This knowledge will be extremely useful in your future mathematical studies. Remember, mathematics is not just about memorizing formulas - it's about understanding concepts and being able to apply them. So, make sure you understand each concept clearly.
Thank you for your attention in today's class. Keep practicing, and don't hesitate to ask questions if you have any doubts. See you in the next class!