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If a is a positive rational number and n is a positive integer such that n a is an irrational number, then n a is called a surd or a radical. Division Algorithm Main Targets To represent the number in Scientific Notation. To convert exponential form to logarithmic form and vice-versa. To understand the rules of logarithms. To apply the rules and to use logarithmic table. It is easier to express these numbers in a shorter way called Scientific Notation, thus avoiding the writing of many zeros and transposition errors.
Napier is placed within a short lineage of mathematical thinkers. That is, the very large or very small numbers are expressed as the product of a decimal number 1 a 1 10 and some integral power of Key Concept Scientific Notation.
A number N is in scientific notation when it is expressed as the product of a decimal number between 1 and 10 and some integral power of To transform numbers from decimal notation to scientific notation, the laws of exponents form the basis for calculations using powers.
Let m and n be natural numbers and a is a real number. The laws of exponents are given below: For a! Step 1: Move the decimal point so that there is only one non - zero digit to its left. Step 2: Count the number of digits between the old and new decimal point.
This gives n, the power of Step 3: If the decimal is shifted to the left, the exponent n is positive. If the decimal is shifted to the right, the exponent n is negative. Example 3. Solution In integers, the decimal point at the end is usually omitted. The decimal point is to be moved 3 places to the left of its original position. So the power of 10 is 3. Solution 0. The decimal point is to be moved four places to the right of its original position.
So the power of 10 is 4. To convert scientific notation to integers we have to follow these steps. Write the decimal number. Move the decimal point the number of places specified by the power of ten: Add zeros if necessary.
Rewrite the number in decimal form. Exercise 3. Represent the following numbers in the scientific notation. Write the following numbers in decimal form.
Represent the following numbers in scientific notation. They were designed to transform multiplicative processes into additive ones. Before the advent of calculators, logarithms had great use in multiplying and dividing numbers with many digits since adding exponents was less work than multiplying numbers. Now they are important in nuclear work because many laws governing physical behavior are in exponential form. Examples are radioactive decay, gamma absorption, and reactor power changes on a stable period.
To introduce the notation of logarithm, we shall first introduce the exponential notation for real numbers. We have already introduced the notation a x , where x is an integer. We knowpthat a n is a positive number whose nth power is equal to a. Now we can see how to define a q , where p is an integer and q is a positive integer.
Notice that p 1. We will not show how a x may be defined for irrational x because the definition of a x requires some advanced topics in mathematics. Key Concept Logarithmic Notation. Let a be a positive number other than 1 and let x be a real number positive, negative, or zero. In both the forms, the base is same. The Rules of Logarithms 1. Product Rule: The logarithm of the product of two positive numbers is equal to sum of their logarithms of the same base.
Quotient Rule: The logarithm of the quotient of two positive numbers is equal to the logarithm of the numerator minus the logarithm of the denominator to the same base. The logarithm of a number in exponential form is equal to the logarithm of the number multiplied by its exponent. Change of Base Rule: If M, a and b are positive numbers and a! State whether each of the following statements is true or false. Solve the equation in each of the following.
Find the value in each of the following in terms of x , y and z. Logarithms to the base 10 are called common logarithms. Therefore, in the discussion which follows, no base designation is used, i. Consider the following table. So, log N is an integer if N is an integral power of What about logarithm of 3. For example, 3. Notice that logarithm of a number between 1 and 10 is a number between 0 and 1 ; logarithm of a number between 10 and is a number between 1 and 2 and so on.
Every logarithm consists of an integral part called the characteristic and a fractional part called the mantissa. For example, log 3. It is convenient to keep the mantissa positive even though the logarithm is negative. Scientific notation provides a convenient method for determining the characteristic.
Thus, the power of 10 determines the characteristic of logarithm. The negative sign of the characteristic is written above the characteristics as 1, 2, etc. For example, the characteristic of 0. Hence, i log Note that the mantissas of logarithms of all the numbers consisting of same digits in same order but differing only in the position of decimal point are the same.
The mantissas are given correct to four places of decimals. A logarithmic table consists of three parts. These columns are marked with serial numbers 1 to 9. We shall explain how to find the mantissa of a given number in the following example. Suppose, the given number is Now Therefore, the characteristic is 1. The row in front of the number 4.
The number is 0. Next the mean difference corresponding to 5 is 0. Thus the required mantissa is 0. Hence, log This table gives the value of the antilogarithm of a number correct to four places of decimal. For finding antilogarithm, we take into consideration only the mantissa. The characteristic is used only to determine the number of digits in the integral part or the number of zeros immediately after the decimal point.
The method of using the table of antilogarithms is the same as that of the table of logarithms discussed above. Since the logarithmic table given at the end of this text book can be applied only to four digit number, in this section we approximated all logarithmic calculations to four digits. From the table, log 4. So, the number contains two digits in its integral part.
Mantissa is 0. From the table, antilog 0. So, the number contains one zero immediately following the decimal point. Taking logarithm on both sides, we get 0.
Write each of the following in scientific notation: Write the characteristic of each of the following i log iv log 0. The mantissa of log is 0. Find the value of the following. Using logarithmic table find the value of the following. Using antilogarithmic table find the value of the following. Points to Remember A number N is in scientific notation when it is expressed as the product of a decimal number 1 a 1 10 and some integral power of Product rule: To use Remainder Theorem.
To use Factor Theorem. To use algebraic identities. To factorize a polynomial. To solve linear equations in two variables. To solve linear inequation in one variable. DIophAntus to A. Diophantus was a Hellenistic mathematician who lived circa AD, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written Arithmetica, a treatise that was originally thirteen books but of which only the first six have survived.
Arithmetica has very little in common with traditional Greek mathematics since it is divorced from geometric methods, and it is different from Babylonian mathematics in that Diophantus is concerned primarily with exact solutions, both determinate and indeterminate, instead of simple approximations.
The language of algebra is a wonderful instrument for expressing shortly, perspicuously, suggestively and the exceedingly complicated relations in which abstract things stand to one another.
Algebra has been developed over a period of years. But, only by the middle of the 17th Century the representation of elementary algebraic problems and relations looked much as it is today.
By the early decades of the twentieth century, algebra had evolved into the study of axiomatic systems. This axiomatic approach soon came to be called modern or abstract algebra.
Important new results have been discovered, and the subject has found applications in all branches of mathematics and in many of the sciences as well. A constant, we mean an algebraic expression that contains no variables at all. If numbers are substituted for the variables in an algebraic expression, the resulting number is called the value of the expression for these values of variables. If an algebraic expression consists of part connected by plus or minus signs, it is called an algebraic sum.
Each part, together with the sign preceding it is called a term. For instance, in the term - 4xz , the coefficient of z 2 y y is - 4x , whereas the coefficient of xz is 4.
A coefficient such as 4, which involves no y 2 2 variables, is called a numerical coefficient. Terms such as 5x y and - 12x y , which differ only in their numerical coefficients, are called like terms or similar terms.
An algebraic expression such as 4rr can be considered as an algebraic expression consisting of just one term. Such a one-termed expression is called a monomial.
An algebraic expression with two terms is called a binomial, and an algebraic expression with three terms is called a trinomial. An algebraic expression with two or more terms is called a multinomial. A term such as - 1 which 2 2 contains no variables, is called a constant term of the polynomial. The numerical coefficients of the terms in a polynomial are called the coefficients of the polynomial.
The coefficients of the polynomial above are 3, 2 and - 1. In adding exponents, one should regard a variable with no exponent as being power one. The constant term is always regarded as having degree zero. The degree of the highest degree term that appears with nonzero coefficients in a polynomial is called the degree of the polynomial. For instance, the polynomial considered above has degree 8. Although the constant monomial 0 is regarded as a polynomial, this particular polynomial is not assigned a degree.
Key Concept Polynomial in One Variable. Here n is the degree of the polynomial and a1, a2, g, an - 1, an are the coefficients of x, x , gx 2 n The three terms of the polynomial are 5x2, 3x and - 1. Binomial Polynomials which have only two terms are called binomials. A binomial is the sum of two monomials of different degrees. A trinomial is the sum of three monomials of different degrees.
A polynomial is a monomial or the sum of two or more monomials. Constant polynomial A polynomial of degree zero is called a constant polynomial.
General form: Linear polynomial A polynomial of degree one is called a linear polynomial. Quadratic polynomial A polynomial of degree two is called a quadratic polynomial. Cubic polynomial A polynomial of degree three is called a cubic polynomial. Example 4. State whether the following expressions are polynomials in one variable or not. Classify the following polynomials based on their degree. Give one example of a binomial of degree 27 and monomial of degree 49 and trinomial of degree If the value of a polynomial is zero for some value of the variable then that value is known as zero of the polynomial.
Key Concept Zeros of Polynomial. Number of zeros of a polynomial the degree of the polynomial. Carl Friedrich Gauss had proven in his doctoral thesis of that the polynomial equations of any degree n must have exactly n solutions in a certain very specific sense. This result was so important that it became known as the fundamental theorem of algebra.
The exact sense in which that theorem is true is the subject of the other part of the story of algebraic numbers. Hence zeros of a polynomial are the roots of the corresponding polynomial equation. Key Concept Root of a Polynomial Equation. Exercise 4. Verify Whether the following are roots of the polynomial equations indicated against them.
Also find the remainder. Find the value of a. An identity is an equality that remains true regardless of the values of any variables that appear within it. We have learnt the following identities in class VIII. Using these identities let us solve some problems and extend the identities to trinomials and third degree expansions.
Using these identities of 4. Using algebraic identities find the coefficients of x2 term, x term and constant term. We have seen how the distributive property may be used to expand a product of algebraic expressions into sum or difference of expressions. In both the terms, ab and ac a is the common factor. Factorize the following expressions: In this section. Split this product into two factors such that their sum is equal to the coefficient of x.
The terms are grouped into two pairs and factorize. The constant term is 2. The factors of 2 are 1, 1, 2 and 2. The constant term is 3. The factors of 3 are 1, 1,3 and 3. Factorize each of the following. Let us consider a pair of linear equations in two variables x and y. The substitution method, the elimination method and the cross-multiplication method are some of the methods commonly used to solve the system of equations.
In this chapter we consider only the substitution method to solve the linear equations in two variables. It is then substituted in the other equation and solved. Find the cost of each. The cost of three mathematics books is the same as that of four science books. Find the cost of each book. From Dharmapuri bus stand if we download 2 tickets to Palacode and 3 tickets to Karimangalam the total cost is Rs 32, but if we download 3 tickets to Palacode and one ticket to Karimangalam the total cost is Rs Find the fares from Dharmapuri to Palacode and to Karimangalam.
Find the numbers. The number formed by reversing the digits is 9 less than the original number. Find the number. Solution Let the tens digit be x and the units digit be y. There is only one such value for x in a linear equation in one variable. We represent those real numbers in the number line. Unshaded circle indicates that point is not included in the solution set.
The real numbers less than or equal to 3 are solutions of given inequation. Shaded circle indicates that point is included in the solution set. Solve the following equations by substitution method. A number consists of two digits whose sum is 9.
The number formed by reversing the digits exceeds twice the original number by Find the original number. Kavi and Kural each had a number of apples. Kavi said to Kural If you give me 4 of your apples, my number will be thrice yours.
Kural replied If you give me 26, my number will be twice yours. How many did each have with them?. Solve the following inequations. Remainder Theorem: Factor Theorem: Main Targets To understand Cartesian coordinate system To identify abscissa, ordinate and coordinates of a point To plot the points on the plane To find the distance between two points Descartes D e s c a r t e s has been called the father of modern philosophy, perhaps because he attempted to build a new system of thought from the ground up, emphasized the use of logic and scientific method, and was profoundly affected in his outlook by the new physics and astronomy.
Descartes went far past Fermat in the use of symbols, in Arithmetizing analytic geometry, in extending it to equations of higher degree. The fixing of a point position in the plane by assigning two numbers - coordinates giving its distance from two lines perpendicular to each other, was entirely Descartes invention. Coordinate Geometry or Analytical Geometry is a system of geometry where the position of points on the plane is described using an ordered pair of numbers called coordinates.
He proposed further that curves and lines could be described by equations using this technique, thus being the first to link algebra and geometry. In honour of his work, the coordinates of a point are often referred to as its Cartesian coordinates, and the coordinate plane as the Cartesian Coordinate Plane.
The invention of analytical geometry was the beginning of modern mathematics. In this chapter we learn how to represent points using cartesian coordinate system and derive formula to find distance between two points in terms of their coordinates.
Conversely, a point P on a number line can be specified by a real number x called its coordinate. The two number lines intersect at the -6 zero point of each as shown in the Fig. Generally the Y horizontal number line is called the x-axis and Fig. The x coordinate of a point to the right of the y-axis is positive and to the left of y-axis is negative. We use the same scale that is, the same unit Y 8 distance on both the axes. To obtain these number, we draw two lines through the point P parallel and hence perpendicular to the axes.
We are interested in the coordinates of the. There are two coordinates: The x-coordinate is called the abscissa and the y-coordinate is called the ordinate of the point at hand. These two numbers associated with the point P are called coordinates of P. They are usually written as x, y , the.
In an ordered pair a, b , the two elements a and b are listed in a specific order. So the ordered pairs a, b and b, a are not equal, i. The terms point and coordinates of a point are used interchangeably. To find the x-coordinate of a point P: To find the y-coordinate of a point P: The coordinate axes divide the plane into four parts called quadrants, numbered counter-clockwise for reference as shown in Fig. The signs of the coordinates are shown in parentheses in Fig.
Let us now illustrate through an example how to plot a point in Cartesian coordinate system. The intersection of these two lines is the position of 5, 6 in the cartesian plane. That is we count from the origin 5 units along the positive direction of x-axis and move along the positive direction of y-axis through 6 units and mark the corresponding point.
This point is at a distance of 5 units from the y-axis and 6 units from the x-axis. Thus the position of 5, 6 is located in the cartesian plane. Example 5. The intersection of these two lines is the position of 5, 4 in the Cartesian plane.
Thus, the point A 5, 4 is located in the Cartesian plane. The intersection of these two lines is the position of - 4, 3 in the Cartesian plane. Thus, the point B - 4, 3 is located in the Cartesian plane. The intersection of these two lines is the position of - 2, - 3 in the Cartesian plane. Thus, the point C - 2, - 3 is located in the Cartesian plane. The intersection of these two lines is the position of 3, - 2 in the Cartesian plane Thus, the point D 3, - 2 is located in the Cartesian plane.
Observe that if we interchange the abscissa and ordinate of a point, then it may represent a different point in the Cartesian plane. What can you say about the position of these points? When you join these points, you see that they lie on a line which is parallel to x-axis. Discuss the type of the diagram by joining all the points. Find the coordinates of the points shown in the Fig. Solution Consider the point A. Hence the coordinates of A are 3, 2.
For any point on the x-axis its y coordinate is zero. Write down the abscissa for the following points. Write down the ordinate of the following points. Plot the following points in the coordinate plane. How is the line joining them situated? The ordinates of two points are each - 6. How is the line joining them related with reference to x-axis? The abscissa of two points is 0. How is the line joining situated?
With rectangular axes plot the points O 0, 0 , A 5, 0 , B 5, 4. What are the coordinates of C? Distance between any Two Points One of the simplest things that can be done with analytical geometry is to calculate the. The distance between two points A and B is usually denoted by AB. Consider the two points A x1, 0 and B x2, 0 on the x-axis.
Consider two points A 0, y1 and B 0, y2. These two points lie on the y axis. Draw AP and BQ perpendicular to x-axis. Draw AP and BQ perpendicular to y-axis. The distance between A and B is equal to the distance between P and Q. The distance between two points on a line parallel to the coordinate axes is the absolute value of the difference between respective coordinates. We shall now find the distance between these two points.
AR is drawn perpendicular to BQ. Given the two points x1, y1 and x2, y2 , the distance between these points is given by the formula: Hence Aliter: Let A and B denote the points 6, 0 and 0, 8 and let O be the origin.
The point 6, 0 lies on the x-axis and the point 0, 8 lies on the y-axis. Solution Let the points be A 4, 2 , B 7, 5 and C 9, 7. Hence the points A, B, and C are collinear. Hence ABC is a right angled triangle since the square of one side is equal to sum of the squares of the other two sides.
Solution Let the points be represented by A a, a , B - a, - a and C - a 3 , a 3. The opposite sides are equal. Hence ABCD is a parallelogram. That is, all the sides are equal. That is, the diagonals are equal. Hence the points A, B, C and D form a square. If the abscissa and the ordinate of P are equal, find the coordinates of P.
Solution Let the point be P x, y. Therefore, the coordinates of P are x, x. Let A and B denote the points 2, 3 and 6, 5.
Find also its radius. Solution suppose C represents the point 4, 3. Let P, Q and R denote the points 9, 3 , 7, - 1 and 1, - 1 respectively.
Hence the points P, Q, R are on the circle with centre at 4, 3 and its radius is 5 units. Solution Let P deonte the point a, b.
Let A and B represent the points 3, - 4 and 8, - 5 respectively.
It is known that the circum-center is equidistant from all the vertices of a triangle. Find the distance between the following pairs of points. Show that the following points form a right angled triangle. Show that the following points taken in order form the vertices of a parallelogram.
Show that the following points taken in order form the vertices of a rhombus. Examine whether the following points taken in order form a square. Examine whether the following points taken in order form a rectangle.
If the distance between two points x, 7 and 1, 15 is 10, find x. Show that 4, 1 is equidistant from the points - 10, 6 and 9, - If the length of the line segment with end points 2, - 6 and 2, y is 4, find y. Find the perimeter of the triangle with vertices i 0, 8 , 6, 0 and origin ; ii 9, 3 , 1, - 3 and origin Find the point on the y-axis equidistant from - 5, 2 and 9, - 2 Hint: A point on the y-axis will have its x coordinate as zero.
Find the radius of the circle whose centre is 3, 2 and passes through - 5, 6. Prove that the points 0, - 5 4, 3 and - 4, - 3 lie on the circle centred at the origin y with radius 5. In the Fig. Give reason. If origin is the centre of a circle with radius 17 units, find the coordinates of any four points on the circle which are not on the axes. Use the Pythagorean triplets The radius of the circle with centre at the origin is 10 units.
Write the coordinates of the point where the circle intersects the axes. Find the distance between any two of such points. Points to Remember Two perpendicular lines are needed to locate the position of a point in a plane.
In rectangular coordinate systems one of them is horizontal and the other is vertical. These two horizontal and vertical lines are called the coordinate axes x-axis and y-axis The point of intersection of x-axis and y-axis is called the origin with coordinates 0, 0 The distance of a point from y-axis is x coordinate or abscissa and the distance of the point from x-axis is called y coordinate or ordinate. Flag for inappropriate content. Related titles. Jump to Page.
Search inside document. Cantors work was the concept of set is vital to mathematical thought and is being used in almost every branch of mathematics. A Solution i 3! For example, i ii rk Let A be the set of even natural numbers less than Reading Notation n A number of elements in the set A the cardinal number of the set A is denoted by n A.
So, the prime factors of 12 are 2, 3. B two sets A and B are said to be equal if they contain exactly the same elements, regardless of order. Note i ii iii every set is a subset of itself i. Find the cardinal number of the following sets i ii iii iv v 7. Which of the following sets are equal? Which of the sets are equal sets? State the reason. Key Concept Universal Set the set that contains all the elements under consideration in a given discussion is called the universal set.
Shade the region Al U B Step 2: Shade the region Bl Al , Bl shaded portion Fig. Theory of Sets Competition Percentage of Students elocution 55 Drawing 45 Both 20 Chapter 2 In this chapter we discuss some properties of real numbers. The smallest natural number is 1, but there is no largest number as it goes up continuously.
The smallest whole number is 0 Remark 1 Every natural number is a whole number. Remark Think and answer! Is zero a positive integer or a negative integer? We have represented rational numbers on the number line.
It is true that there is a pattern in this decimal expansion, but no block of digits repeats endlessly and so it is not recurring. Key concept Irrational Number A number having a non-terminating and non-recurring decimal expansion is called an irrational number.
Note In fact, we can generate infinitely many non-terminating and non-recurring decimal expansions by replacing the digit 8 in 1 by any natural number as we like. Find any three irrational numbers between Find any two irrational numbers between 3 and 3. These are square roots of rational 3 numbers, which cannot be expressed as squares of any rational number.
Key Concept 2, 3, 7 etc. In the table given below both the columns A and B have irrational numbers. Solution The orders of the given irrational numbers are 2, 3 and 4. Now we convert each irrational number as of order Now, we convert each irrational number as of order Descending order: Express the following surds in its simplest form. Express the following as pure surds. Which is greater? Cantors work was the concept of set is vital to mathematical thought and is being used in almost every branch of mathematics.
In mathematics, sets are convenient because all mathematical structures can be regarded as sets. Understanding set theory helps us to see things in terms of systems, to organize things into sets and begin to understand logic.
In chapter 2, we will learn how the natural numbers, the rational numbers and the real numbers can be defined as sets. In this chapter we will learn about the concept of set and some basic operations of set theory.
We often deal with a group or a collection of objects, such as a collection of books, a group of students, a list of states in a country, a collection of coins, etc. Set may be consider of as a mathematical way of representing a collection or a group of objects. The objects of a set are called elements or members of the set. The main property of a set in mathematics is that it is well-defined.
This means that given any object, it must be clear whether that object is a member element of the set or not. Which of the following collections are well-defined? Therefore, 4 is not a set. Generally, sets are named with the capital letters A, B, C, etc. Reading Notation! A 3 is an element of A, written as 3! Fill in the blank spaces with the appropriate symbol!
A iii A Solution i 3! This is known as the Descriptive form of specification. For example, i ii iii the set of all natural numbers. For example, i ii rk Let A be the set of even natural numbers less than By convention, the elements in a set should not be repeated.
Let A be the set of letters in the word CoFFee, i. So, in roster form of the set A the following are invalid. Representation of sets in Different Forms Descriptive Form the set of all vowels in english alphabet the set of all odd positive integers less than or equal to 15 the set of all perfect cube numbers between 0 and Example 1. So, the set contains the elements 1, 2, 3, 4, 5, 6, 7, 8. Reading Notation n A number of elements in the set A the cardinal number of the set A is denoted by n A.