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Maths project

R
Ruchi Dixit

The document discusses operations on real numbers, including: 1) Addition and subtraction of real numbers follows rules based on sign, where numbers with the same sign are added and different signs are subtracted. 2) Multiplication and division of real numbers results in a positive number if the signs are the same, and negative if different. 3) Properties like commutativity, associativity, identity, inverse and distribution apply to real numbers as they do to other types of numbers.

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1 | P a g e 1. Introduction Real Number The collection of all rational numbers and irrational numbers together is the set of real numbers. A real number is either a rational number or an irrational number. In a division, if remainder becomes zero after certain stage, then the decimal expansion is terminating. If the remainder never becomes zero but repeats after certain stage, then the decimal expansion is non-terminating recurring. A rational number can be expressed as its decimal expansion. The decimal expansion of a rational number is either terminating or non-terminating recurring. The decimal expansion of an irrational number is non-terminating non-recurring. Every real number can be represented on a number line uniquely. Conversely every point on the number line represents one and only one real number. The process of visualisation of representing a decimal expansion on the number line is known as the process of successive magnification. The sum, difference and the product of two rational numbers is always a rational number. The quotient of a division of one rational number by a non-zero rational number is a rational number. Rational numbers satisfy the closure law under addition, subtraction, multiplication and division. They also satisfy the commutative law and associative law under addition and multiplication. Operation on Real Number: The sum, difference, multiplication and division of irrational numbers are not always irrational. Irrational numbers do not satisfy the closure property under addition, subtraction, multiplication and division. Real numbers satisfy the commutative, associative and distributive laws. These can be stated as: Commutative law of addition: a + b = b + a Commutative law of multiplication: a b = b a Associative law of addition: a + (b + c) = (a + b) + c Associative law of multiplication: a (b c) = (a b) c Distributive law: a (b + c) = (a b) + (a c) or (a + b) c = (a c) + (b c)
2 | P a g e Real numbers can be represented on the number line. The square root of any positive real number exists and that also can be represented on number line. The sum or difference of a rational number and an irrational number is an irrational number. The product or division of a rational number with an irrational number is an irrational number. Some of the basic identities involving square roots are: If a, b, c and d are positive real numbers, ab = ab ab = ab (a + b)(a b) = a b (a + b)(a b) = a2 b (a + b)(c + d) = ac + ad + bc + bd (a + b)2 = a + 2ab + b If the product of two irrational numbers is a rational number, then each of the irrational numbers is called the rationalising factor of the other. The process of multiplying an irrational number with its rationalising factor to get the product as a rational number is called rationalisation of the given irrational number.
3 | P a g e 2. History : Simple fractions have been used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" ("The rules of chords") in,c. 600 BC, include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians since Manava (c. 750690 BC), who were aware that the square roots of certain numbers such as 2 and 61 could not be exactly determined. Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. The Middle Ages brought the acceptance of zero, negative, integral, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects, which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. The Egyptian mathematician Ab笛 Kmil Shuj ibn Aslam (c. 850930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots. In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard. In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones. In the 18th and 19th centuries there was much work on irrational and transcendental numbers. Johann Heinrich Lambert (1761) gave the first flawed proof that cannot be rational; Adrien-Marie Legendre (1794) completed the proof, and showed that is not the square root of a rational number. Paolo Ruffini (1799) and Niels Henrik Abel (1842) both constructed proofs of the AbelRuffini theorem: that the general quantic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots. variste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois Theory. Joseph Liouville (1840) showed that neither e nor e2 can be a root of an integer quadratic, and then established the existence of transcendental numbers; Georg Cantor (1873) extended and greatly simplified this proof.[8] Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David (1893), and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871. In 1874 he showed that the set of all real numbers is uncountable infinite but the set of all algebraic numbers is countable infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. See Cantor's first uncountability proof.
4 | P a g e 3. Observation: Operations on Real Numbers (positive and negative) Real numbers include numbers on both sides of the number line. We need to be able to perform arithmetic operations (+, -, x, 歎) on all real numbers, whether they are positive or negative. . Positive and Negative Numbers Remember that on the number line, values increase in positive value when we move right, and numbers increase in negative value when we move left. If we think of positive and negative numbers in terms of money, we can consider positive value as assets and negative value as debt. When we accumulate more assets we have more money, but accumulation of debt sends us further into the negative direction. Add a Positive Amount so, if we are adding a positive amount to a number, we move to the right along the number line.
5 | P a g e Add a Negative Amount If we add a negative amount to a number, we move that number of units to the left. Rules for Addition of Real Numbers We can make some general rules so we dont have to draw a number line for every problem: When adding numbers with like signs, we add their absolute values and use the sign of the original addends. 2.5 + 3 = |2.5| + |3| = 5.5 -1 + (-4) = - (|-1| + |-4|) = - (1 + 4) = -5 Subtraction Subtraction is the opposite of addition, so we can restate any subtraction problem as adding the inverse (additive inverse) of the number. This is helpful when we are working with negative and positive numbers. 5 2 = 5 + (-2) = 3 -1.25 2.5 = -1.25 + (-2.5) = -3.75 -1 (-1.3) = -1 + 1.3 = 0.3 Multiplication We know that when we multiply two positive numbers we get a positive result. 2 * 4 = 8 but what if one or both of the numbers are negative? Think in terms of money. If ` 2 is deducted from your bank account 4 times, you are down `8. (-2) * 4 = -8 however, if that deduction were reversed 4 times, then you would be ahead `8. (-2) * (-4) = 8
6 | P a g e Division Here is an example of division involving negative numbers, Lets say you have a debt of `30, represented by -30 Now, two good friends come along and offer to help pay the debt and split it 3 ways (-30) 歎3 = -10 You are still in debt, but only `10 now. Now, if the debt is reversed as well as divided, the problem looks like this: (-30) 歎 (-3) = 10 The result is a credit to each rather than debt. Generalization We can make some general rules for multiplication and division of real numbers. If the signs are the same (both positive or both negative) we get a positive result. 2 * 2 = 4 (-2) * (-2) = 4 4 歎 2 = 2 (-4) 歎 (-2) = 2 If the signs are different, then the result is negative. 2 * (-2) = -4 (-2) * 2 = -4 4 歎 (-2) = -2 (-4) 歎 2 = -2 Multiple factors Note that if we have an odd number of negatives, the result will be negative and if we have an even number (or zero) negatives, the result will be positive Impact on Exponents What happens when we raise a negative number to some power? = (-2) * (-2) * (-2) * (-2) * (-2) = -32 we have an odd number of negative factors = (-2) * (-2) * (-2) * (-2) * (-2) * (-2) = 64 we have an even number of negative factors. To generalize: When raising a negative number to a power, if the exponent is odd, the result will be negative; if the exponent is even, the result will be positive.
7 | P a g e Commutative Property - interchange or switch the elements Example shows commutative property for addition: X + Y = Y + X Think of the elements as "commuting" from one location to another. "They get in their cars and drive to their new locations." This explanation will help you to remember that the elements are "moving" (physically changing places). = Associative Property- regroup the elements Example shows associative property for addition: (X + Y) + Z = X + (Y + Z) The associative property can be thought of as illustrating "friendships" (associations). The parentheses show the grouping of two friends. In the example below, the red girl (y) decides to change from the blue boyfriend (x) to the green boyfriend (z). "I don't want to associate with you any longer!" Notice that the elements do not physically move, they simply change the person with whom they are "holding hands" (illustrated by the parentheses.) Identity Property- What returns the input unchanged? X + 0 = X Additive Identity X * 1 = X Multiplicative Identity Try to remember the "I" in the word identity. Variables can often times have an "attitude". "I am the most important thing in the world and I do not want to change!" The identity element allows the variable to maintain this attitude.
8 | P a g e Additive Identity is 0. Multiplicative Identity is 1. Inverse Property- What brings you back to the identity element using that operation? X + -X = 0 Additive Inverse X * 1/X = 1 Multiplicative Inverse Think of the inverse as "inventing" an identity element. What would you need to add (multiply) to this element to turn it into an identity element? The Additive Inverse is the negation of the element. ..The Multiplicative Inverse is one divided by the element.
9 | P a g e Distributive Property- multiply across the parentheses. Each element inside the parentheses is multiplied by the element outside the parentheses. a(b + c) = a*b + a*c Let's consider the problem 3(x + 6). The number in front of the parentheses is "looking" to distribute (multiply) its value with all of the terms inside the parentheses. = 3x + 18 4. Result This project has made attempt to show how operations on real number works. Project is good for understanding the concepts.

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Maths project

  • 1. 1 | P a g e 1. Introduction Real Number The collection of all rational numbers and irrational numbers together is the set of real numbers. A real number is either a rational number or an irrational number. In a division, if remainder becomes zero after certain stage, then the decimal expansion is terminating. If the remainder never becomes zero but repeats after certain stage, then the decimal expansion is non-terminating recurring. A rational number can be expressed as its decimal expansion. The decimal expansion of a rational number is either terminating or non-terminating recurring. The decimal expansion of an irrational number is non-terminating non-recurring. Every real number can be represented on a number line uniquely. Conversely every point on the number line represents one and only one real number. The process of visualisation of representing a decimal expansion on the number line is known as the process of successive magnification. The sum, difference and the product of two rational numbers is always a rational number. The quotient of a division of one rational number by a non-zero rational number is a rational number. Rational numbers satisfy the closure law under addition, subtraction, multiplication and division. They also satisfy the commutative law and associative law under addition and multiplication. Operation on Real Number: The sum, difference, multiplication and division of irrational numbers are not always irrational. Irrational numbers do not satisfy the closure property under addition, subtraction, multiplication and division. Real numbers satisfy the commutative, associative and distributive laws. These can be stated as: Commutative law of addition: a + b = b + a Commutative law of multiplication: a b = b a Associative law of addition: a + (b + c) = (a + b) + c Associative law of multiplication: a (b c) = (a b) c Distributive law: a (b + c) = (a b) + (a c) or (a + b) c = (a c) + (b c)
  • 2. 2 | P a g e Real numbers can be represented on the number line. The square root of any positive real number exists and that also can be represented on number line. The sum or difference of a rational number and an irrational number is an irrational number. The product or division of a rational number with an irrational number is an irrational number. Some of the basic identities involving square roots are: If a, b, c and d are positive real numbers, ab = ab ab = ab (a + b)(a b) = a b (a + b)(a b) = a2 b (a + b)(c + d) = ac + ad + bc + bd (a + b)2 = a + 2ab + b If the product of two irrational numbers is a rational number, then each of the irrational numbers is called the rationalising factor of the other. The process of multiplying an irrational number with its rationalising factor to get the product as a rational number is called rationalisation of the given irrational number.
  • 3. 3 | P a g e 2. History : Simple fractions have been used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" ("The rules of chords") in,c. 600 BC, include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians since Manava (c. 750690 BC), who were aware that the square roots of certain numbers such as 2 and 61 could not be exactly determined. Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. The Middle Ages brought the acceptance of zero, negative, integral, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects, which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. The Egyptian mathematician Ab笛 Kmil Shuj ibn Aslam (c. 850930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots. In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard. In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones. In the 18th and 19th centuries there was much work on irrational and transcendental numbers. Johann Heinrich Lambert (1761) gave the first flawed proof that cannot be rational; Adrien-Marie Legendre (1794) completed the proof, and showed that is not the square root of a rational number. Paolo Ruffini (1799) and Niels Henrik Abel (1842) both constructed proofs of the AbelRuffini theorem: that the general quantic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots. variste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois Theory. Joseph Liouville (1840) showed that neither e nor e2 can be a root of an integer quadratic, and then established the existence of transcendental numbers; Georg Cantor (1873) extended and greatly simplified this proof.[8] Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David (1893), and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871. In 1874 he showed that the set of all real numbers is uncountable infinite but the set of all algebraic numbers is countable infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. See Cantor's first uncountability proof.
  • 4. 4 | P a g e 3. Observation: Operations on Real Numbers (positive and negative) Real numbers include numbers on both sides of the number line. We need to be able to perform arithmetic operations (+, -, x, 歎) on all real numbers, whether they are positive or negative. . Positive and Negative Numbers Remember that on the number line, values increase in positive value when we move right, and numbers increase in negative value when we move left. If we think of positive and negative numbers in terms of money, we can consider positive value as assets and negative value as debt. When we accumulate more assets we have more money, but accumulation of debt sends us further into the negative direction. Add a Positive Amount so, if we are adding a positive amount to a number, we move to the right along the number line.
  • 5. 5 | P a g e Add a Negative Amount If we add a negative amount to a number, we move that number of units to the left. Rules for Addition of Real Numbers We can make some general rules so we dont have to draw a number line for every problem: When adding numbers with like signs, we add their absolute values and use the sign of the original addends. 2.5 + 3 = |2.5| + |3| = 5.5 -1 + (-4) = - (|-1| + |-4|) = - (1 + 4) = -5 Subtraction Subtraction is the opposite of addition, so we can restate any subtraction problem as adding the inverse (additive inverse) of the number. This is helpful when we are working with negative and positive numbers. 5 2 = 5 + (-2) = 3 -1.25 2.5 = -1.25 + (-2.5) = -3.75 -1 (-1.3) = -1 + 1.3 = 0.3 Multiplication We know that when we multiply two positive numbers we get a positive result. 2 * 4 = 8 but what if one or both of the numbers are negative? Think in terms of money. If ` 2 is deducted from your bank account 4 times, you are down `8. (-2) * 4 = -8 however, if that deduction were reversed 4 times, then you would be ahead `8. (-2) * (-4) = 8
  • 6. 6 | P a g e Division Here is an example of division involving negative numbers, Lets say you have a debt of `30, represented by -30 Now, two good friends come along and offer to help pay the debt and split it 3 ways (-30) 歎3 = -10 You are still in debt, but only `10 now. Now, if the debt is reversed as well as divided, the problem looks like this: (-30) 歎 (-3) = 10 The result is a credit to each rather than debt. Generalization We can make some general rules for multiplication and division of real numbers. If the signs are the same (both positive or both negative) we get a positive result. 2 * 2 = 4 (-2) * (-2) = 4 4 歎 2 = 2 (-4) 歎 (-2) = 2 If the signs are different, then the result is negative. 2 * (-2) = -4 (-2) * 2 = -4 4 歎 (-2) = -2 (-4) 歎 2 = -2 Multiple factors Note that if we have an odd number of negatives, the result will be negative and if we have an even number (or zero) negatives, the result will be positive Impact on Exponents What happens when we raise a negative number to some power? = (-2) * (-2) * (-2) * (-2) * (-2) = -32 we have an odd number of negative factors = (-2) * (-2) * (-2) * (-2) * (-2) * (-2) = 64 we have an even number of negative factors. To generalize: When raising a negative number to a power, if the exponent is odd, the result will be negative; if the exponent is even, the result will be positive.
  • 7. 7 | P a g e Commutative Property - interchange or switch the elements Example shows commutative property for addition: X + Y = Y + X Think of the elements as "commuting" from one location to another. "They get in their cars and drive to their new locations." This explanation will help you to remember that the elements are "moving" (physically changing places). = Associative Property- regroup the elements Example shows associative property for addition: (X + Y) + Z = X + (Y + Z) The associative property can be thought of as illustrating "friendships" (associations). The parentheses show the grouping of two friends. In the example below, the red girl (y) decides to change from the blue boyfriend (x) to the green boyfriend (z). "I don't want to associate with you any longer!" Notice that the elements do not physically move, they simply change the person with whom they are "holding hands" (illustrated by the parentheses.) Identity Property- What returns the input unchanged? X + 0 = X Additive Identity X * 1 = X Multiplicative Identity Try to remember the "I" in the word identity. Variables can often times have an "attitude". "I am the most important thing in the world and I do not want to change!" The identity element allows the variable to maintain this attitude.
  • 8. 8 | P a g e Additive Identity is 0. Multiplicative Identity is 1. Inverse Property- What brings you back to the identity element using that operation? X + -X = 0 Additive Inverse X * 1/X = 1 Multiplicative Inverse Think of the inverse as "inventing" an identity element. What would you need to add (multiply) to this element to turn it into an identity element? The Additive Inverse is the negation of the element. ..The Multiplicative Inverse is one divided by the element.
  • 9. 9 | P a g e Distributive Property- multiply across the parentheses. Each element inside the parentheses is multiplied by the element outside the parentheses. a(b + c) = a*b + a*c Let's consider the problem 3(x + 6). The number in front of the parentheses is "looking" to distribute (multiply) its value with all of the terms inside the parentheses. = 3x + 18 4. Result This project has made attempt to show how operations on real number works. Project is good for understanding the concepts.
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