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# Number Systems

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### Number Systems

1. 1. Maths Project NUMBER SYSTEMS
2. 2. REAL NUMBERS
3. 3. Real Numbers Rationa l Irrational
4. 4. INTRODUCTION
5. 5. In mathematics, a real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 (1.41421356… the square root of two, an irrational algebraic number) and π (3.14159265…, a transcendental number).
6. 6. Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. The reals are uncountable, that is, while both the set of all natural numbers and the set of all real numbers are infinite sets.
7. 7. BASIC PROPERTIES
8. 8. A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. Real numbers are used to measure continuous quantities. They may be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as
9. 9. More formally, real numbers have the two basic properties :The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication.
10. 10. The second says that if a nonempty set of real numbers has an upper bound, then it has a real least upper bound. The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property.
11. 11. It is divided into two parts :- Rational And Irrational
12. 12. Real Numbers
13. 13. RATIONAL NUMBERS
14. 14. INTRODUCTION
15. 15. In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number.
16. 16. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number
17. 17. Rational Numbers are divided into three main parts :Integers Whole Numbers Natural Numbers
18. 18. 1. INTEGERS
19. 19. An integer is a number that can be written without a fractional or decimal component. For example, 21, 4, and −2048 are integers; 9.75, 5½, and √2 are not integers. The set of integers is a subset of the real numbers, and consists of the natural numbers (0, 1, 2, 3, ...) and the negatives of the non-zero natural numbers (−1, −2, −3, ...).
20. 20. 2. WHOLE NUMBERS
21. 21. Whole number is collection of positive numbers and zero. Whole number also called as integer. The whole number is represented as {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ….}. The set of whole numbers may be finite or infinite. The finite defines the numbers in the set are countable. Infinite set means the numbers are uncountable. . Zero is neither a fraction nor a decimal, so zero is an
22. 22. 3. NATURAL NUMBERS
23. 23. In mathematics, the natural numbers are those used for counting and ordering . Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. The natural numbers had their origins in the words used to count things, beginning with the number 1.
24. 24.  The addition (+) and multiplication (×) operations on natural numbers have several algebraic properties:  Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.  Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.
25. 25.  Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.  Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.  Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c)
26. 26. No zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0.
27. 27. IRRATIONAL NUMBERS
28. 28. INTRODUCTION
29. 29. In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero. Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating
30. 30. It has been suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava (c. 750 – 690 BC) believed that the square roots of numbers such as 2 and 61 could not be exactly determined. However, historian Carl Benjamin Boyer states that "...such
31. 31. THANKING YOU Name :- Hardik Agarwal Class :- IX C Roll No. :- 10