Number System, Mathematics Assignment Help

mathematics - Number System, Mathematics

Numbers and numerals: A number is an idea which answers the question "how many objects in a collection?"

The numerals are mere words or symbols by means of which numbers are represented.

Natural numbers: The number 1, 2, 3,..... used for counting are called natural numbers.

Whole number: All counting numbers together with 'zero' are called whole numbers and

Integers: The numbers 0, 1, - 1, 2, - 2, 3, - 3, .... are called integers, numbers 1, 2, 3, ..... are called positive integers and the numbers - 1, - 2, - 3, .... are called negative integers. Zero is called zero integers and is neither positive nor negative. The set of integers is denoted by 1 or Z.

Rational number: A number of the form p/q where p and q are integers and q ¹ 0 is called a rational number.

Every integer is a rational number and every fraction is a rational number.

Irrational numbers: Number which are' not rationals are called irrational numbers. Thus, irrational numbers cannot be written in the form  where, p. q. Î I and q ¹ 0. Note: An irrational number is a non terminating and non-repeating decimal.

Real numbers: The union of the set of rational numbers and the set of irrational numbers is the set of real numbers. Its set is denoted by R.

Note:

(1)      Completeness properties of number-line: There is one-to-one correspondence between the set of real numbers and the set of all points on the number line.

(2)      Absolute value: Absolute or modulus value of a real number x, denoted by |x|, is defined as

|x| = { x if x>= 0 or -x if x < 0 }

 (3)     Properties of R

Laws of Addition:

(i)       a + b Î R, for all a, b Î R (Closure law)

(ii)      a + b = b + a, all, a, b Î R (Commutative law)

(iii)     a + (b + c) = (a + b) + c; for all a, b, c Î R (Associative law)

(iv)     There exists an identity element o in R such that a + 0 = 0 + a = a (Existence of additive identity)

(v)      For every a Î R, there exists an additive inverse in R, denoted by -a, such that a + (-a) = (-a) + a = 0.

(Existence of Additive inverse)

 

Laws of Multiplication:

(i)       ab Î R, for all a, b Î R (Closure law)

(ii)      a × b = b × a, for all a, b Î R (Commutative law)

(iii)     a × (b × c) = (a × b) × c: for all a, b, c Î R (Associative law)

(iv)     There exists a unity element in R, denoted by 1 such that a × 1 = 1 × a = a (Existence of unity element)

(v)      For every a (¹ 0) Î R, there exists a multiplicative inverse in R, denoted by 1/2 such that a*1/a*1/a = 1

(Existence of multiplicative inverse).

Distributive Laws:

(i)       a × (b + c) = a × b + a × c: for all a, b c Î R

(ii)      (b + c) × a = b × a + c × a; for all a, b, c Î R.

 

(4)      Order relations in R: The order relations in R are used exactly in the same way as in the set Q of rational numbers.

If x, y Î R, then one and only one of the following conditions must be satisfied.

(i)       x = Y (ii) x > Y (iii)   x < y (Law of Trichotomy)

          (ii)      If x, y, Z Î R, then x < y and y < Z

Þ X < z. (Transitivity)

          (iii)     If x, y, Z Î R and x < y, then x + Z < Y + Z and conversely.

(iv)     If x, y, Z Î R and y < Z, then for x > 0, xy < xz.

Multiple: A number which is an exact time of a given number is called the multiple.

Factor: A number which divides the given number exactly is called factor of the given number.

Prime number: It is a natural number, other than 1, which has no other factor except 1 and itself, e.g., 2, 3, 5, 7, 11 are prime numbers.

Composite numbers: It is a natural number different from 1, which has at least one factor other than 1 and itself, e.g., 4, 6, 8, 9, 10, 12 are composite numbers.

Note:

Zero and one are neither prime nor composite numbers.

Division Algorithm:

Dividend = Divisor × Quotient + Remainder.

Tests of Divisibility:

(i)       A number is divisible by 2, if its last digit is divisible by 2.

(ii)      A number is divisible by 3, if the sum of digits is divisible by 3.

(iii)     A number is divisible by 4, if its last two digits represent number divisible by 4.

(iv)     A number is divisible by 5, if it ends in 5 or 0.

(v)      A number is divisible by 6, if it is divisible by 2 and 3.

(vi)     A number is divisible by 8, if its last three digits represent a number divisible by 8.

(vii)    A number is divisible by 9, if the sum of digits is divisible by 9.

(viii)    A number is divisible by 10, if it ends with 0.

(ix)     A number is divisible by 11, if the difference between the sum of the digits in odd places and the sum of the digits in even places is either divisible by 11 or is equal to zero.

Unique factorization theorem:

Every natural number greater than 1 can be factorised into primes and except for order of factors, the factorisation is unique.

H.C.F.: The Highest Common Factor (H.C.F.) of two or more given numbers is the greatest number which divides each of given numbers exactly.

L.C.M. The Least Common Multiple (L.C.M.) of two given numbers is that lowest number which is divisible exactly by each of the given numbers.

Note: Product of two numbers = Product of their H.C.F and L.C.M.

Note:

(1)      The absolute value of an integer is the integer without considering its siqn, Absolute value of zero is taken as zero.

(2)      The integer 'zero' is less than every positive integer and greater than every negative integer.

At a glance:

 

Representation of rational numbers as decimals: We know that every rational number when expressed in decimal form is expressible either in terminating or in repeating decimal form.

Representation of a terminating decimal as a rational number: Rule: Put 1 under the decimal point in the denominator and annex as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and put the fraction in the simplest form.

Recurring or Repeating Decimals: In such decimals a digit or a block of digits repeats itself again and again. We represent such decimals by putting a bar on repeated digit or digits.

(i)        Pure Recurring Decimals: Decimal in which all the figures after the decimal point are repeated, is known as a pure recurring decimal.

 

(ii)       Mixed Recurring Decimals: A decimal in which at least one figure after the decimal point repeated and then a figure or a set of figures is repeated, is known as a mixed recurring decimal.

Note:

If the denominator of a rational number (in standard form) contains no prime factors, other than 2 and 5, then this rational number can always be expressed in terminating decimal form. Otherwise, it is expressed as a non terminating repeating decimal.

Properties:

(1)      The set of rational number is closed for addition, multiplication and subtraction. Addition (or multiplication) of two rational numbers is a rational number.

(2)      The sets of positive and negative rational numbers are closed for division.

(3)      Division of two rational numbers is a rational number.

(4)      The sets of rational numbers can be represented by points on the number line.

Properties of Rational Numbers : " a, b, C Î Q.

(i)       " a Î Q, 0 Î Q s. t. a + 0 = 0 + a = a [0 is known as additive identity]

(ii)      " a Î Q, - a Î Q such that a + (-a) = (-a) + a = 0 [Additive inverse]

(iii)     " a Î Q, 1 Î Q, such that 1 a = a. 1 = a [Multiplicative identity]

(V)     " a, b, c Î Q,

a. (b + c) = a. b + a. c.

Also (a + b). c = a. c + b. c [Distributive law]

Order relation in Q.

(i)       Law of trichotomy:  " a, b c Î Q one and only one of the following possibilities holds: a > b, a = b or a < b.

(ii)      Law of transitivity: a > b and b > C Þ a > c.

(iii)     Monotone law for addition: a > b Þ a + c > b + c " a, b, c Î Q.

(iv)     Monotone law for multiplication: a > b, C > 0 Þ ac > bc " a, b, c Î Q.

 

Q as ordered field: The set of rational numbers is an ordered field with respect to operations of addition and multiplication.

Archimedean property: Between any two distinct rational numbers a and b, there lies an infinite number of rational numbers. 

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