Questions and Answers
What is the set of numbers that includes all rational and irrational numbers?
What is the purpose of Euclid's Division Lemma?
What is a characteristic of irrational numbers?
What is the decimal expansion of a rational number?
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What is a repeating decimal?
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What rule should be followed when performing operations on real numbers?
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What is the general form of a polynomial?
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What is the degree of the polynomial 3x^4 + 2x^2 + x?
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What is a monomial?
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What is the Remainder Theorem?
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What is an algebraic identity?
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What is a zero of a polynomial?
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Study Notes
Real Numbers
Introduction
 Real numbers are a set of numbers that include all rational and irrational numbers.
 They can be represented on a number line.
Euclid's Division Lemma
 If
a
andb
are positive integers, then there exist unique integersq
andr
such thata = bq + r
, where0 ≤ r < b
.  This lemma is used to find the HCF (Highest Common Factor) of two numbers.
The Fundamental Theorem of Arithmetic
 Every composite number can be expressed as a product of prime numbers in a unique way, except for the order in which they are written.
 This theorem is used to find the prime factorization of a number.
Irrational Numbers
 Irrational numbers are real numbers that cannot be expressed as a ratio of integers (e.g., √2, π, e).
 They can be represented as nonterminating, nonrepeating decimals.
Decimal Expansion of Rational Numbers
 The decimal expansion of a rational number is either terminating or nonterminating, recurring.
 Examples: 1/2 = 0.5 (terminating), 1/3 = 0.333... (nonterminating, recurring)
Repeating Decimals
 A repeating decimal is a decimal that has a block of digits that repeats infinitely.
 Examples: 1/3 = 0.333... (repeating decimal), 1/7 = 0.142857... (repeating decimal)
Operations on Real Numbers
 Real numbers can be added, subtracted, multiplied, and divided, following the usual rules of arithmetic.
 The order of operations (PEMDAS) should be followed when performing operations on real numbers.
Properties of Real Numbers
 Commutative property:
a + b = b + a
,a × b = b × a
 Associative property:
(a + b) + c = a + (b + c)
,(a × b) × c = a × (b × c)
 Distributive property:
a × (b + c) = a × b + a × c
 Existence of additive and multiplicative identities (0 and 1, respectively) and inverse elements.
Real Numbers
 Real numbers include rational and irrational numbers and can be represented on a number line.
Euclid's Division Lemma
 The lemma states that for positive integers
a
andb
, there exist unique integersq
andr
such thata = bq + r
, where0 ≤ r < b
.  This lemma is used to find the Highest Common Factor (HCF) of two numbers.
The Fundamental Theorem of Arithmetic
 Every composite number can be expressed as a product of prime numbers in a unique way, except for the order in which they are written.
 This theorem is used to find the prime factorization of a number.
Irrational Numbers
 Irrational numbers are real numbers that cannot be expressed as a ratio of integers (e.g., √2, π, e).
 They can be represented as nonterminating, nonrepeating decimals.
Decimal Expansion of Rational Numbers
 The decimal expansion of a rational number is either terminating or nonterminating, recurring.
 Examples: 1/2 = 0.5 (terminating), 1/3 = 0.333... (nonterminating, recurring)
Repeating Decimals
 A repeating decimal is a decimal that has a block of digits that repeats infinitely.
 Examples: 1/3 = 0.333... (repeating decimal), 1/7 = 0.142857... (repeating decimal)
Operations on Real Numbers
 Real numbers can be added, subtracted, multiplied, and divided, following the usual rules of arithmetic.
 The order of operations (PEMDAS) should be followed when performing operations on real numbers.
Properties of Real Numbers
 Commutative property:
a + b = b + a
,a × b = b × a
.  Associative property:
(a + b) + c = a + (b + c)
,(a × b) × c = a × (b × c)
.  Distributive property:
a × (b + c) = a × b + a × c
.  Existence of additive and multiplicative identities (0 and 1, respectively) and inverse elements.
Polynomials
 A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
 The general form of a polynomial is:
a_n x^n + a_(n1) x^(n1) +...+ a_1 x + a_0
.  The degree of a polynomial is the highest power of the variable (x) in the polynomial.
 Polynomials can be classified into three types: monomial, binomial, and trinomial.
 A monomial is a polynomial with only one term.
 A binomial is a polynomial with two terms.
 A trinomial is a polynomial with three terms.
Remainder Theorem
 If
p(x)
is a polynomial andp(a) = 0
, then(x  a)
is a factor ofp(x)
.  If
p(x)
is divided by(x  a)
and the remainder isr
, thenp(a) = r
.
Factor Theorem
 If
(x  a)
is a factor ofp(x)
, thenp(a) = 0
.  If
p(a) = 0
, then(x  a)
is a factor ofp(x)
.
Algebraic Identities

(x + y)^2 = x^2 + 2xy + y^2
. 
(x  y)^2 = x^2  2xy + y^2
. 
x^2 + y^2 = (x + y)^2  2xy
. 
(x + y)(x  y) = x^2  y^2
.
Zeroes of a Polynomial
 A zero of a polynomial
p(x)
is a value ofx
that makes the polynomial equal to zero.  The number of zeroes of a polynomial is equal to its degree.
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Description
Explore the basics of real numbers, Euclid's Division Lemma, and the Fundamental Theorem of Arithmetic. Learn how to represent real numbers and apply theorems to find the HCF of two numbers.