Grade 9 Mathematics
Simone Lau
Numbers
Mathematical numbers:
Natural numbers, also known as counting numbers, are the numbers beginning with 0, with each
successive number greater than its predecessor by 1. If the set of natural numbers is denoted by
N, then
N = {0,1,2,3,…}
Whole numbers are similar to natural numbers. Therefore, if the set of whole numbers is
denoted by W, then
W = {0,1,2,3,…}
Integers are whole numbers that can be both positive and negative. If the set of integers is
denoted by Z, then
Z = {…, -2,-1,0,1,2,… }
Rational numbers are the numbers that can be represented as the quotient of two integers p
and q, where q is not zero. If the set of rational numbers is denoted as Q, then
Q = {all x, where x = p/q, p and q are integers, q is not zero}
Rational numbers can be represented as:
1. Integers: (4/2) = 2, (12/4) = 3
2. Fractions: 3/4, 13/3
3. Terminating Decimals: (3/4) = 0.75, (6/5) = 1.2
4. Repeating Decimals: (13/3) = 4.333…., (4/11) = 0.363636
Conversely, irrational numbers are the numbers that cannot be represented as the quotient of
two integers, i.e., irrational numbers cannot be rational numbers and vice-versa. If the set of
irrational numbers is denoted by H, then
H = {all x, where there exists no integers p and q such that x = p/q, q is not zero}
Real numbers are numbers that are either rational or irrational, which basically means all
numbers. If the set of real numbers is denoted by R, then
R = {all x in Q or H}
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Grade 9 Mathematics
Simone Lau
Numbers
Real Number System:
The real number system is comprised of the set of real numbers and the arithmetic
operations of addition and multiplication (subtraction, division, and other operations are
derived from these two).
Real Number Line
The set of real numbers can be pictorially shown by the real number line. It is a straight
line, whose origin is designated by the number 0, and continues in both directions.
Properties
All real numbers have the following properties:
1. Reflexivity
For any real number a, a = a.
Ex: 3 = 3, y = y, x + z = x + z (x,y,z are real numbers)
2. Symmetry
For any real numbers a and b, if a = b, then b = a.
Ex: If 3 = 1 + 2, 1 + 2 = 3
3. Transivity
For any real numbers a, b and c, if a = b and b = c, then a = c.
Ex: If 2 + 3 = 5 and 5 = 1 + 4, then 2 + 3 = 1 + 4.
4. Substitution
For any real numbers a and b, if a = b, then a may be replaced by b, and b may be
replaced by a, in any mathematical statement without changing then meaning of
the statement.
Ex: If a = 3 and a + b = 5, then 3 + b = 5.
5. Trichotomy
For any real numbers a and b, one and only one of the following conditions
holds:
(1) a is greater than b ( a > b)
(2) a is equal to b
( a = b)
(3) a is less than b
( a < b)
Example: 3 < 4 , 4 + 2 = 6 , 7 > 5
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Grade 9 Mathematics
Simone Lau
Numbers
Polynomials
A monomial is a term containing variables with only non-negative integers as exponents.
A monomial refers to a polynomial with just one term. Similarly, a polynomial with two
terms is a binomial; a polynomial with three terms is a trinomial.
The degree of a polynomial is the degree of the monomial (in the polynomial) with the
highest degree.
Examples of polynomials:
(1) 3x is a monomial of degree 1.
(2) 3x + 4y3 is a binomial of degree 3.
Numerical Representations in Algebra
Equations
An algebraic equation is a mathematical statement equating two algebraic expressions,
i.e., it states that one algebraic expression is equal to another algebraic expression.
The following additive and multiplicative properties apply to equations:
Addition Property of Equality:
For all real numbers a, b, and c,
a = b if and only if a + c = b + c.
Multiplication Property of Equality:
For all real numbers a, b, and c, where c ≠ 0, a = b if and only if ac = bc.
Inequalities
An algebraic inequality is a mathematical statement comparing two unequal algebraic
expressions. It states that one algebraic expression is greater than (>), less than (<),
greater than or equal to (≥), or less than or equal to (≤).
The following additive and multiplicative properties apply to inequalities.
Addition Property of Inequality:
For all real numbers a, b, and c, a > b if and only if a + c > b + c.
Multiplication Property of Inequality:
(1) If c > 0, then a > b if and only if ac > bc.
(2) If c < 0, then a > b if and only if ac < bc.
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Grade 9 Mathematics
Simone Lau
Numbers
Exponents
The concept of exponentiation, which represents "repeated multiplication" for integer
exponents, can be extended to the real numbers. The following laws for exponents apply
to all real numbers (and thus to all integers):
1.
2.
3.
4.
5.
6.
7.
xmxn = xm+n
xm/xn = xm-n
(xm) n = xmn
(xy) m = xmym
(a/b) m = am/bm
x0 = 1
b-n = 1/(b n)
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