Closure
a + b is a real number; when you add two real numbers, the result is also
a real number
Example: 3 and 7 are both real numbers, 3+7=10 and the sum , 10, is also a
real number.
a – b is a real number; when you subtract two real numbers the result is
also a real
number.
Example: 2 and 5 are both real numbers, 2 – 5 = -3, and the difference,
-3, is also a real
number.
(a)(b) is a real number; when you multiply two real numbers, the result
is also a real
number.
Example: 9 and -2 are both real numbers; (9)(-2) = -18, and the product ,
-18 is also a
real number.
a / b is a real number when b ≠ 0; when you divide two real numbers, the
result is also a
real number unless the denominator (divisor) is zero.
Example: -10 and 5 are both real numbers, -10 / 5 = -2, and the quotient,
-2, is also a real
number.
Commutative
a + b = b + a; you can add numbers in either order and get the same
answer.
Example: 2 + 6 = 8 and 6 + 2 = 8 so 2 + 6 = 6 + 2
(a)(b) = (b)(a); you can multiply numbers in either order and get the
same answer.
Example: (7)(10) = 70 and (10)(7) = 70 so (7)(10) = (10)(7)
a – b ≠ b – a; you cannot subtract in any order and get the same answer.
Example: 4 – 6 = -2, but 6 – 4 = 2. There is no commutative property for
subtraction .
a / b ≠ b/a; you cannot divide in any order and get the same answer.
Example : 4/2 = 2, but 2/4 =.5 so there is no commutative property for
division.
Associative
(a + b) + c = a + (b + c); you can group any numbers in any arrangement
when adding
and get the same answer.
Example: (1 + 2) + 3 = 3 + 3 = 6 and 1 + (2 + 3) = 1 + 5 = 6 so (1 + 2) +
3 = 1 + (2 + 3).
(ab)c = a(bc); you can group any numbers in any arrangement when
multiplying and get
the same answer.
Example: (2 x 6)3 = (12)3 = 36 and 2(6 x 3) = 2(18) = 36 so (2 x 6)3 =
2(6 x 3)
The associative property does not work for subtraction or division.
Identities
a + 0 = a; zero is the identity for addition because adding zero does not
change the
original number.
Example: 7 + 0 = 7 and 0+7 = 7.
a(1) = a; one is the identity for multiplication because multiplying by one
does not
change the original number.
Example: 21(1) = 21 and (1)21=21.
Identities for subtraction and division become a problem. It is true that 29 – 0
= 29, but 0
– 29 = -29, not 29. This is also the case for division because 4/1 = 4, but 1/4
= .25, so the
identities do not hold when the numbers are reversed.
Inverses
a + (-a) = 0; a number plus its additive inverse (the numbers with the
opposite sign ) will
always equal zero.
Example: 6 + (-6) = 0 and (-6) + 6 = 0. The exception is zero because 0 +
0 = 0 already.
a(1/a) = 1; a number time its multiplicative inverse or reciprocal ( the
number written as
a fraction and flipped ) will always equal one.
Example: 5(1 / 5) = 1. The exception is zero because zero cannot be
multiplied by any
number and result in a product of one /
Distributive Property
• a(b + c) = ab + ac or a(b - c) = ab - ac; each term in the parentheses
must be multiplied
by the term in front of the parentheses
Example: 4(5 + 7) = 4(5) + 4(7) = 20 + 28 = 48. This is a simple
Example and the
distributive property is not required to find the answer. When the problem
involves a
variable however, the distributive property is a necessity.
Example: 4(5a + 7) = 4(5a) + 4(7) = 20a + 28.
Properties of Equality
Reflexive : a = a; both sides of the equation are identical
Example: 6+k = 6+k
Symmetric: If a = b then b = a. This property al lows you to exchange the
two sides of an
equation.
Example: 4a – 7 = 9 - 7a + l5 becomes 9 - 7a + 15 = 4a - 7.
Transitive: If a = b and b = c then a = c. This property allows you to
connect
statements which are each equal to the same common statement.
Example: 5a - 6 = 9k and 9k = a + 2; you can eliminate the common term 9k
and connect
the following into one equation: 5a - 6 = a+2.
Addition Property of Equality: If a = b then a + c = b + c. This property
allows you to
add any number or algebraic term to any equation as long as you add it to both
sides to
keep the equation true.
Example: 5 = 5; if you add 3 to one side and not the other the equation
becomes 8 = 5
which is false, but if you add 3 to both sides you get a true equation 8 = 8.
Also, 6a + 2 =
14 becomes 6a + 2 + (-2) = 14 + (-2) if you add -2 to both sides. The result is
the
equation 6a = 12
Multiplication Property of Equality: If a = b then ac = bc when c ≠ 0.
This property
allows you to multiply both sides of an equation by any nonzero value .
Example: If 4a = -24, then (4a)(0.25 ) = (-24)(0.25) and a = -6. Notice
that both sides of
the = were multiplied by 0.25.
DEFINITIONS
Natural or Counting numbers: {1, 2, 3, 4, 5,…}
Whole numbers: {0, 1, 2 , 3,…}
Integers:{....-4, -3, -2, -1, 0, 1, 2, 3, 4...}
Rational numbers: {p/q | p and q are integers, q ≠ 0}; the sets of
Natural numbers, Whole
numbers, and Integers, as well as numbers which can be written as proper or
improper
fractions, are all subsets of the set of Rational numbers.
Examples:
Irrational numbers: {x | x is a real number but is not a Rational number
}; the sets of
Rational numbers and Irrational numbers have no elements in common and are
therefore
disjoint sets.
Real numbers: {r | r is the coordinate of a point on a number line }; the
union of the set
of Rational numbers with the set of Irrational numbers equals the set of Real
numbers.
Imaginary numbers: {ai | a is a real number and i is the number whose
square is -1 }; i²
= -1: the sets of Real numbers and Imaginary numbers have no elements in common
and
are therefore disjoint sets.
Complex numbers: {a + bi | a and b are real numbers and i is the number
whose square is
-1 }; the set of Real numbers and the set of Imaginary numbers are both subsets
of the set
of Complex numbers .
Examples: 4 + 7i and 3 -2i are complex numbers.
OPERATIONS OF REAL NUMBERS
Vocabulary
Total or Sum is the answer to an addition problem. The numbers
added are called
addends
Example: In 5 + 9=14, 5 and 9 are addenda and 14 is the total or sum.
Difference is the answer to a subtraction problem. The number subtracted
is called the
subtrahend. The number from which the subtrahend is subtracted is called the
minuend.
Example: In 25 - 8 = 17, 25 is the minuend, 8 is the subtrahend, and 17
is the difference.
Product is the answer to a multiplication problem. The numbers multiplied
are each
called a factor.
Example: In 15 x 6 = 90, 15 and 6 are factors and 90 is the product.
Quotient is the answer to a division problem. The number being divided is
called the
dividend. The number that you are dividing by is called the divisor. If there is
a number
remaining after the division process has been completed, that number is called
the
remainder.
Example: In 45 ÷ 5 = 9 , which may also be written as 45/5,45 is the
dividend, 5 is the
divisor and 9 is the quotient.
An Exponent indicates the number of times the base is multiplied by
itself; that is, used
as a factor.
Example: In 5², 5 is the base and 2 is the exponent, or power, and 5² =
(5)(5) = 25, notice
that the base, 5, was multiplied by itself 2 times.
A Prime number is a natural numbers greater than 1 having exactly two
factors, itself and
one.
Examples: 7 is prime because the only two natural numbers that multiply
to equal 7 are 7
and 1; 13 is prime because the only two natural numbers that multiply to equal
13 are 13
and 1.
Composite numbers are natural numbers that have more than two factors.
Examples: 15 is a composite number because 1, 3, 5, and 15 all multiply
in some
combination to equal 15; 9 is composite because 1. 3, and 9 all multiply in some
combination to equal 9.
The Greatest Common Factor (GCF)or Greatest Common Divisor (GCD)
of a set of
numbers is the largest natural number that is a factor of each of the numbers in
the set;
that is, the largest natural number that will divide into all of the numbers in
the set
without leaving a remainder.
Example: The greatest common factor (GCF) of 12,30 and 42 is 6 because 6
divides
evenly into 12,30, and 42 without leaving remainders.
The Least Common Multiple (LCM) of a set of numbers is the smallest
natural number
that can be divided (without remainders) by each of the numbers in the set.
Example: The least common multiple of2, 3, and4 is 12 because although 2,
3, and 4
divide evenly into many numbers including 48, 36, 24, and 12, the smallest is
12.
The Denominator of a fraction is the number in the bottom; that is, the
divisor of the
indicated division of the fraction.
Example: In 5/8, 8 is the denominator and also the divisor in the
indicated division.
The Numerator of a fraction is the number in the top; that is, the
dividend of indicated
division of the fraction.
Example: In 3/4, 3 is the numerator and also the dividend in the
indicated division.
