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Addition
Chapter 2
Associative property
Chapter 2
Closure property
Chapter 2
Commutative property
Chapter 2
Composite number
Chapter 2
Natural numbers (counting numbers)
Chapter 2
Distributive property for
multiplication over addition
Chapter 2
Divisibility
Chapter 2
Divisor
Chapter 2
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Factor
Chapter 2
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A property of grouping that applies to certain operations
(addition and multiplication, for example, but not to
subtraction or division): If a, b, and c are natural
numbers, then
(a 1 b) 1 c 5 a 1 (b 1 c)
One of the fundamental undefined operations applied to the
set of counting numbers.
and
(ab)c 5 a(bc)
Chapter 2
States that the order in which two numbers are added
makes no difference; that is (if we read from left to right):
If a and b are natural numbers, then
a1b5b1a
Chapter 2
A set S is closed for an operation  if a  b is an element of
S for all elements a and b in S.
and
ab 5 ba
Chapter 2
The positive integers;
 5 {1, 2, 3, 4, 5, . . .}.
Chapter 2
A number that has two or more prime factors.
Chapter 2
Chapter 2
If m and d are natural numbers, and if there is a natural
number k so that m = d  k, we say that d is a divisor of m, d
is a factor of m, d divides m, and m is a multiple of d.
We denote this relationship by d|m.
If a, b, and c are real numbers, then a(b 1 c) 5 ab 1 ac
and (a 1 b)c 5 ac 1 bc for the basic operations. That is,
the number outside the parentheses indicating a sum or
difference is distributed to each of the numbers inside the
parentheses.
Chapter 2
Chapter 2
Each of the numbers multiplied to form a product is called
a factor of the product.
The quantity by which the dividend is to be divided. In ,
b
b is the divisor.
Chapter 2
Chapter 2
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a
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Factoring
Chapter 2
Fundamental theorem of arithmetic
Chapter 2
Multiplication
Chapter 2
Number of divisors
Chapter 2
Prime factorization
Chapter 2
Prime number
Chapter 2
Property of closure for multiplication
Chapter 2
Sieve of Eratosthenes
Chapter 2
Subtraction
Chapter 2
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Factor tree
Chapter 2
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Every natural number greater than 1 is either a prime or
a product of primes, and its prime factorization is unique
(except for the order in which the factors appear).
The process of determining the factors of a product.
Chapter 2
Chapter 2
For a 2 0, multiplication is defined as follows:
Every natural (counting) number greater than 1 has at least
two distinct divisors, itself and 1.
a 3 b means b 1 b 1 b 1 c 1 b
a addends
If a 5 0, then 0 3 b 5 0.
Chapter 2
Chapter 2
A prime number is a natural number that has exactly two
divisors.
The factorization of a number so that all of the factors
are primes and so that their product is equal to the given
number.
Chapter 2
Chapter 2
A method for determining a set of primes less than some counting number
n. Write out the consecutive numbers from 1 to n. Cross out 1, since it is
not classified as a prime number. Draw a circle around 2, the smallest prime
number. Then cross out every following multiple of 2, since each is divisible
by 2 and thus is not prime. Draw a circle around 3, the next prime number.
Then cross out each succeeding multiple of 3. Some of these numbers,
such as 6 and 12, will already have been crossed out because they are also
multiples of 2. Circle the next open prime, 5, and cross out all subsequent
multiples of 5. The next prime number is 7; circle 7 and cross out multiples
of 7. Continue this process until you have crossed out the primes up to !n.
All of the remaining numbers on the list are prime.
Chapter 2
The representation of a composite number showing the
steps of successive factoring by writing each new pair of
factors under the composite.
Chapter 2
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Let  be the set of natural (or counting) numbers. Let a and
b be any natural numbers. Then
ab is a natural number
We say  is closed for multiplication.
Chapter 2
The operation of subtraction is defined by:
a 2 b 5 x means a 5 b 1 x
Chapter 2
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Greatest common factor (g.c.f.)
Chapter 2
Relatively prime
Chapter 2
Least common multiple (l.c.m.)
Chapter 2
Zero
Chapter 2
Whole numbers
Chapter 2
Integers
Chapter 2
Absolute value
Chapter 2
Division
Chapter 2
Division of integers
Chapter 2
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Division by zero
Chapter 2
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Two integers are relatively prime if they have no common
factors other than 1.
The greatest common factor (g.c.f.) of a set of numbers is
the largest number that divides (evenly) into each of the
numbers in the given set.
Chapter 2
Chapter 2
The number that separates the positive and negative
numbers; it is also called the identity for addition or the
additive identity; that is, it satisfies the property that
x10501x5x
The least common multiple (l.c.m.) of a set of numbers
is the smallest number that each of the numbers in the set
divides into evenly.
Chapter 2
Chapter 2
Composed of the natural numbers, their opposites,
and zero;
The positive integers and zero;
 5 {. . . , 23, 22, 21, 0, 1, 2, 3, . . . }.
Chapter 2
 5 {0, 1, 2, 3, . . . }.
Chapter 2
If a, b, and z are integers, where b 2 0, then division a 4 b
a
is written as and is defined in terms of multiplication.
b
a
5 z means a 5 bz
b
The absolute value of x, denoted by 0 x 0 , is defined as
0x0 5 e
x,
2x,
if x $ 0
if x , 0
Chapter 2
Chapter 2
In the definition of division, a ÷ b, b 2 0 because if
b 5 0, then bx 5 0, regardless of the value of x, and
therefore could not equal a nonzero number a. On the
0
other hand, if a 5 0, then 5 1 checks from the definition,
0
0
and so also does 5 2, which means that 1 5 2, another
0
contradiction. Thus, division by 0 is never possible.
The quotient of two integers is the quotient of the absolute
values, and is positive if the given integers have the same
sign, and negative if the given integers have opposite signs.
Furthermore, division by zero is not possible and division
into 0 gives the answer 0.
Chapter 2
Chapter 2
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Rational number
Chapter 2
Proper fraction
Chapter 2
Improper fraction
Chapter 2
Reduced
Chapter 2
Fundamental property of fractions
Chapter 2
Least common denominator
Chapter 2
Perfect squares
Chapter 2
Square numbers
Chapter 2
Pythagorean theorem
Chapter 2
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Leg (side)
Chapter 2
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A fraction for which the numerator is less than the
denominator.
The set of rational numbers, denoted by , is the set of all
numbers of the form
a
b
where a and b are integers, and b 2 0.
a is called the numerator and b is called the denominator.
A rational number is also called a fraction.
Chapter 2
Chapter 2
If the greatest common factor of the numerator and
denominator of a given fraction is 1, then we say the
fraction is in lowest terms or reduced.
A fraction for which the numerator is greater than the
denominator.
Chapter 2
Chapter 2
The smallest number that is exactly divisible by each of the
given numbers.
If both the numerator and denominator are multiplied or
divided by the same nonzero number, the resulting fraction
will be the same.
Chapter 2
Chapter 2
Numbers that are squares of the counting numbers: 1, 4, 9,
16, 25, 36, 49, 64, 81, 100, 121, 144, 169, . . . .
Since 12 5 1, 22 5 4, 32 5 9, . . . , the perfect squares are
1, 4, 9, 16, 25, 36, 49, . . . .
Chapter 2
Chapter 2
One of the two sides of a right triangle that are not the
hypotenuse.
If a triangle with legs a and b and hypotenuse c is a right
triangle, then a2 1 b2 5 c2. Also, if a2 1 b2 5 c2, then
the triangle is a right triangle.
Chapter 2
Chapter 2
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Hypotenuse
Chapter 2
Square root
Chapter 2
Positive square root
Chapter 2
Irrational number
Chapter 2
The number 
Chapter 2
Radical form
Chapter 2
Laws of square roots
Chapter 2
Real numbers
Chapter 2
Unit distance
Chapter 2
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Number line
Chapter 2
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The square root of a nonnegative number n is a number so
that its square is equal to n. In symbols, the positive square
root of n, denoted by !n, is defined as that number for
which
The longest side in a right triangle.
Chapter 2
Chapter 2
A number that can be expressed as a nonrepeating,
nonterminating decimal; the set of irrational numbers is
denoted by .
The symbol !x is the positive number that, when
multiplied by itself gives the number x. The symbol “! ”
is always positive.
Chapter 2
Chapter 2
The ! symbol in an expression such as !2. The number
2 is called the radicand, and an expression involving a
radical is called a radical expression.
A number that is defined as the ratio of the circumference
of any circle to its diameter. It cannot be written in exact
decimal form but it is a number between 3.1415 and
3.1416.
Chapter 2
Chapter 2
!n !n 5 n
There are 4 laws of square roots.
The set of real numbers, denoted by , is defined as the
union of the set of rationals and the set of irrationals.
( 1 ) "0 5 0
( 3 ) "ab 5 "a"b
( 2 ) "a2 5 a
(4)
a
Åb
5
"a
"b
Chapter 2
Chapter 2
A line used to display a set of numbers graphically (the axis
for a one-dimensional graph).
The distance between 0 and 1 on a number line.
Chapter 2
Chapter 2
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Dense set
Chapter 2
Real number line
Chapter 2
Identity property for addition
Chapter 2
Identity for addition (additive identity)
Chapter 2
Identity for multiplication
(multiplicative identity)
Identity property for multiplication
Chapter 2
Chapter 2
Inverse property for addition
Chapter 2
Opposite (additive inverse)
Chapter 2
Reciprocal (multiplicative inverse)
Chapter 2
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Mathematical modeling
Chapter 2
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A line on which points are associated with real numbers in
a one-to-one fashion.
A set of numbers with the property that between any two
points of the set, there exists another point in the set that is
between the two given points.
Chapter 2
Chapter 2
The number 0 has the property 0 1 n 5 n 1 0 for any real
number n, and is called the identity for addition.
The number 0 (zero) has a special property for addition that
allows it to be added to any real number without changing
the value of that number. This property is called the identity
property for addition of real numbers.
Chapter 2
Chapter 2
There exists in  a number 1, called one, so that
13a5a315a
For any a [ R. The number one is called the identity for
multiplication or the multiplicative identity.
The number 1 has the property 1  n 5 n  1 for any real
number n, and is called the identity for multiplication.
Chapter 2
Chapter 2
Opposites x and 2x are the same distance from 0 on the
number line but in opposite directions; 2x is also called
the additive inverse of x. Do not confuse the symbol “2”
meaning opposite with the same symbol used to mean
subtraction or negative.
For each a [ R, there is a unique number (−a) [ R,
called the opposite (or additive inverse) of a, so that
a 1 (2a) 5 2a 1 a 5 0
Chapter 2
Chapter 2
An iterative procedure that makes assumptions about realworld problems to formulate the problem in mathematical
terms. After the mathematical problem is solved, it is tested
for accuracy in the real world and revised for the next step
in the iterative process.
Chapter 2
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The reciprocal of n is 1/n, also called the multiplicative
inverse of n.
Chapter 2
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