Associative property

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1 Addition Associative property Closure property Commutative property Composite number Natural numbers (counting numbers) Distributive property for multiplication over addition Divisibility Divisor Factor

2 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) and (ab)c 5 a(bc) One of the fundamental undefined operations applied to the set of counting numbers. 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 and a 1 b 5 b 1 a ab 5 ba A set S is closed for an operation if a b is an element of S for all elements a and b in S. The positive integers; 5 {1, 2, 3, 4, 5,...}. A number that has two or more prime factors. 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. 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 a b, b is the divisor.

3 Factoring Fundamental theorem of arithmetic Multiplication Number of divisors Prime factorization Prime number Property of closure for multiplication Sieve of Eratosthenes Subtraction Factor tree

4 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. Every natural (counting) number greater than 1 has at least two distinct divisors, itself and 1. For a 2 0, multiplication is defined as follows: 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. 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. 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. 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. The representation of a composite number showing the steps of successive factoring by writing each new pair of factors under the composite. The operation of subtraction is defined by: a 2 b 5 x means a 5 b 1 x

5 Greatest common factor (g.c.f.) Relatively prime Least common multiple (l.c.m.) Zero Whole numbers Integers Absolute value Division Division of integers Division by zero

6 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. 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 x x 5 x 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. Composed of the natural numbers, their opposites, and zero; 5 {..., 23, 22, 21, 0, 1, 2, 3,... }. The positive integers and zero; 5 {0, 1, 2, 3,... }. If a, b, and z are integers, where b 2 0, then division a 4 b is written as a 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 x, if x $ 0 0 x 0 5 e 2x, if x, 0 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 other hand, if a 5 0, then checks from the definition, 0 and so also does 0 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.

7 Rational number Proper fraction Improper fraction Reduced Fundamental property of fractions Least common denominator Perfect squares Square numbers Pythagorean theorem Leg (side)

8 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. 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. 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. Numbers that are squares of the counting numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169,.... Since , , ,..., the perfect squares are 1, 4, 9, 16, 25, 36, 49,.... 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 a 2 1 b 2 5 c 2. Also, if a 2 1 b 2 5 c 2, then the triangle is a right triangle.

9 Hypotenuse Square root Positive square root Irrational number The number Radical form Laws of square roots Real numbers Unit distance Number line

10 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!n!n 5 n The longest side in a right triangle. 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. 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 and 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 (2) "a 2 5 a a (3) "ab 5 "a"b (4) Å b 5 "a "b 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.

11 Dense set Real number line Identity property for addition Identity for addition (additive identity) Identity property for multiplication Identity for multiplication (multiplicative identity) Inverse property for addition Opposite (additive inverse) Reciprocal (multiplicative inverse) Mathematical modeling

12 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. 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. There exists in a number 1, called one, so that 1 3 a 5 a a 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. 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 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. The reciprocal of n is 1/n, also called the multiplicative inverse of n.

SEVENTH EDITION and EXPANDED SEVENTH EDITION

SEVENTH EDITION and EXPANDED SEVENTH EDITION Slide 5-1 Chapter 5 Number Theory and the Real Number System 5.1 Number Theory Number Theory The study of numbers and their properties. The numbers we use to