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# Number Theory Terms

TermDefinitionExample
WHOLE NUMBERS The set of all counting numbers beginning with 0 (non-negative integers) 0, 1, 2, 3, ...
INTEGERS The set of all positive and negative counting numbers, including 0 ...-2, -1, 0, 1, 2, ...
RATIONAL NUMBERS The set of all numbers that can be expressed as a fraction .5, -4, 37, 2/9
IRRATIONAL NUMBERS the set of all numbers that CANNOT be expressed at a fraction pi, sq root of a prime number
DIVISIBLE The condition where one natural number divides evenly into another natural number with no remainder 56 is divisible by 7, because 7 divides into 56 8 times with no remainder
FACTORING The process of writing a natural number as the product of two or more natural numbers. (the numbers that make up the product are called factors.) 3 * 6 = 18 3 and 6 are factors
PRODUCT the result of multiplying two or more numbers together. 3 * 6 = 18 18 is the product
PRIME NUMBER A natural number that has exactly two factors, 1 and itself 2, 3, 5, 7. 11, 13
COMPOSITE NUMBER A natural number that has three or more factors 4, 6, 8, 9, 12
PRIME FACTORIZATION The process of expressing a number as the product of all prime numbers 28 = 2 * 2 * 7
MULTIPLE The product of a give number and any whole number An integer (b) is a multiple of an integer (a), if there exist an integer c such that ac=b 22, 36 or multiples of 2
TERMINATING DECIMALS Decimals that have a definite number of digits 2.4 6.95, 3
REPEATING DECIMALS Decimals in which one or more digits repeat forever .3333..., .70707....
NONREPEATING DECIMALS Decimals that do not end nor repeat square root of any prime number pi
POWER An expression that has base and an exponent 2^3, x^2
REMAINDER (R) For integers a and b, with b >0, there exists a unique integer (q) and remainder (r) such that a=bq + r, where 0≤r≤b.
ABSOLUTE VALUE the absolute value of a real number is its distance from zero on the number line. The absolute value of any real number, a, written as |a|, is defined as: |a| = a if a ≥ 0 and |a| = -a if a < 0.
CLOSURE OF A SET
COMBINATION The number of ways of picking k unordered outcomes from n possibilities. The number of combinations of n distinct objects taken r at a time is C(n,r)= n!/(n-r)!r! ORDER DOESN'T MATTER SO AB = BA The number of ways of selecting a committee of 2 people out of 4 people is C(4,2) = 4! / (4-2)!2! = 4*3*2*1 / (2!)(2!) = 6 ways
COMMON FACTOR in algebra a polynomial that is a factor of two or more polynomials. That is, each of the polynomials is divisible by the common factor. x – 1 is a common factor of x2 + x -2 and x2 -6x + 5, since each polynomial is divisible by x – 1.
COMPLEX NUMBER a number of the form a + bi, where a and b are real numbers and i is the imaginary unit equal to the square root of -1. a is called the real part and b is called the imaginary part
COMPLEX PLANE A coordinate plane used for graphing complex numbers. horizontal axis is real axis and vertical axis is the imaginary axis.
CONJUGATE OF COMPLEX NUMBER for any complex number z = a + bi, its conjugate z is equal to a - bi. In the complex plane the points representing a complex number and its conjugate are mirror images with respect to the real axis the conjugate of 2 + 3i is 2 - 3i
COORDINATE PLANE a 2-D region determined by a pair of axes.
DECIMAL EXPANSION a decimal form of a rational number can be obtained by dividing the denominator into the numerator.
DENSITY OF NUMBER LINE between any two real numbers there is always another real number
DIVISIBILITY RULE FOR 7 take the last digit off the number, double it and subtract the double number from the remaining number. If the result is divisible by 7, then original number is divisible by 7. 14 4 doubled is 8, 1 - 8 = -7 -7 is divisible by 7 so 14 is divisible by 7.
DIVISIBILITY RULE FOR 8 if the number formed by the last 3 digits is divisible by 8. 3624 is divisible by 8 because 624 is divisible by 8
DIVISIBILITY RULE FOR 11 if difference between the sum of the odd-numbered digits and sum of even numbered digits, counted from right to left, is divisible by 11 528 8 + 5 = 13 (sum of odd-numbered digits) 2 (sum of even-numbered digits) 13-2=11
DIVISOR an integer a is a divisor of an integer b if there exists an integer c such that ac=b
EQUIVALENCE PROPERTIES OF EQUALITY Reflexive (a=a) Symmetric (if a=b then b = a) Transitive (a=b, b=c then a = c)
EVEN NUMBER an even number is an integer that that is divisible by 2, thus, all even numbers can be written in the form 2n, where n is an integer
FACTOR an integer a is said to be a factor of an integer b, if there exists an integer c such that ac=b. For example, 6 is a factor of 42, 6 is not a factor of 44.
FINITE SET a set that contains exactly n elements, where n is a natural number.
Trichotomy Either a<0, a=0, or a>0
Identity element of addition and multiplication a x 1 = a = 1 x a a + 0 = a = 0 + a
Identity Property of addition the sum of any number (a) and zero is a. a + 0 = a
Identity Property of multiplication the product of any number (a) and 1 is a.
Imaginary Number a number of the form bi, where b is a real number and b ≠ 0 and i = square root of -1.
infinite set a set which doesn't have a finite cardinality. The set of natural numbers is the smallest infinite set
INTEGER PART the largest integer less than or equal to x. Denoted as [x] [2.8]= 2 [-2.1] = -3
INVERSE ELEMENT OF ADDITION An element a' is the additive inverse of the element a, if a + a' = 0 = a' + a. i.e. a number plus its additive inverse equals the additive identity 0. the additive inverse of 5 is -5.
INVERSE PROPERTY OF MULTIPLICATION The product of a number and its multiplicative inverse is equal to1 (the multiplicative identity). For every non zero real number a, there exists a real number a−1 , such that a.a−1 =1.
INVERSE PROPERTY OF ADDITION The sum of a number and its additive inverse is equal to 0 (the additive identity).For every real number a there exists another real number -a, such that a + (-a) = 0.
NUMBER THEORY The mathematical study of integers and their generalization
ODD NUMBER An integer that is not divisible by 2; thus, all odd numbers can be written in the form 2n + 1, where n is an integer.
PERFECT SQUARE a number that is the square of a whole number
PERMUTATION a permutation of n objects can be thought of as all the possible ways of their arrangement or rearrangement. The number of permutations of n objects taken r at a time’ is denoted by n!/(n-r)! Order does matter: AB and BA are two different solu For example there are 6 permutations of A, B, C taken two at a time: AB, AC, BA, BC, CA, and CB.
POLYNOMIAL A function of the form P(x)=a(n)x^n + a(n-1)x^n-1 + a(1)x + a(0) for all real x, where the coefficients are real numbers and n a non negative integer. If 0(n)≠0, P(x) is called a real polynomial of degree n.
PYTHAGOREAN THEOREM gives the relationship between the lengths of the sides of a right angled triangle. The theorem states: In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
QUALITATIVE REASONING the application of reasoning based on non-numerical data such as problem solving, reasoning skills, and communication.
QUANTITATIVE REASONING The application of reasoning based on numerical data such as graphs and tables
REAL NUMBERS the set of irrational numbers together with the set of rational numbers
RECTANGULAR ARRAY a set of elements arranged into rows and columns such as a matrix.
ROOT Let f(x) = 0 be an equation that involves the indeterminate x. A root of the equation is a value c such that f(c) = 0
SCIENTIFIC NOTATION a way to express a very large or very small number as the product of a number between 1 and 10 and a power of 10. the scientific notation for 2,500,000 is 2.5 • 10^6
VENN DIAGRAM A method of displaying relations between subsets of some universal set. The universal set E is represented by the interior of a rectangle, and the subsets of E are represented by regions inside this by simple closed curves.
DIVISIBILITY PROPERTIES We denote divisibility using a vertical bar: a | b means "a divides b". For example, we can write 643|123456
NATURAL NUMBERS The set of all counting numbers beginning with 1 (positive integers0 1, 2, 3, 4,...
Created by: sstoveland