PART 2.1: NUMBER THEORY X.1
IMPORTANT SETS OF NUMBERS:
Natural Numbers: The set N1 = {1,2,3,...} represents all the whole counting numbers. Occasionally, this sequence starts from zero: N 0 = {0,1,2,3,...} Integers: The set of whole numbers, negative whole numbers and zero.
Z = {0,±1,±2,±3,...}
Z = {...,−3,−2,−1,0,1,2,3,...} or
Rational Numbers: A rational number is any number that can be expressed as the fraction
p of two integers, with the q
denominator q not equal to zero. The decimal expansion of a rational number always terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over indefinitely. In converse is also true, any repeating or terminating decimal represents a rational number. Every rational number can be written as The set is denoted by
r=
(r − 1) + (r + 1) 2
Q
Irrational Numbers: An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero. The following test can determine if a number is rational or irrational:
(
)
1 x is rational lim lim cos 2 n (m!×π × x ) = m →∞ n →∞ 0 x is irrational The square roots of all natural numbers which are not perfect squares are irrational.
Algebraic Numbers: The set formed by the union of both rational and irrational numbers. Any solution to a polynomial of any finite degree with integer coefficients is an algebraic number. The set is denoted by A
Transcendental Numbers: These are numbers that can not that are not algebraic (ie: can not be formed from a finite degree polynomial with integer coefficients). The most prominent examples of transcendental numbers are π and e. Almost all number are transcendental but examples are difficult to prove.
Real Numbers: The set of all numbers on the number line. Rational and Irrational, Algebraic and Transcendental. The set is denoted by ℜ
Imaginary Numbers: Numbers that when squared give a negative result. Eg: i is imaginary as The set is denoted by I
Complex Numbers:
i 2 = −1
A combination of a real and an imaginary number in the form a + bi , where a & b are real, and i is imaginary. The set is denoted by C . All numbers and the sets above are special cases of complex numbers.
4.2
FUNDAMENTALS OF ARITHMETIC:
Fundamental Theory of Arithmetic: Any integer greater than 1 is either a prime number, or can be expressed as a unique product of prime numbers
Axioms of Real Numbers: Additive axioms. For every x,y,z in R, we have 1. x + y = y + x 2. 3.
(x + y ) + z = x + ( y + z ) 0+ x = x+0= x (− x ) + x = x + (− x ) = 0
4. Multiplicative axioms. For every c,d,x,y in R, we have 5. 0 x = 0 6. 1x = x
( )
( )
7. cd x = c dx Distributive axioms. For every c,d,x,y in R, we have 8. c x + y = cx + cy
(
9.
)
(c + d )x = cx + dx (c+d)
Indeterminate Expressions: ∞ = DNE ∞ 0 = DNE 0 0 × ∞ = DNE ∞ − ∞ = DNE 0 0 = DNE ∞ 0 = DNE 1∞ = DNE Lagrange’s Theorem: That every natural number can be written as the sum of four square integers. Eg: 59 = 7 2 + 32 + 12 + 0 2 Ie: x = a 2 + b 2 + c 2 + d 2 ; a, b, c, d , x ∈ N 0
X.2
FRACTIONS:
Fractions are ratios or quotients of two integer numbers. A simple fraction can be written in the form
n . ‘n’ is called the numerator and ‘d’ the denominator. Another format for a fraction is the mixed d n where ‘w’ represents the whole number part. In higher mathematics, simple number x = w d a fractions are almost exclusively used. Any integer ‘a’ can be expressed as the fraction 1 x=
Simple Fractions to Mixed Numbers: When n > d , there is a whole part to the fraction: n ÷ d = w remainder y n y =w d d Mixed Numbers to Simple Fractions: n n wd n wd + n w = w+ = + = d d d d d Reciprocals of a Fraction: n d The reciprocal of is d n Addition:
d1 = d 2 n1 n2 n1 + n2 + = d1 d 2 d1 When d1 ≠ d 2 n1 n2 n1d 2 n2 d1 n1d 2 + n2 d1 + = + = d1 d 2 d1d 2 d1d 2 d1d 2 When
Subtraction: When d1 = d 2 n1 n2 n1 − n2 − = d1 d 2 d1 When d1 ≠ d 2 n1 n2 n1d 2 n2 d1 n1d 2 − n2 d1 − = − = d1 d 2 d1d 2 d1d 2 d1d 2 Multiplication: n1 n2 n1 × n2 × = d1 d 2 d1 × d 2
Division: n1 n2 n1 d 2 n1 × d 2 ÷ = × = d1 d 2 d1 n2 d1 × n2 Simplifying a Fraction: By the Fundamental Theorem of Arithmetic, write out the numerator as the product of prime numbers. And write out the denominator as the product of prime numbers. Cancelling like terms and then recalculating the numerator and denominator yields the simplified fraction.
n1 a1 × a 2 × a3 × ... n2 = = d1 b1 × b2 × b3 × ... d 2
X.3 1 k
(
ROOT EXPANSIONS: kx ± ky
) =( 2
(
x± y
x± y=
1 k
x± y=
x ± 1 y y
x± y=
1 x ± xy x
kx ± ky
(
2
)
2
2
)
2
x y x ± y = k ± k k
X.4
)
2
FACTORIAL:
Definition of the Factorial Operation: Explicitly: n!= n × ( n − 1) × ( n − 2) × ... × 2 × 1 =
n
∏k k =1
if n = 0 1 Or by recurrence: n!= (n − 1)!×n if n > 0
Table of Factorials: 0! 1! 2! 3! 4! 5! 6! 7! 8! 9! 10!
1 (by definition) 1 2 6 24 120 720 5040 40320 362880 3628800
11! 12! 13! 14! 15! 16! 17! 18! 19! 20!
39916800 479001600 6227020800 87178291200 1307674368000 20922789888000 355687428096000 6402373705728000 121645100408832000 2432902008176640000
n!≈ 2π × n
Approximation:
n+
1 2
× e −n
(within 1% for n>10)
Double Factorial: When n is odd:
k
(2k )!
i =1
2 k k!
n!! = (2k − 1)!! = ∏ (2i − 1) = k
When n is even:
n!!= (2k )!!= ∏ (2i ) = 2k k! i =1
Table of Double Factorials: 0!! 1!! 2!! 3!! 4!! 5!! 6!! 7!! 8!! 9!! 10!!
1 (by definition) 1 2 3 8 15 48 105 384 945 3840
11!! 12!! 13!! 14!! 15!! 16!! 17!! 18!! 19!! 20!!
10395 46080 135135 645120 2027025 10321920 34459425 185794560 654729075 3715891200
Superfactorial: n
n
k =1
k =1
sf (n) = 1!×2!×3!×... × n!= ∏ k! = ∏ k n−k +1 = 1n × 2 n−1 × 3 n−2 × ... × (n − 1) × n 2
Table of Superfactorials: sf(0) sf(1) sf(2) sf(3) sf(4) sf(5) sf(6) sf(7) sf(8) sf(9) sf(10)
1 (by definition) 1 2 12 288 34560 24883200 125411328000 5056584744960000 1834933472251080000000 6658606584104730000000000000
Hyperfactorial: n
H (n) = ∏ k k = 11 × 2 2 × 33 × ... × (n − 1)
n −1
× nn
k =1
Table of Hyperfactorials: H(0) H(1) H(2) H(3)
1 (by definition) 1 4 108
H(4) H(5) H(6) H(7) H(8) H(9) H(10)
27648 86400000 4031078400000 3319766398771200000 55696437941726500000000000 21577941222941800000000000000000000 215779412229418000000000000000000000000000000