Math Riddles!!!

 I am challenging my classmates to answer these riddles. Have fun !


 RIDDLE 101: You have a bag with 'N' strings in it. You randomly grab two ends and tie them together until there are no more loose ends.

In the end, what is the expected number of loops (strings tied to their own end)? 

  

  RIDDLE 102: You have a bag with 'N' strings in it. You randomly grab two ends and tie them together until there are no more loose ends.

In the end, what is the expected number of loops (strings tied to their own end)? 

 RIDDLE 103: What is the next number in the sequence? 1 11 21 1211 111221 312211

 RIDDLE 104: Thomas has missed an excessive number of days of school, so he must meet with Principal Davis. Mr. Davis asks him "Why on Earth have you missed so many days?"

Thomas replies "There just isn't enough time for school. I need 8 hours of sleep a day, which adds up to about 122 days a year. Weekends off is 104 days a year. Summer vacation is about 60 days. If I spend about an hour on each meal, that's 3 hours a day or 45 days a year. I need at least 2 hours of exercise and relaxation time each day to stay physically and mentally fit, adding another 30 days.

Add all of that up and you get about 361 days. That only leaves 4 days for school."

The principal knows Thomas is full of it, but can't figure out why. Where is Thomas going wrong?

 RIDDLE 105: What's the angle between minute hand and hour hand at a quarter past three?

REAL NUMBERS

Introduction to  Real Numbers

A symbol of the set of real numbers(ℝ)

In mathematics, a real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, such as theinteger −5 and the fraction 4/3, and all the irrational numbers such as √2 (1.41421356…, the square root of two, an irrational algebraic number) and π(3.14159265…, a transcendental number). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and complex numbers include real numbers.

Real numbers can be thought of as points on an infinitely long number line.

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th century mathematics. The currently standard axiomatic definition is that real numbers form the unique Archimedean complete totally ordered field (R ; + ; · ; <), up to an isomorphism,^[1] whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or certain infinite "decimal representations", together with precise interpretations for the arithmetic operations and the order relation. These definitions are equivalent in the realm of classical mathematics.

The reals are uncountable; that is, while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers: the cardinality of the set of all real numbers (denoted  and called cardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted ). The statement that there is no subset of the reals with cardinality strictly greater than  and strictly smaller than  is known as the continuum hypothesis. It is known to be neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

For more readings : 

http://hotmath.com/hotmath_help/topics/number-systems.html  

Properties of Real Numbers

 We use properties of real numbers in manipulating algebraic expressions, given that our variables most often represent real numbers.  Having a solid understanding of these properties is useful in developing fluency in algebraic processes.Closure Properties ofReal Numbers1) Thesumof any two real numbers is a real number.
(In other words, ifaandbare real, then so isa + b.)

2) Theproductof any two real numbers is a real number.
(In other words, ifaandbare real, then so isab.)Commutative Properties of Real Numbers1) Letaandbbe real numbers, thena + b = b + a.

2) Letaandbbe real numbers, thena·b = b·a.

Associative Properties of Real Numbers

1) Leta,b, andcbe real numbers, then (a + b) +c= a +(b+c).

2) Leta,b, andcbe real numbers, then (a·b)·c= a·(b·c).Identity Properties of Real Numbers1) There is a unique real number, 0, such that for all real numbersa,a+ 0 = 0 +a=a.  We say that 0 is the additive identity.
2) There is a unique real number, 1, such that for all real numbersa,a·1 = 1·a=a.  We say that 1 is the multiplicative identity.Inverse Properties of Real Numbers1) For all real numbersa, there exists a unique real number, denoted -a, such thata+ (-a) = 0.  We say that–ais the additive inverse ofa.2) For all real numbersa, wherea0, there exists a unique real number, denoted, such that= 1.  We say thatis the multiplicative inverse of a.
Multiplicative Property of Zero

For every real numbera,a·0 = 0·a= 0.Division Property of Zero

For every real numbera, wherea0, 0a= 0.Distributive Properties of Real Numbers

1) Leta,b, andcbe real numbers, thena·(b + c)= a·b + a·c.

2) Leta,b, andcbe real numbers, then(b + c)·a= b·a+c·a. 

for more readings : 
http://www.math.com/school/subject2/lessons/S2U2L1DP.html
http://www.regentsprep.org/regents/math/algebra/AN1/properties.htm

The beauty of mathematics and the love of God