MATHEMATICS ENCYCLOPEDIA

Question of the suspension bridge The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m . Find the length of a supporting wire attached to the roadway 18 m from the middle . Think of the problem as a real problem that you have to calculate the length of the wire. It is a very interesting problem that helps one to understand where mathematics is used in real life situation. It would be easy to visualize the problem if we have a diagram. So, let us draw a diagram. Now let us plot a coordinate graph for this diagram If we now solve using the coordinates of the parabola would get the answer.  Let us take the point ( 50, 24 ) and solving we get  Again using the point ( 18, y ) and solving we get  This is the length between...

What is a weird number ? In number theory, weird number is a natural number that is abundant but not semi-perfect. The sum of proper divisors of the number is greater than the number , but no subset of those divisors sum to the number itself. The smallest weird number is 70 . Its proper divisors are 1, 2, 5, 7, 10, 14 and 35 which adds to 74. Examples of weird numbers are  70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690 etc. No odd weird number is known.

WHAT IS A PROOF ? Proof of a mathematical statement consists of sequence of statements, each statement being justified with a definition or an axiom or a preposition that is previously established by the method of deduction using only the allowed logical rules. Thus each proof is a chain of deductive arguments each of which has its premises and conclusions . Many a times, we prove a preposition directly from what is given in the preposition. But some times it is easier to prove an equivalent preposition rather than proving the preposition itself. This leads to, two ways of proving a preposition directly or indirectly and the proofs obtained are called direct proofs and indirect proof and further each has three different ways of proving which is discussed below. Direct proof Straight forward proof: It is a chain of arguments which leads directly from what is given or assumed, with the help of axioms, definitions or already proved theorems, to what is to be proved using r...

What is an Euler trail ? An Euler trail ( Euler path ) is a trail in a finite graph which visits every edge exactly once. An Euler trail is a walk on the edges of a graph which uses each edge exactly once. Euler trail is also called an Euler chain , Euler walk , Euler path . Here is the puzzle as you could see in the picture. Your job is to cross each line exactly once without lifting your pen.  In the following figure consider every marked line as a single line and each must be crossed once without lifting you pen. Here are some examples already done, but in each case, one line is left ( marked red ) to be crossed . See the figures. Have you got the solution ? well, this is impossible to do however hard you try.  Explanation Consider the figure below and find the numbers of line reaching the centre of each box. In the figure you could see that there are three centre points with odd number o...

Square puzzle How can you cut a 3/4th of a square into 4 equal pieces ? Suppose you have a square as shown in figure And you have cut out a quarter of the square.  Now again you want to cut the remaining bigger piece ( red piece ) into 4 equal parts, how could  this  be done ? This is not as hard as you are thinking. See the figure and discover some more puzzles like this.

What is golden ratio ? In mathematics, two quantities are in the golden ratio, if their ratio is same as the ratio of their sum to the larger of the two quantities . Let a and b be two quantities such that a > b > 0, then algebraically it is written as follows, Golden ratio is denoted by Greek letter phi given below: The value of golden ratio is, Golden ratio is a very important concept in mathematics which is used in designing geometrical shapes , in the construction of architectural structures etc. Golden ratio is an irrational number. Other names of golden ratio are golden number, golden section, golden mean, golden proportion, golden cut, divine section, extreme and mean ratio etc. Golden ratio is related to the fibonacci series . If you take any two successive ( one after the other ) fibonacci numbers, their ratio is very close to the golden ratio . Calculation of the golden ratio From the definition of golden ratio, w...

Oxford dictionary definition: A quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another. Cambridge dictionary definition: Something physical such as force that has size and direction. A vector is a quantity having both magnitude and direction , such as displacement, velocity, force, acceleration etc. What is magnitude ? Magnitude: Magnitude means the size of a mathematical object, e.g. when you say a force of 5N is applied in the west direction, it means the magnitude of force is 5 and the direction of force is west. A directed line segment is a vector denoted by arrow over AB ( see figure ) and read as 'vector AB' The arrow indicates the direction of vector . The point A from where the vector AB  starts is called its initial point and the point B where it ends is called its terminal point .  The distance between initial and terminal points of a vector is called th...