MATHEMATICS ENCYCLOPEDIA

Slope of a line A line in a coordinate plane forms two angles with the x-axis , which are supplementary.  The angle θ made by the line l with positive direction of x-axis and measured anti-clockwise is called the inclination of the line. Obviously 0° ≤ θ ≤ 180°. We observe that line parallel to x-axis, or coinciding with x-axis, have inclination of 0°.  The inclination of a vertical line ( parallel to or coinciding with y-axis ) is 90°. If θ is the inclination of a line l, then tanθ is called the slope or gradient of the line l. The slope of a line having inclination 90°  is not defined. The slope of a line is denoted by m. Thus m = tanθ, θ ≠ 90°. It may be observed that slope of x-axis is zero and slope of y-axis is not defined . Slope of a line when coordinates of any two points on the line are given  Let P ( x, y ) and Q ( x, y ) be two points on a line l with inclination θ. Then slope is given by ...

Have you ever wondered what is the speed of light ? Well this one of the most interesting questions in the world of Mathematics and Physics . The speed of light in vaccum is a fundamental constant . The most interesting fact about this speed is that if an object moves with this speed in one frame, it has the same speed in any other frame. This the key fact on which the special theory of relativity is based . The speed of light is defined to be exactly 299,792,458 m/s. Perhaps the first attempt to measure the speed of light was made by Galileo . Two experimenter A and B, each having a lantern and a shutter, stands on two small hills.  The shutter can cover or uncover the lantern. Initially, both the lanterns are covered . One of the persons A uncovers his lantern. The second person B uncovers his lantern when he sees the light from the lantern of A. The first person A covers his lantern when he sees the light from the lantern of B.  The time el...

Maria Agnesi ( 1718-1799 ) was the first woman to write a Mathematics handbook and the first woman appointed as a Mathematics Professor at a university. She was an Italian Mathematician, pholosopher . She was a sleep walker and could do mathematical problems in her sleep. Often in the morning she would find that she had sleepwalked to her desk and completed a problem she had been unable to solve the last night.

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...