Mainly Maths Here

  The Fibonacci When one of  "the most talented Western mathematicians of the Middle Ages", Leonardo Bonnaci, created the Fibonacci sequence, it became widely popular and since then, it has held its value as one of the  most famous mathematical concepts to ever exist amongst learners. It has impacted the world immensely, its influence ranging from finance and coding theories to even the shape of a simple flower! The Sequence: In the Fibonacci, each term is made up of the sum of its previous two terms.  N= (N-1) + (N-2) where 'N' is the term number. The  Fibonacci  sequence goes as follows: 0,1,1,2,3,5,8,13,21,34,55..... For example, lets take the 7th term which is 8. This implies that in our case N=7. The 2 terms before it, the 5th and 6th terms are 3 and 5  respectively , which add up to 8.  The next term would be the sum of 34 and 55, so the 12th term is 89!  The sequence goes on till   infinity . Applications in real life: T...

  Infinity---------------------------------------------------- What is infinity? Have you ever wondered how many times can one go around a circle? How many points there are on a line? How many circles are there in a sphere? The answer is infinity! Many refer to infinity as if it was a gigantic number; it's not. In fact, infinity is just never ending. Something that keeps going on. Infinity isn't a real value, it isn't measurable or quantifiable, one cannot actually reach infinity. Infinity plus one is still infinity! There are numerous paradoxes to explain the concept of infinity, one example is pi. Pi is a never ending number. It starts like 3.141592653 ...... The decimals here 'lead to infinity'! Practically, infinity doesn't hold a lot of value, but it is frequently used in concepts like geometry, cosmology, logic and computing, and it helps in the development of mathematical theories as well.  Working with infinity While working with infinity, interesting th...

  You must have come across 'e' in many calculators, it's not just any other alphabet but actually a number! Just like pi! The Euler's number is one of the most popular numbers in math. It is an irrational number, it never ends and 'e' is a common way of denoting it's value of 2.182818284590.....   Question Suppose you have a box of chocolates and you drop them all on the floor, the probability of each chocolate going back in the wrong spot is 1/e! The more the chocolates, the closer the answer to 1/e! Uses 'e' is used a lot in logarithms, it also has something to do with the 'ln' on your calculator. Practically, 'e' is used in data science, growth and population models, and physics amongst other concepts. The Graph When you differentiate a curve, you get an equation stating the gradient of every point on the curve, but when you differentiate a y=e^x curve, the gradient at any point will always be e^x! The  f(x)=e^x graph never actual...

  Trigonometry revolves around finding the length/ angle of a part of a triangle using various methods. One can use the basic formulas to find the values of a right angled triangle but we can use other formulas for non-right angled triangles. Important Formulas:  This formula is used to find the area of any triangle  ↑ In all of these formulas, the uppercase letters represent the angle and the lowercase letters represent the side opposite to the corresponding angle. Now let's work with a question! We can first use the cosine rule to find one the angles. Let us start by finding out the value of the angle opposite 6.5cm.  So we use the cosine rule, and substitute the values: a=6.5 b=4.3 c=5.2 and cosA is unknown.  We make cosA the subject of the formula and solve. Then we find A using the inverse cos. A= 85.8 Now we have an angle and a side. We can use the sin rule to find the rest of the angles: sin85.8/6.5= sinB/4.3= sinC/5.2 We get the values of the angles as A...

  In a graph, there are numerous points which join to form various polygons. These polygons may be irregular in shape, with any number of sides. The area of these polygons can be found using different methods! If we have a regular polygon, we can use that polygon's formula to find the area. So if we have a hexagon, we can just use the formula :                                              a^2(3√3)/2 This is the formula to find the area of a regular hexagon. 'a' represents the length of a side. There is a different formula for different regular polygons.  The side length of a polygon can be found using the distance formula, where we calculate the distance between two points. But, what if the polygons are not regular? If they have different side lengths and different angles on each side? The formula generally used here is commonly known as the sh...

   Differentiation Differentiation is a method of finding the gradient of any point in a polynomial graph. Polynomial graphs are graphs of quadratic, cubic, quartic, etc. equations. By differentiating the polynomial equation, we create another general equation that gives us the gradient of any point on the graph.  To find the gradient of a specific point on a graph, we usually take the point and draw a tangent from that point. Then we make a right angled triangle of that tangent and use Pythagoras Theorem to find the gradient( the hypotenuse). This method is too long and drawing tangents make it susceptible to human error. This is why we use differentiation!  How to differentiate: We do it in parts. We take the power of the value and multiply it to the coefficient. Then we subtract the power by one. For example: If y = x 4 , dy/dx = 4x 3 If y = 2x 4 , dy/dx = 8x 3 If y = x 5  + 2x -3 , dy/dx = 5x 4  - 6x -4 dy/dx is the 'symbol' of differentiation. Question...

  SIMULTANEOUS EQUATIONS:-   Often, solving a problem requires an equation. Sometimes there is more than one unknown variable in an equation.  For example: 3x + 6y = 15 Here, there can be multiple combinations of answers of x and y, and trial and error is not an efficient way of solving the equation. This is when other equations are given to help find the unknown variables algebrically. Let us take another equation and calculate 'simultaneously'. 2x - 2y = -4 Now let us write the equations together: 3x + 6y = 15______________equation 1 2x - 2y = -2_______________equation 2 Now, we multiply both the equations in such a way that the coefficient of one variable becomes equal: 6x + 12y = 30 we multiplied the first equation by 2. 6x - 6y = -6 we multiplied the second equation by 3. The coeeficient of 'x' becomes 6 in both! Now we can subtract the equations to get rid of the 'x'. All we do in this step is change the sign of every corresponding variable: (6x-6x) + (12y...

     Arithmetic Sequences An arithmetic sequence is a sequence made by adding/subtracting a constant from the previous number. Generally, this constant is termed as 'd', as it represents the difference between any two numbers of the sequence. 'a' is the first term of the sequence and 'n' is the term number. Examples of an Arithmetic Sequence: 1,3,5,7,9.... 8,6,4,2,0,-2.... 0,1,2,3,4,5,6,7.... The formula to find the term of an arithmetic sequence at a specific term number is given by: a+ (n-1)d This is known as the formula to find the nth term so for example: 2,4,6,8,10... is a sequence. Here to calculate the 59th term we do: 2+(59-1)2 2+(58*2) = 118! If you notice, the terms go as: So even if you have any two terms in the sequence, you can find the difference using simultaneous equations! And from there the nth term!

  Middle Term Splitting Apart from plugging in the coefficients in the formula which solves quadratic equations, there are simpler ways of solving quadratic equations, without the use of calculators! The method described here is known as mid-term splitting.  Benefits: This method requires much less calculation and effort. It is solved in a quicker and cleaner method. Drawbacks:  Unlike the quadratic formula, the middle term splitting method only works in certain cases.  How does it work: a x²  + bx + c This method will only work if there are two numbers that add up to 'b' and have a product of 'a*c' What we do is we write the equation in another form and factorise it into two parts which equal zero. The examples below will explain better! Here, 6 and -5 are the two numbers needed. They are the set of numbers which add up to the origibal 'b' whoch was 1, and form a product of -30( a*c)! After that the equation was just factorised. Since this would equal to zero, ...

  EXPONENTIAL GRAPHS                                    Exponential means a very rapid increase. An exponential graph shows this through a very steep curve. An exponential graph is formed from the equation of : f(x)=a^x This equation is never crosses the x axis, this is because a to the power of anything would never equal to 0 or become negative, the graph goes on till 'infinty', unless numbers are be added and/or multiplied to change the curve. These are some exponential graphs: It might be unclear from the naked eye but the graphs don't intercept/touch the x-axis but keep going closer towards it!  As you can see, the greater the number, the steeper the graph. All graphs cross the point (1,0) as anything to the power of 0 ( the x value) will be 1( the y value)! But what happens if we multiply the 'x' and not the number in the equation? The graph of 2^2x is steeper!  If we add/ ...

  Horizontal Asymptotes   Many graphs, especially exponential graphs have asymptotes. Asymptotes are curves that do not touch a line but go really close to it. In the graph of y=2^x, the line y=0 is an asymptote as the graph doesnt touch it but keeps going closer to it till infinity. Horizontal asymptotes are asymptotes that are horizontally placed, in the graph of y=2^x, the asymptote y=0 is a horizontal asymptote! To find the horizontal asymptote of an equation, we need to consider only the values with the highest degree of x in the numerator and the denominator. The highest degree of x is the x value with the highest power. Worked Examples: Here 'M' has a degree of 2 while the highest degree in the numerator is just 1. So as M>N, there is no horizontal asymptote. Here 'M' is 0 and 'N' is 1, since M<N, the horizontal asymptote is y=0 Here both M and N have a degree of 1. So we divide the coeefficients and get 2/6=1/3, the horizontal asymptote therfore is...

  If we have the equation of a line, and want to find the equation of a line perpendicular to that, that passes through a certain point, we can derive it from the original equation! The gradient of the perpendicular is the reciprocal of the original gradient! To find the perpendicular, we need the gradient and a point it passes through to find the constant/ y-intercept. So, let us take an equation: y=2x+7 The gradient of the perpendicular would be the reciprocal of the gradient here, that is reciprocal of 2 which is -1/2! Lets say that the perpendicular passes through the points (8,6) Now we can find the complete equation: 6= -0.5(8) + c Therefore, c = 10! The equation of the perpendicular to y=2x+7 that passes through the point 8.6 is  y=-0.5x + 10 Worked Examples: Question: Solution:  m=-5 2= -40 +c c=42 y=-5x+42

  Inverse Functions An inverse of a function is a another function that reverses the operations performed by the original function. It is used to get back the original value from the final answer we got using the first function! Finding the inverse of a function is extremely simple. It requires only two steps! 1. Switch 'x' and 'y'( the two variables) 2. Make 'y' the subject of the equation You have the inverse function now! Worked example: Here as you can see, the 'x' and 'y' variables were swapped and then we made 'y' the subject again to give us the inverse! Many functions exist, one-one, one-many, many-one. An inverse function only exists in one-one functions, where only one 'x' value gives a single 'y' value. Questions: Find the inverse of each of the four functions below. Solutions:                                                         ...

                                  Vectors  Vectors tell us the distance of a point in a graph from the origin. Vectors are made up of two numbers written vertically between a set of brackets.  Here, x and y are two numbers. X represents how much a point has moved along the x axis, horizontally. And y represents how much that point has moved up, vertically. If x or y are negative, it means that the point has shifted left or down respectively. Basically, the coordinates of the point! We can use this to solve various questions for example; if we need to find the distance of a point from its origin or multiply the vectors of two points or add them, out of other operations. Distance: To find the distance of a point from its origin we simply use the  Pythagoras's theorem.  We know that the point has shifted horizontally by x and vertically by y, from the origin. The distance  therefor...