good day welcome to the first lesson of the subject mathematics in the modern world we are shara assassis and donald in cape bayern to discuss to you the first chapter let's first define mathematics for you what is mathematics math is defined as the study of the relationships among numbers quantities and shapes for example you are to put a cover on a can but for you to know how much paper you would need you have to find first the surface area of the can or the cylinder this shows the relationship among numbers and shapes now can you think of another example that shows relationship among numbers quantities and shapes okay another definition of mathematics is it enhances our critical thinking skills reasoning special thinking and creativity most of us every time we try to answer a certain mathematical problem we tend to think of a way to solve the problem and if for instance that that certain way didn't work we would find another solution again and we will never stop until we get to the right answer this is somehow how mathematics helps us to be a better thinker and lastly mathematics helps us organize patterns and regularities in the world which will be discussed on the next slides okay so we have pattern and numbers in nature and the world so we have uh types of pattern first is symmetry or the symmetrical pattern it is a design or pattern that is identical on both halves when folded you may notice that when this butterfly is folded it will have two identical halves just like the other examples presented okay so that is for symmetrical pattern next pattern is spiral pattern which is defined as curved pattern that focuses on a certain point and a series of circular shapes that revolve around it you may notice that some of the examples presented are from the natural environment the reason why plants use a uh spiral form like the plant like this plant presented is because they are constantly trying to grow but stay secure third is fractal pattern fractal pattern are built from simple repeated shapes that are reduced in size when repeated the two best examples are romanesco broccoli and the spider web and last but not the least desolations tessellations are created with identical shapes which fit together with no gaps so we have here pineapple and the beehive as the example we will now proceed with the fibonacci sequence fibonacci sequence is discovered by an italian mathematician named leonardo pisano his nickname was fibonacci which roughly means son of bonacci and november 23 is the fibonacci day so later on i will explain to you why is that so okay so the sequence goes like this 0 1 1 2 3 5 8 13 21 and so on so what can you notice in the fibonacci sequence try to think what can you notice about this sequence okay each number in the sequence is the sum of the two numbers which preceded so you may notice that we started from zero and one so that is the start of the sequence and then for us to identify what will be the next uh value we would add these two numbers so we have zero plus one and the resulting value is the next number in the sequence which is one then for us to know to say the um next term again we would add one plus one so the answer is two then we add one plus two the answer is three we we add two and three the answer is five and so on so that is how the sequence goes and why is november 23 the fibonacci day um you may notice that the digits we have in november 23 is one one two three and here in the sequence the first four non-zero digits are one one two three so that is why uh i mean which corresponds to the fibonacci days so that is why um november 23 is the fibonacci day okay you can also check this link to know what is the magic of fibonacci numbers next we have the golden ratio golden ratio is denoted by fee which approaches a value of 1.618034 okay so next is the relationship between the fibonacci sequence and the golden ratio it is said that the ratio of any two successive fibonacci numbers is very close to the golden ratio referred to and represented as fee which is i said earlier approximately equal to 1.618034 okay so where to find the ratio of the two successive fibonacci numbers so we would let a be the smaller number from the sequence and you would let b be the larger number from the sequence so we have 2 and 3 we will find the ratio b over a which is equal to 1 to 1.5 which is uh quite close to the golden ratio then let's try another consecutive two consecutive fibonacci numbers 3 and 5. so five divided by three it's b over eight we have this um quotient which is quite equal to the golden ratio so what can you notice in this uh table so you can notice that the bigger the pair of the fibonacci numbers considered the closer the approximation and you may also ask if there is a chance that the ratio of the two successive fibonacci numbers will be equal to the golden ratio the answer is no it will just always be close to the golden ratio but never equal next we have the pattern and regularities in the world as organized by mathematics okay the first one is the motion of a pendulum the motion of a pendulum shows that the time it takes to swing back to its original position can be explained by mathematics through regularities in motion and the second one is the reflection in a plane mirror which shows that the regularity in size and distance you could see the same size as the object in the mirror can be mathematically explained by the law of reflection okay i have your question what do you think is the application of mathematics in your chosen course okay i want you to reflect on that and that ends the first chapter thank you for listening and i hope that you learned