Please say in a few words what is differentiation and what is integration to a fresh mind trying to understand what are these. Thanks
Good challenging questions. Thanks for asking.
If two are more related quantities affect at least one other quantity of an object, then how any change in one quantity will affect the other is the differentiation. For example, age and strength, weight and PB, speed and consumption etc. So, if you say carry a big box on your shoulder and tries to climb an Indian Railway station stairs, the strain on you will increase due to your age and you may even fall back. Ask a simple question: If a tank has 50 gallons of water and if the spout can discharge a cup (say 16 Oz) water how much time it will take to empty the tank? Here the rate at which water is discharged and the water pressure will be involved. Not a linear time say, 6400 Ozs/ (16/128) US measurement!
When liquids, gases, sound etc., which can not be counted, you add a very small quantity at a time to reduce the error in measurement and arrive at an approximate measurement. Since the error is involved in the measurement, you say, between this limit(of measurement) this will be the value. So, when you say I am driving at 55KM per hour, it is not 55Km rather it could be between 54.2 to say 56.5. That is why, in integration, we use graphical visual method to measure things. In Arithmetic and Algebra in general, you get an approximate numerical measure and any error will be on the decimal place or fraction.
So, even though you need arithmetic methods to calculate, you still can use other measures such as, Algebra, Calculus and so on, but must come back to arithmetic if a numerical value is needed.
So, arithmetic is the stem of an umbrella and all other branches are attached to it like the ribs and cover over the stem.
If you look at any text book, they will start with a definition, but not with any real life example. It is difficult but not impossible. Based on the observation and coming with a theory is “inductive reasoning”. Newton found his theorem in that way. Actually it is claimed that he stole “Acharya Kande’s – Gujarat scholar) theorem. See:the PDF.
Now, most text books starts with a theorem and provide proof with an example. This is what children memorize but can not solve real life problems. Thus neither the teacher has the inductive reasoning nor they teach with once in a while brilliant teacher who change your life.
So, I was very angry at many of my math teachers and started working on my own to understand mathematics (not all higher level math).
Here is another example: Ask any one to define “What is statistics” in this forum and I will give the definition later.
Understanding Algebra was tricky. There were no definition until `10th century when decimal system was established by al-Khwarizmi in today’s Iraq, decimal notation originally created by Ariya Bhatta for astronomical calculation, expanded by Brahma Gupta teaching Profit-loss to Arab traders and so on. It might have been known to others but documentation is lacking. So, 1760 text book is still used with new text books – old wine in new bottle. There is one great scholar Dr. Venkatasubramanian at IIT Bombay is the authority on this subject.