I have always wondered what does trigonometry, calculus, logarithms solve real world problems? Where do they apply in real life? Is there any simple book where I can understand it?
[Math] How to relate calculus, trigonometry etc in real life
calculuslogarithmsreference-requestsoft-questiontrigonometry
Related Solutions
What's 128 times 512?
For those of us who grew up in the digital age and so have memorized the table of powers of 2 to many exponents (I know them to exponent 16 off top of my head), this is easy using logarithms base 2: $$128 * 512 = 2^7 * 2^9 = 2^{16} = 65536 $$ Notice: this is nothing more nor less than a logarithm calculation. Let me lay it out in a different way. $$\log_2(128) = 7 $$ $$\log_2(512) = 9 $$ $$\log_2(128 * 512) = \log_2(128) + \log_2(512) = 7 + 9 = 16 $$ So, $$128 * 512 = 2^{16} = 65536 $$ At all stages, I have consulted a memory device (in this case, my brain) to do the calculations.
Okay, now let me turn to your problem. $$734 * 213 $$ You say "the log usage doesn't give the correct result". Well, if you want all 6 digits of the answer to be completely correct, that's true.
But wait, how did I even know that the answer was 6 digits? I knew this because, in my head and very rapidly, I did an estimated calculation that amounts to the following logarithm calculation (it takes me 50 times longer to write this out than it took to do it in my head): \begin{align*} \log_{10}(734) &= \log_{10}(100 * 7.34) = \log_{10}(100) + \log_{10}(7.34) = 2 + \log_{10}(7.34) \\ \log_{10}(213) &= \log_{10}(100 * 2.13) = \log_{10}(100) + \log_{10}(2.13) = 2 + \log_{10}(2.13)\\ \log_{10}(734 * 213) &= \log_{10}(734) + \log_{10}(213) = 2 + \log_{10}(7.34) + 2 + \log_{10}(2.13) \\ &= 4 + \log_{10}(7.34) + \log_{10}(2.13) \\ &= 4 + \log_{10}(7.34 * 2.13) \end{align*} Now I apply this to do an estimate. $$4 + \log_{10}(7 * 2) < 4 + \log_{10}(7.34 * 2.13) < 4 + \log_{10}(8*3) $$ $$4 + \log_{10}(14) < \log_{10}(734 * 213) < 4 + \log_{10}(24) $$ $$\log_{10}(10000*14) < \log_{10}(734 * 213) < \log_{10}(10000*24) $$ $$\log_{10}(140000) < \log_{10}(734 * 213) < \log_{10}(240000) $$ and so I conclude that $$140000 < 734 * 213 < 240000 $$ Hence I know that $734 * 213$ is a six digit number, in fact I know quite a bit more.
This is the real value of logarithms: it lets you do estimated calculations very, very quickly. The better logarithm table you have, the better and more accurate your calculation will be.
Best Answer
I am gonna go with two examples I commonly give to my students.
Trigonometry: This one is historical. Indians in 6th century were able to work out the distance ratio between moon-earth and sun-earth by realizing that when they see a half moon, the angle between earth-moon-sun has to be a right angle and they can use their trigonometric functions (which they knew already) to work out what the distance to the sun is, relative to the distance to the moon.
Source of the picture, more info on this
Calculus: A much more vague example, but one that has had success with non-mathematical people has been a simple idea of how derivatives might be used in the real world. If one knows that derivatives relate to the slope, then I begin by drawing a plane with air flowing around it, such as this:
Planes that cannot fly are not a good idea and so is trial-and-error method of building twenty planes and see which one does the best. That's where the maths come in, calculating the airflow (Which is demonstrated here by the slope given by the derivative, although in reality it's of course not quite that easy - mathematically speaking).
add: Logarithms Watch this numberphile video for a nice example.
Hope this gives some illustration of the endless ways mathematics can be used in the "real world"