Andrew Thangaraj
Aug-Nov 2020
What are numbers used for?
To count or enumerate
To quantify or measure
To represent information or data
Year | Total | Change |
---|---|---|
2020 | 1,380,004,385 | 13,586,631 |
2019 | 1,366,417,754 | 13,775,474 |
2018 | 1,352,642,280 | 13,965,495 |
2017 | 1,338,676,785 | 14,159,536 |
2016 | 1,324,517,249 | 14,364,846 |
2015 | 1,310,152,403 | 15,174,247 |
2010 | 1,234,281,170 | 17,334,249 |
2005 | 1,147,609,927 | 18,206,876 |
2000 | 1,056,575,549 |
Year | Total | Change |
---|---|---|
2020 | 126,476,461 | -383,840 |
2019 | 126,860,301 | -341,891 |
2018 | 127,202,192 | -300,533 |
2017 | 127,502,725 | -260,540 |
2016 | 127,763,265 | -221,868 |
2015 | 127,985,133 | -111,444 |
2010 | 128,542,353 | 43,247 |
2005 | 128,326,116 | 160,388 |
2000 | 127,524,174 |
Landscape | 2006 | 2010 | 2014 | 2018 |
---|---|---|---|---|
Shivalik Gangetic | 297 | 353 | 485 | 646 |
Central India and the Eastern Ghats | 601 | 601 | 688 | 1,033 |
Western Ghats | 402 | 534 | 776 | 981 |
North East Hills & Brahmaputra Plains | 100 | 148 | 201 | 219 |
Sunderbans | NA | 70 | 76 | 88 |
Total Tiger Population in India | 1,411 | 1,706 | 2,226 | 2,967 |
JUN 2020 | MAR 2020 | DEC 2019 | SEP 2019 | JUN 2019 | |
---|---|---|---|---|---|
Sales | 10,659 | 11,754 | 11,089 | 10,844 | 11,269 |
Other Income | 55 | 166 | 291 | 302 | 279 |
Total Income | 10,714 | 11,920 | 11,380 | 11,146 | 11,549 |
Total Expenditure | 32,459 | 19,553 | 14,179 | 44,580 | 14,557 |
EBIT | -21,745 | -7,633 | -2,799 | -33,434 | -3,008 |
Interest | 3,803 | 4,177 | 3,722 | 3,654 | 3,723 |
Tax | 0 | -99 | 17 | 13,962 | -1,799 |
Net Profit | -25,548 | -11,711 | -6,538 | -51,051 | -4,932 |
Scale | Range (m) | Examples |
---|---|---|
Subatomic | \(10^{−35}\) | Quantum foam |
\(10^{−18}-10^{−15}\) | Proton, neutron | |
Atomic to cellular | \(10^{−15}-10^{−12}\) | Atomic nucleus |
\(10^{−12}-10^{−9}\) | Hydrogen atom | |
\(10^{−9}-10^{−6}\) | DNA helix, virus | |
Cellular to human | \(10^{−6}-10^{−3}\) | Bacteria, hair diameter |
\(10^{−3}-10^0\) | Mosquito, violin | |
Human to astronomical | \(10^0-10^3\) | Human, Eiffel Tower |
\(10^3-10^6\) | Everest, large asteroid | |
Astronomical | \(10^6-10^9\) | Moon, Earth |
\(10^9-10^{12}\) | Sun, Earth’s orbit | |
\(10^{12}-10^{15}\) | Solar System | |
\(10^{15}-10^{18}\) | Light-year, nearest star | |
\(10^{21}-10^{24}\) | Milky Way, nearest galaxy | |
\(10^{24}\) | visible universe |
Cricket
Run rate, strike rate, ball speed
Athletics
Time (races), length (throws)
Laptops are common in every sport today
All information is represented, stored and transmitted as numbers
ASCII/unicode: convert letters/text into numbers
Music/audio: sample and quantize the signal
Image/video: digital sensors and display
Computers store numbers in binary format and perform all operations on them using binary arithmetic
Binary is base 2 number system: 0 and 1
Today’s digital world: everything is bits
Basic skills
Ordering of numbers
Operations
addition, subtraction
multiplication, division
Advanced skills
Observe trends
Predict, plan
Do not worry about what the number represents
Computations with numbers are universal
Study numbers and operations with numbers
Comparison, Addition, Multiplication, Division
Study more difficult problems
There are some interesting applications for difficult problems!
Natural numbers
0,1,2,3,4,…
Integers
…,-4,-3,-2,-1,0,1,2,3,4,…
Rational numbers (includes terminating and recurring decimals)
Examples: 1/2, -13/57, 34.2345, -111.343434….
Irrational numbers (non-terminating, non-recurring decimals)
Examples: \(\sqrt{2}\), \(\pi\) etc.
Addition, multiplication
Simple and direct methods learnt in school
Number of steps is (roughly) the number of digits or its square
Division with remainder
Long division learnt in school
Number of steps is (roughly) the square of number of digits
Factor \(51\)
\(3\times 17\)
Factor \(6887\)
\(71\times 97\)
Factor 1522605027 9225333605 3561837813 2637429718 0681149613 8068865790 8494580122 9632589528 9765400035 0692006139 (RSA-100 with 100 digits)
3797522793 6943673922 8088727554 4562785456 5536638199
×
4009469095 0920881030 6837352927 6146838921 4899724061
(took few days in 1991 in a parallel computing facility, and takes about one hour today on a powerful server-class computer)
310 7418240490 0437213507 5003588856 7930037346 0228427275 4572016194 8823206440 5180815045 5634682967 1723286782 4379162728 3803341547 1073108501 9195485290 0733772482 2783525742 3864540146 9173660247 7652346609 (RSA-640 with 193 digits)
1634733 6458092538 4844313388 3865090859 8417836700 3309231218 1110852389 3331001045 0815121211 8167511579 x 1900871 2816648221 1312685157 3935413975 4718967899 6851549366 6638539088 0271038021 0449895719 1261465571
(80 CPUs took 3 months in 2005)
2140324650 2407449612 6442307283 9333563008 6147151447 5501779775 4920881418 0234471401 3664334551 9095804679 6109928518 7247091458 7687396261 9215573630 4745477052 0805119056 4931066876 9159001975 9405693457 4522305893 2597669747 1681738069 3648946998 7157849497 5937497937 (RSA-250 with 250 digits)
64135 2894770715 8027879019 0170577389 0848250147 4294344720 8116859632 0245323446 3023862359 8752668347 7087376619 2558569463 9798853367
×
3337 2027594978 1565562260 1060535511 4227940760 3447675546 66784520987 0238417292 1003708025 7448673296 8818775657 1898625803 6932062711
(Feb 2020: 2700 core-years with a 2.1GHz core processor)
https: established using RSA public key cryptosystem
RSA: Uses difficulty of factoring long numbers
One of the biggest applications of number theory
610+ digit numbers are used today
Numbers are omnipresent in everyday life
Practice: decimal for human reading and binary for computer arithmetic
Theory: study numbers in an abstract way and develop non-obvious applications