Numbers in Everyday Life

Andrew Thangaraj

Aug-Nov 2020

Numbers are everywhere

What are numbers used for?

To count or enumerate

To quantify or measure

To represent information or data

Population: India

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

Population: Japan

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

Population: Tigers in India

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

Money: Company quarterly income statement (in Cr.)

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

Measure length (Source: Wikipedia)

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

Sports statistics

Cricket

Run rate, strike rate, ball speed

Athletics

Time (races), length (throws)

Laptops are common in every sport today

Computers, digital world and numbers

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

Number skills

Basic skills

  • Ordering of numbers

  • Operations

    • addition, subtraction

    • multiplication, division

Advanced skills

  • Observe trends

  • Predict, plan

Theory of numbers

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!

World of Numbers

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.

Easy operations with natural numbers

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

Factoring a natural number

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)

Longer numbers?

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)

RSA public key cryptosystem

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

Summary

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