# remove all variblesrm (list =ls())#################################################################################################################### A couple of introductory comments about basic math###################################################################################################################--------------------------------------------# how many zeros are there in a power of 10?#--------------------------------------------# The number of zeros (before the decimal point) in a# POSITIVE power of 10 is the same as the exponent10^1# 10 (one zero)
[1] 10
10^2# 100 (two zeros)
[1] 100
10^3# 1000 (three zeros)
[1] 1000
10^4# 10000 (four zeros)
[1] 10000
# etc ...# The number of zeros after the decimal point in a# NEGATIVE power of 10 is one less than the absolute value of the exponent10^-1# 0.1 (no zeros after the decimal point)
[1] 0.1
10^-2# 0.01 (one zero after the decimal point)
[1] 0.01
10^-3# 0.001 (two zeros after the decimal point)
[1] 0.001
10^-4# 0.0001 (three zeros after the decimal point)
[1] 1e-04
# etc ...#-------------------------------------------------------# Intro to Scientific Notation in R## Multiplying numbers by powers of 10#-------------------------------------------------------# If you multiply a number by a POSITIVE POWER of 10 the decimal point# will move to the RIGHT by the number of positions as expressed by the exponent.# Example:1.234*10^0# 1.234 : 10^0 is 1 so this doesn't change the first number
[1] 1.234
1.234*10^1# 12.34 : 10^1 is 10 so this moves the decimal to the right by 1 position
[1] 12.34
1.234*10^2# 123.4 : 10^2 is 100 so this moves the decimal to the right by 1 position
[1] 123.4
1.234*10^3# 1234.0
[1] 1234
1.234*10^4# 12340.0
[1] 12340
# etc ...# If you multiply a number by a NEGATIVE POWER of 10 the decimal point# will move to the LEFT by the number of positions as expressed by the exponent.# Example:123.4*10^-1# 12.34 move decimal point 1 position to the left
[1] 12.34
123.4*10^-2# 1.234 move decimal point 2 positions to the left
[1] 1.234
123.4*10^-3# 0.1234 move decimal point 3 positions to the left
[1] 0.1234
123.4*10^-4# 0.01234 move decimal point 4 positions to the left
[1] 0.01234
# etc ...# These types of calculations can be used as a shorthand to write very large numbers# and very small numbers in a very compact way. For example: ## 9.234 * 10^100 is shorthand for # the very very large number 9234000000...(total of 97 zeros)...0000# i.e. 97 zeros after the 9234## 9.234 * 10^-100 is shorthand for# the very very small number 0.00000......(99 zeros)...0009234# i.e. 99 zeros between the decimal point and the 9234## The above calculations are examples of "Scientific notation".# Scientific notation is used as a shorthand for writing very big numbers# (and very small numbers - see below)#---------------------------------------------------------------------# R has a shorthand notation for writing these types of calculations.# Instead of writing 1.2345*10^6, you could instead write 1.2345e6## The "e" in the number stands for "exponent". The "e" is understood# to be read as "times ten to the power of". The number after the "e"# is the exponent for the power of 10.# EXAMPLE - all of the following are the same exact number:#---------------------------------------------------------------------1234500# this is the same
[1] 1234500
1.2345*10^6# this is the same 1234500
[1] 1234500
1.2345e6# this is the same
[1] 1234500
# By default R will display values in scientific notation if the number is# very very big or very small. For example:# very large number - by default, R shows this value in scientific notation12300000000# 1.23e+10
[1] 1.23e+10
# very small number - by default, R shows this value in scientific notation0.000000123# 1.23e-07
[1] 1.23e-07
# We can also write scientific notation in our code if we like.1.23e+9/1e+8# 12.3 , same as 1230000000 / 100000000 but a easier to read :)
[1] 12.3
1.2e-4# 0.00012
[1] 0.00012
0.0000000000123==1.23e-11
[1] TRUE
0.00000000001231==1.23e-11
[1] FALSE
# If R returns some very large numbers or some very small numbers in a vector,# the ALL of the numbers in the vector will be displayed in scientific notation.# Some typical arithmetic - nothing surprising.100/c(100, 20) # 1 5
[1] 1 5
# Same code as above but the last value will be very large so ALL of the# results are expressed in scientific notation.100/c(100, 20) # 1 5
[1] 1 5
100/c(100, 20, 0.001) # 1e+00 5e+00 1e+05
[1] 1e+00 5e+00 1e+05
# numbers are all low enough to display without scientific notation3^ (1:5) # 3 9 27 81 243
[1] 3 9 27 81 243
# Some of the larger numbers are displayed in scientific notation so# R displays ALL of the numbers in scientific notation3^ (1:30) # 3.00e+00 9.00e+00 2.70e+01 8.10e+01 2.43e+02 etc ...
#-----------------------------------------------------# MORAL OF THE STORY - don't become alarmed## Occasionally, you will see R displaying numbers in # scientific notation. Don't become confused. Understand# that these are just "regular numbers" being displayed in # a more concise format. Any math that is done with these# numbers is the same as if you did the same math with the # equivalent non-scientific-notation format.#-----------------------------------------------------#----------------------------------------------------------# PRACTICE#----------------------------------------------------------# QUESTION - Write the value 4.98e+5 as a "regular" number# QUESTION - Write the value of 4.98e-3 as a "regular" number# what are the values of the following expressions?## 1e-2 + 2e-1## 9.876e5## 5.23e4 + 1000### What will R display for the following numbers?## 12340000000000 (ten zeros)## 0.0000000000123 (ten zeros)########################################################################## 2022 - BEREN - ONLY COVERED UP TO HERE IN THIS FILE# YOU ARE NOT RESPONSIBLE FOR THE MATERIAL BELOW THIS POINT IN THIS FILE################################################################################################################################################### 2022 - WILF - ONLY COVERED UP TO HERE IN THIS FILE# YOU ARE NOT RESPONSIBLE FOR THE MATERIAL BELOW THIS POINT IN THIS FILE################################################################################################################################################################################################### Roundoff error in Decimal (base-10) and Binary (base-2)########################################################################################################################## The numbers what we humans use every day are made up of 10 different digits i.e 0,1,2 .. 9# These numbers are therefore called "base-10" numbers or "decimal" numbers.## It may sound strange, but in order to perform many calculations, computers internally use numbers# that only have 1 and 0 as digits. These numbers are known as "base-2" or "binary" numbers. When computers display numbers to # people, the numbers are converted to the "base-10" (or "decimal") numbers that we are# used to seeing.## Right now we will NOT discuss the intricate details of how # computer work with Binary (base-2) numbers.# (For more info. you can refer to the powerpoint on Canvas.)# HOWEVER, you should understand the following ...## The fractions that can be converted to an exact number with a decimal point are # different for base-2 (aka binary) numbers and for base-10 (aka decimal) numbers. # For binary numbers (i.e. numbers that computers use internally)# fractions that can be reduced to have a power of 2 in the denominator# can be represented by a terminating decimal point value. # Other fractions cannot. ## This issue leads to "roundoff" error very frequently when doing math# in computer programming- similar to 1/3 + 1/3 + 1/3 ...## R masks this issue by displaying what you might think a number is.# However, the actual value for the number might be something else. # # You can use the print.default (or just print) function with the # digits argument to display the ACTUAL value that R stores for a number0.1
[1] 0.1
print(0.1, digits=22)
[1] 0.1000000000000000055511
0.3
[1] 0.3
print(0.3, digits=22)
[1] 0.2999999999999999888978
0.6
[1] 0.6
print(0.6, digits=22)
[1] 0.5999999999999999777955
1.1
[1] 1.1
print(1.1, digits=22)
[1] 1.100000000000000088818
# This issue can lead to "roundoff" errors. There are ways to deal with # this but it can get a little involved for this early in the course.# for right now, just accept it. We'll revisit this issue later.# It is well known that the fraction 1/3 cannot be directly represented by # a number with a decimal point. The best we can do is 0.333... You would need# an infinite number of 3's after the decimal point to accurately represent the # fraction 1/3. Therefore 0.333 + 0.333 + 0.333 is NOT the same as 1/3 + 1/3 + 1/3.# Similarly 1/11 is 0.090909... You cannot represent one eleventh exactly using# a finite number of digits.# In binary (i.e. base-2) a similar problem exists. However, the numbers that cannot# be represented exactly are different from those in decimal. For example, if you # would convert 1/5 to binary, you would not get an exact representation. # Therefore R will not be able to represent the value of 1/5 exactly # using standard "numeric" values.oneFifth =1/5oneFifth # R displays a rounded version of the value
[1] 0.2
print(1/5 , digits=22) # you can see the actual value stored by specifying the # of digits after the decimal point
[1] 0.2000000000000000111022
print(1/5*1, digits=22)
[1] 0.2000000000000000111022
print(1/5*2, digits=22)
[1] 0.4000000000000000222045
print(1/5*3, digits=22)
[1] 0.6000000000000000888178
print(1/5*4, digits=22)
[1] 0.8000000000000000444089
print(1/5*5, digits=22)
[1] 1
# The same is true for 1/10oneTenth =1/10oneTenth
[1] 0.1
print(oneTenth, digits=22)
[1] 0.1000000000000000055511
print(oneTenth *1 , digits =22)
[1] 0.1000000000000000055511
print(oneTenth *2 , digits =22)
[1] 0.2000000000000000111022
print(oneTenth *3 , digits =22)
[1] 0.3000000000000000444089
print(oneTenth *4 , digits =22)
[1] 0.4000000000000000222045
print(oneTenth *5 , digits =22)
[1] 0.5
print(oneTenth *6 , digits =22)
[1] 0.6000000000000000888178
print(oneTenth *7 , digits =22)
[1] 0.7000000000000000666134
print(oneTenth *8 , digits =22)
[1] 0.8000000000000000444089
print(oneTenth *9 , digits =22)
[1] 0.9000000000000000222045
print(oneTenth *10 , digits =22)
[1] 1
############################################################################################################################################################################### EXPLORING R's VARIOUS OPTIONS## R has over 80 different options that you can set. For most of the time we will# choose to just leave the defaults that are set when R is installed. However, # sometimes it is helpful to change some of these options. # options() # see all of R's options and their values## When working with roundoff error issues it is helpful sometimes to automatically# see more digits after the decimal point when you perform R commands. # You can do that through R's options. See the following for more info.## options() # see all of R's options and their values## names(options()) # see just the names of the options without their values## getOption("digits") # see the value of the digits option (default for new installation is 7)## options(digits=22) # set the number of max # of digits that R will display for a number################################################################################################################################################################################options() # see all of R's options and their values
getOption("digits") # see the value of the digits option (default for new installation is 7)
[1] 7
options(digits=22) # set the number of max # of digits that R will display for a number# You can see now that these numbers have roundoff error1/5
[1] 0.2000000000000000111022
1/10
[1] 0.1000000000000000055511
# These numbers do not1/2
[1] 0.5
1/4
[1] 0.25
1/8
[1] 0.125
# set digits option back to the defaultoptions(digits=7) # set the number of max # of digits that R will display for a number# You don't see the roundoff error anymore but it is still there ...1/5
[1] 0.2
1/10
[1] 0.1
# A somewhat more involved explanation ...############################################################################################################################## Floating point numbers and "roundoff errors"#### (A "floating point number" refers to a number that ## contains a decimal point. The term "floating point" is## a reference to a technique that# computers use to process numbers with decimal points in them.## We will not cover exactly why they## are called "floating point" numbers, but the reason is ## related to the fact that multiplying a number by a power of 10## simply moves the decimal point.)############################################################################################################################# Some fractions that we write do not have an exact representation# as floating point numbers. For example, many people will write# 1/3 as 0.333, however there really should be an infinite number of 3's in the floating point# version. 0.333 is just a rough approximation. 0.3333 is closer to 1/3# and 0.33333 (five 3's) is even closer. However, you will NEVER get# an exact representation of 1/3 unless you write an infinite number# of 3's after the decimal point.## Therefore 1/3 + 1/3 + 1/3 = 1# When converted into floating point numbers becomes, approximately ## 0.333 + 0.333 + 0.333 = 0.999 (which looks wrong!)## Some fractions can be converted exactly, eg.# 1/2 is 0.5 exactly# 1/4 is 0.25 exactly# 1/5 is 0.2 exactly# 1/8 is 0.125 exactly# 1/10 is 0.1 exactly# 1/16 is 0.0625 exactly # 1/25 is 0.04 exactly## In general if a fraction can be reduced to an equivalent fraction whose# denominator is a power of 2 , a power of 5 or a power of 10, then the # fraction can be represented by an exact terminating decimal.#------------------------------------------------------------------# # Base-10 (aka Decimal) numbers vs Base-2 (aka Binary) numbers# #------------------------------------------------------------------# Languages such as Hebrew, Arabic and English use different alphabets.# However, some languages such as French, Spanish and English use the same# alphabet. However, the words they form often mean different things in the # different languages. For example the word "pie" in English is a tasty treat# but "pie" in Spanish means a foot. In order to really understand what the word# "pie" means you need to know if the word is Spanish or English!!!# In a similiar way, there are different "numeric languages". # "Numeric languages", such as "Roman numerals", use different "alphabets".# For example, the number that we know as one hundred and twenty three, i.e. 123# is written in Roman numerals as CXXIII.## However, some "numeric languages" use the same digits that we recognize but# in a different way. The internal workings of computers are designed to use# a different "numeric language" than the numbers that people are used to seeing.# Internally, computers process information using a "numeric language" known as# "base-2" or "binary". For example, the number "one hundred and twenty three", i.e. 123, # can be written in binary (or base-2) as 01111011. Every base-2 (or binary) number# is written using just zeros and ones. For example, the binary number# 1001100100100101 is the equivalent of the decimal (base-10) number# 39205 (i.e. thirty nine thousand two hundred and five).## When a computer gets a decimal (base-10) number from a peson, the computer first converts# the decimal (base-10) number into a binary (base-2) number. The computer then# performs any processing it needs to do. Finally, the result is translated back# into the decimal form and displayed to the person.#----------------------------------------------------------------------------------# # Decimal (base-10) and binary (base-2) numbers can represent the exact same integers.##----------------------------------------------------------------------------------# Any integer that we can write in base-10 (or decimal) can also be written, in a different# form in base-2 (or binary). For example, the following are the decimal and binary# equivalents for a few numbers:## DECIMAL (base-10) BINARY (base-2)# 0 0 # 1 1# 2 10# 3 11# 4 100# 5 101# 6 110# 7 111# 8 1000#----------------------------------------------------------------------------------# # Decimal (base-10) can NOT precisely represent some "real" numbers (i.e. fractions)# and there will be "roundoff" error.##----------------------------------------------------------------------------------# In the number system we use every day, some fractions can be perfectly represented# by "real" numbers (i.e. numbers with decimal points) while others cannot.# For EXAMPLE:# 1/2 is EXACTLY 0.5# 1/4 is EXACTLY 0.25# 1/5 is EXACTLY 0.2# 1/8 is EXACTLY 0.124# 1/10 is EXACTLY 0.1## However, the following fractions CANNOT be represented EXACTLY by a number with a decimal point.# These fractions lead to "repeating digits after the decimal point"# EXAMPLE:# 1/3 is 0.333... (the 3's repeat forever)# 1/6 is 0.1666... (the 6's repeat forever)# 1/7 is 0.142857142857... (the 142857's repeat forever)# 1/9 is 0.111... (the 1's repeat forever)#----------------------------------------------------------------## Binary (base-2) can NOT represent other "real" numbers##----------------------------------------------------------------# Binary numbers have similar limitations but for DIFFERENT fractions.# Any fraction that has a power of 2 in the denominator can be represented# exactly in binary. However, other fractions cannot.# For example:## 1/2 is EXACTLY 0.1 (in binary)# 1/4 is EXACTLY 0.01 (in binary)# 1/8 is EXACTLY 0.001 (in binary)# 1/16 is EXACTLY 0.0001 (in binary)# 3/4 is EXACTLY 0.11 (in binary)# 3/8 is EXACTLY 0.011 (in binary)# (any fraction for which the denominator is a power of 2 can be represented exactly in binary)## However:## 1/3 in binary is 0.010101... (the 01's repeat forever)# 1/5 in binary is 0.00110011... (the 0011's repeat forever)# 1/6 in binary is 0.0010101... (the 01's repeat forever)# 1/10 in binary is 0.000110011... (the 0011's repeat forever)# (fractions whose denomiators are not a power of two cannot be represented exactly in binary)