22  20. scientific notation ~(abbridged version)~

22.1 Introductory comments about some basic math

# remove all variables
rm (list = ls())

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##  A couple of introductory comments about basic math
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#--------------------------------------------
# how many zeros are there in a power of 10?
#--------------------------------------------

# 10^2 is 100
100     # 100 is the same as 10^2
[1] 100
10^2    # 10^2 is the same as 100
[1] 100
# 10^3 is 1000
1000    # 1000 is the same as 10^3
[1] 1000
10^3    # 10^3 is the same as 1000
[1] 1000
# etc ...
# In summary, you can figure out a power of 10, by adding the number
# of zeros after the 1 that match the exponent, e.g. 10^2 is 100 (2 zeros)
# This works similarly (but not exactly the same) for negative exponents of 10

# 10^-1 is 0.1
0.1      # 0.1 is the same as 10^-1
[1] 0.1
10^-1    # 10^-1 is the same as 0.1
[1] 0.1
# 10^-1 is 0.1
0.01     # 0.01 is the same as 10^-2
[1] 0.01
10^-2    # 10^-2 is the same as 0.01
[1] 0.01
# etc...
# The number of zeros after the decimal point is one less than the absolute value
# of the exponent of 10.

22.2 What is “Scientific Notation”

“Scientific notation” is used as a shorthand for writing very big numbers (and very small numbers - see below)

“Scientific notation” is not an “R” concept. It is a concept that is used by scientists and other people who work with numbers, whether they are using R or not. It is simply a technique to make it easier to work with very large and very small numbers. (Keep reading …)

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:

1.2345 * 10^0     # 10^0 is 1 so this doesn't change the first number
[1] 1.2345
1.2345 * 10^1     # 10^1 is 10 so this moves the decimal to the right by 1 position
[1] 12.345
1.2345 * 10^2     # 10^2 is 100 so this moves the decimal to the right by 1 position
[1] 123.45
1.2345 * 10^6     # This moves the decimal point to the RIGHT by 10 positions
[1] 1234500

The above calculations are all examples of “Scientific notation”. Scientific notation is used as a shorthand for writing very big numbers (and very small numbers - see below)

22.3 Scientific Notation in R

#---------------------------------------------------------------------
# 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.
# EXMAPLE - all of the following are the same exact number:
#---------------------------------------------------------------------

1234500            # this is the same
[1] 1234500
1.2345 * 10^6      # this is the same
[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. For example:

12345000000 # by default, R will show this value in scientific notation
[1] 1.2345e+10
#-------------------------------------------------------
# Negative exponents of 10 move the decimal to the LEFT
#-------------------------------------------------------

# 0.00123 is the same as 0.123 * 10^-4

0.0123      # this is the same value as below
[1] 0.0123
1.23*10^-2   # this is the same value as above
[1] 0.0123
# R will display very small numbers using scientific notation also.
# The following is a very small number (there are ten zeros).
# R will display this in scientific notation

0.0000000000123   # same as 1.23e-11
[1] 1.23e-11
1.23e-11          # we can write that directly also
[1] 1.23e-11

22.4 MORAL OF THE STORY - don’t become alarmed

#-----------------------------------------------------
# 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.
#-----------------------------------------------------

22.5 –PRACTICE–

Question - what are the values of the following expressions?

# Part (a)     1e-2 + 2e-1
1e-2          # this is just 0.01
[1] 0.01
2e-1          # this is 0.2
[1] 0.2
# ANSWER
1e-2 + 2e-1   # so this is 0.21
[1] 0.21
# Part (b)     9.876e5
9.876e5
[1] 987600
# Part (c)     5.23e4 + 1000
5.23e4 + 1000
[1] 53300

Question - What will R display for the following numbers?

# part (a)   12340000000000   (ten zeros)
12340000000000 # (ten zeros)
[1] 1.234e+13
# part (b)   0.0000000000123 (ten zeros)
0.0000000000123 # (ten zeros)
[1] 1.23e-11