commutative

Examples of commutative in the following topics:

• Optional Collaborative Classrom Exercise

  • Nineteen people were asked how many miles, to the nearest mile they commute to work each day.The data are as follows:
  • True or False: Three percent of the people surveyed commute 3 miles.If the statement is not correct, what should it be?
  • What fraction of the people surveyed commute 5 or 7 miles?
  • What fraction of the people surveyed commute 12 miles or more?

• Basic Operations

  • The commutative property describes equations in which the order of the numbers involved does not affect the result.
  • Addition and multiplication are commutative operations:
  • As with the commutative property, addition and multiplication are associative operations:

• The Rural Rebound

  • Often, these communities are commuter towns or bedroom communities.
  • Commuter towns are primarily residential; most of the residents commute to jobs in the city.
  • In general, commuter towns have little commercial or industrial activity of their own, though they may contain some retail centers to serve the daily needs of residents.
  • Although most exurbs are commuter towns, most commuter towns are not exurban.
  • This may happen especially where commuter towns form because workers in a region cannot afford to live where they work and must seek residency in another town with a lower cost of living.

• Variability in linear combinations of random variables

  • Suppose John's daily commute has a standard deviation of 4 minutes.
  • What is the uncertainty in his total commute time for the week?
  • Thus, the variance of the total weekly commute time is: variance = 1 2 × 16 + 1 2 × 16 + 1 2 × 16 + 1 2 × 16 + 1 2 × 16 = 5 × 16 = 80
  • The standard deviation for John's weekly work commute time is about 9 minutes.
  • However, if John walks to work, then his commute is probably not affected by any weekly traffic cycle.

• Linear combinations of random variables

  • For instance, the amount of time a person spends commuting to work each week can be broken down into several daily commutes.
  • It takes John an average of 18 minutes each day to commute to work.What would you expect his average commute time to be for the week?
  • We were told that the average (i.e. expected value) of the commute time is 18 minutes per day: E(X i ) = 18.
  • Would you be surprised if John's weekly commute wasn't exactly 90 minutes or if Elena didn't make exactly $152?
  • For John's commute time, there were five random variables – one for each work day – and each random variable could be written as having a fixed coefficient of 1:

• The Identity Matrix

  • Note that the definition of [I][I] stipulates that the multiplication must commute, that is, it must yield the same answer no matter in which order multiplication is done.
  • This stipulation is important because, for most matrices, multiplication does not commute.
  • There is no identity for a non-square matrix because of the requirement of matrices being commutative.

• Superposition of Fields

  • Vector addition is commutative, so whether adding A to B or B to A makes no difference on the resultant vector; this is also the case for subtraction of vectors.
  • Their sum is commutative, and results in a resultant vector c.

• Addition, Subtraction, and Multiplication

  • = $ac + bidi + bci + adi$ (by the commutative law of addition)
  • = $ac + bdi^2 + (bc + ad)i$ (by the commutative law of multiplication)

• Inverses of Composite Functions

  • The functions $g$ and $f$ are said to commute with each other if $g ∘ f = f ∘ g$.
  • In general, the composition of functions will not be commutative.

• Working from Home

  • With modern telecommunication technology, no longer is it necessary for employees to undergo a daily commute to a central place of work.
  • Many telecommuters work from home, while others, who are occasionally referred to as "nomad workers" or "web commuters", work from coffee shops or other locations.
  • Working from home can free up the equivalent of 15 to 25 workdays a year from time that would have otherwise been spent commuting, and save between $4,000 and $21,000 per year in travel and work-related costs.
  • Entrepreneurs choose to run businesses from home for a variety of reasons, including lower business expenses, personal health limitations, and a more flexible schedule due to the lack of a commute.