On Equations of Intersecting, Coinciding, and Parallel Lines

When we have two lines on a plane, there are three possibilities:

  • the lines will never meet (parallel)
  • the lines will meet at one point (intersecting)
  • the lines will meet at infinitely many points (coinciding).

As for the third case, coinciding means that lines which are on top of each other.

In Algebra, we have learned that a line can be represented with an equation. The equations which represent lines are called linear equations. We have learned that linear equations can be represented by y = mx + b, where m and b are real numbers.

We can examine the three cases mentioned above in terms of equations. Can we determine if lines are parallel, intersecting, or coinciding based on equations only?  » Read more

How to Represent Inequalities in One Variable

The inequality x > 3 means all numbers greater than 3. The set of numbers that makes the inequality true is called the solution set of the inequality. An instance of this numbers such as 4 is a solution to x > 3.

The graph of x > 3 is shown on the number line below. The “empty circle” means that the solution does not include 3. The highlighted part of the number line (blue ray) means that it includes all real numbers greater than 3 and the arrow indicates that it goes up to infinity.

representing inequalities

Another notation to represent x > 3 is (3, ∞). The number on the left side is the lower bound and the right side is the upper bound. This notation means from 3 up to infinity. In addition, the symbol ( denotes that 3 is not included in the solution. Therefore, the interval (5,7) means all real numbers from 5 to 7, not including 5 and 7. The infinity symbol always takes the ( and ) symbols.  » Read more

Sad News: John Nash dies at 86

Perhaps the saddest news this week are the deaths of famous mathematician John Nash, 86, and wife Alicia, 82 in a car accident last Saturday, May 23. Nash was known for his work in Economics (Game Theory) particularly on the Nash equilibrium as well as his contributions real algebraic geometry.  He received the Nobel Prize for Economics in 1994, a Double Helix Medal in 2010, and an Abel Prize this year.

John Nash

image via Wikipedia

The life of John Nash, particularly his battle with paranoid schizophrenia, was detailed in Sylvia Nasar’s A Beautiful Mind which was later made into a  Hollywood film where Nash was portrayed by Russell Crowe.

Many people expressed their grief on Nash’s death through Twitter and other social media sites. His death is truly a big loss to the mathematics community.

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