# Category Archives: Mathematics

Mathematics uses many concepts in threes. The first structure mathematically is a triangle. There are acute, right, and obtuse angles. Trigonometry is the study of the relationship of the sides of a triangle. Have your heard of Pascal's Triangle?

# Percentage of integers containing digit 3

What percentage of all integers contains at least one instance of the digit three? For example, 13, 31, 33 and 103 all contain the digit "three" at least once. ANSWER!   Answer: How Many Threes? 100% of all integers contain at least one three. What?!? How can this be? The solution is so surprising, it is difficult, if not impossible to believe that 100% of integers contain the digit three at least once.

# equivalence relation

 In set theory, a relation that is reflexive, symmetric, & transitive on a set X is called an equivalence relation on X. SImply put as a graph, reflexive me ans there is a loop, symetric means for every directed edge from v to w, there is also a directed edge from w to v, and transitive means for a directed edge x to y, and y to z, there is also an edge from x to z.

# The Matrix

## The Matrix

More than 22 centuries ago, Aristosthenes gave the world his delivery on the prime numbers. Many have wondered about the nature of prime numbers, and many deliveries have been written about prime numbers since. In my book "In Search of a Cyclops" (published in 2000 as "The Proof of Nothing," with both versions available at pentapublishing.com), I take a different look at prime numbers. I use their sequencing as the explanatory basis for all numbers, and through the prime numbers I was able to discover a special matrix of all numbers. It is a delivery that does not give the number 3 the most important position. But the intriguing and controversial matrix does explain the numerous occurrences of threes we find all around us.

# minus sign with a fraction

The minus sign for a negative fraction (a rational number) can be in one of three places. It can be in front of the fraction, assigned to the numerator, or assigned to the denominator. In all three cases, the fraction is negative.

# Mean, Mode, Median

### Average

Three types of averages are the mean, mode and median. The weekly salaries of six employees at McDonalds are \$140, \$220, \$90, \$180, \$140, \$200. For these six salaries, find: (a) the mean (b) the median (c) the mode.

List the data in order: 90, 140, 140, 180, 200, 220

Mean:    90+ 140+ 140+ 180 + 200 + 220 =
6

Median:    90,140,140,180,200,220
The two numbers that fall in the middle need to be
averaged.     140 + 180 = 160
2

Mode:   The number that appears the most is 140

# Magic Square

A magic square is an arrangement of the numbers from 1 to n^2 (n-squared) in an nxn matrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same. It is not hard to show that this sum must be n(n^2+1)/2.

# Pascal’s Triangle

Pascal’s Triangle was originally developed by the ancient Chinese, but Blaise Pascal was the first person to discover special patterns contained inside the triangle. They teach his ideas in various schools online in math courses. You probably also heard of this guy from your high school math teacher.

Triangular numbers appear in Pascal’s Triangle. In fact, the 3rd diagonal of Pascal’s Triangle gives all triangular numbers as shown below:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1