Union and Intersection

Almost every Mathematics test has at least one problem involving Unions and Intersections.

The Union of two sets is the set of elements which are in either set.

Example: A = {1,2,3}, B = {3,4,5}. The UNION of A and B, written AU B = {1, 2, 3, 4, 5}. Notice how none of the numbers are repeated in the final set. Though there is a 3 in set A and a 3 in set B, there is only one 3 in set A U B.


The Intersection of two sets is the set of elements which are in all of the selected sets.
Example: A = {3,5,9}, B = {5, 9, 11}. The INTERSECTION of A and B, written A ∩ B = {5, 9}. Again, multiple numbers are not repeated.

Permutations and Combinations

   

Numbers and Means! (*)

Abundant numbers: Numbers where the sum of the factors, excluding the number itself, are greater than the number. (Also know as Excessive numbers. Every multiple of a Perfect number or Abundant number will also be an Abundant number.)
EX:  12 < (1+2+3+4+6)

Deficient numbers: Numbers where the sum of the factors, excluding the number itself, are less than the number.
EX: 10 > (1+2+5)

Perfect numbers: Numbers where the sum of the factors, excluding the number itself, equal the number.
EX: 6 = (1+2+3)

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Geometric mean: If you have n numbers, multiply the numbers together and take the nth root.
EX: Geometric mean of 6 and 7 = (6x7)^1/2

Harmonic mean: If you have n numbers, take the reciprocal of each number, add these fractions together, and divide n by this total.
EX: Harmonic mean of 6 and 7 = 2/(1/6 + 1/7)

Arithmetic mean: If you have n numbers, add the numbers together and divide by n.
EX: Arithmetic mean of 6 and 7 = (6+7)/2

Mathematicians You Should Know!

There is usually a question on every UIL Mathematics test concerning famous mathematicians. We've begun to compile a list of some of those men and women who appear the most frequently, and their major contributions to the field of mathematics.

Sonya Kovalevsky - she was the first major female Russian mathematician, is responsible for important contributions to analysis, differential equations and mechanics, and is the first woman appointed to a full professorship in Northern Europe.

Ada Byron Lovelace - daughter of Lord Byron, she is chiefly known for her work on Charles Babbage's analytical engine. She is sometimes called the "World's First Computer Programmer".

Benoit Mandelbrot - Father of fractal geometry.

Georg Cantor - Father of set theory.

Hypatia - First female mathematician.

Freda Porter - One of the few American Indian women who have earned a Ph.D. in mathematics, and a member of the Lumbee tribe.

Amalie "Emmy" Noether - German-born; Daughter of Max Noether and sister of Fritz Noether; 2nd woman to recieve a doctorate in the field of mathematics; contributed greatly to abstract algebra such as: ring, groups, and fields.

Karen E Smith - Born in New Jersey; recognized for research in algebraic geometry and communicative algebra at University of Michigan; received 2001 Ruth Lyttle Satter Prize given every two years to a woman for an outstanding contribution to mathematics.

Velvet Dow - Born on this day and contributes infinite amounts of knowledge onto her students whom love her very much. Mainly known for being extremely wise and knowledgeable. Sometimes seen wearing her beard of wisdom. Happy Birthday! :)

Properties of Real Numbers!

One type of problem that appears on nearly every UIL Mathematics test deals with different properties of real numbers. We've compiled a list of the properties that appear most often:

1) Assosiative Property - Changing the grouping of numbers does not change the result of the equation. This works for both addition and multiplication. For example:

(4 + 7) + 13 = 4  + (7+13) = 24
(a + b) + c = a + (b + c) = a + b + c

2) Commutative Property - Multiplying/adding numbers in any order still produces the same result. For example:

7*4 = 4*7 = 28
ab = ba

8+9 = 9+8
a + b = b + a

3) Distributive Property - Multiplying a single term times two or more terms inside a set of parentheses produces the same result as multiplying a single term times each number in the parentheses. For example:

3(4+5) = 3(4) + 3(5) = 27
a(b+c) = a(b) + a(c)

4) Identity Property - Adding a zero to any number does not change the number; multiplying a number by 1 does not change the number. (Hint: The number will retain its identity!) For example:

5 + 0 = 5
a + 0 = a

7 * 1 = 7
a * 1 = a

5) Inverse Property - For every real number not equal to zero, the Inverse Property is another real number, called the Inverse Number. There are two types of Inverse Properties: Additive Inverses and Multiplicative Inverses. For example:

Additive Inverse: 1 + (-1) = 0  (1 is the number, -1 is the Inverse Number.)

Multiplicative Inverse: 6 * (1/6) = 1 (6 is the number, 1/6 is the Inverse Number.)