Monthly Archives: August 2013

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Basic Inequalities

Inequalities are solved similar to equations, with rules of inequality. For example, the solving of 3x+4<7 is as follows:

3x + 4 < 7

-4  -4

3x       < 3

div 3          div 3

x         < 1

This can be graphed on a number line with a dot where the 1 is. For a greater-than or less-than inequality, an open dot(o) is used to represent the number. For a greater-than-or-equal-to or less-than-or-equal-to inequality, use a closed dot(&#149).

This would be graphed:

   < ——- circ

               1

Note the open dot over 1.

In an inequality, one major difference is that when dividing or multiplying by a negative number, the inequality sign is switched.

frac{ x }{text{-}3} leq -3
ast text{-}3       ast text{-}3
x     geq  9
Notice the flipped sign. When multiplying both sides by -3, the ≤ becomes ≥.
This would be graphed as:
                bullet ———->

               9


Compound Inequalities

Inequalities can be combined with and or or.

An and inequality has multiple inequality signs. For example:

1 < x < 10

Inequality operations can still be performed, but must affect all sections. When dividing or multiplying by a negative number, you must flip the signs.

And inequalities can also be graphed, by connecting two dots. For example, 1 < x < 10 is graphed:
        O——O

        1        10

Or inequality:
An or inequality is two inequalities joined by an or

x  3
Which is graphed as two separate inequalities.


Interval Notation

Interval notation is a way of describing inequalities as a set of numbers from a to b.

For example, x>3 would be described as:
(3 text{ , } infty)
Because it starts at 3 and goes to infinity.
A parenthesis is used for both sides because it approaches but never reaches 3 and infinity. If the inequality was x≤5 the interval notation would be:
(text{-}infty text{,} 5]
Now a square bracket is by 5 because x can reach 5.
For or inequalities, two interval notation groups are used, separated by or.

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A Boolean is a value that is true or false. It is used in algebra, set theory, and computer programming. In fact, it is the building block of any computer. In computer programming, a Boolean is represented by a 0(false), or a 1(true).

Boolean Logic was developed by George Boole in the mid 1800’s. It has several gates, which are operations on values. The gates are:

  • NOT: Returns the opposite, so 1 returns 0 and vice versa.
  • AND: If both values are 1, the output is 1. Otherwise, the output is 0.
  • OR: If either or both values are 1, the output is 1. If both are zero, the output is 0.
  • NOR(OR + NOT): If either or both are 1, the output is 0. If both are zero, the output is 1
  • NAND(AND + NOT): If both values are 1, the output is 0. Otherwise, the output is 1.
  • XOR(Exclusive OR): If both values are the same, the output is 1. Otherwise, the value is 1.
  • XNOR(Exclusive NOR): If both values are the same, the output is the opposite of the values. Otherwise, the output is 0.

Boolean values are binary, or base-2. Therefore, the digit places are powers of two. So 1 is one 20 , but 10 is one 21, or 2.

This leads to interesting addition.
0+0=0
0+1=1
1+1=10
10+1=11
11+1=100

A Boolean memory value is called a bit. 8 bits make a byte. A byte therefore has 256 different possible values. Other storage values are:

Kilo (K)
2^10 = 1,024 bytes
Mega (M)
2^20 = 1,048,576
Giga (G)
2^30 = 1,073,741,824
Tera (T)
2^40 = 1,099,511,627,776
Peta (P)
2^50 = 1,125,899,906,842,624
Exa (E)
2^60 = 1,152,921,504,606,846,976
Zetta (Z)
2^70 = 1,180,591,620,717,411,303,424
Yotta (Y)
2^80 = 1,208,925,819,614,629,174,706,176

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Today I introduce Quiz Articles! This is how it works: I will post an article on a random subject of science. I will give you a few days to learn the material. Then, I will remove the article from the website and replace it with a quiz about the article. Have fun and good luck!

Article:
Scientists have recently discovered that the four fundamental forces act upon special fields. Certain things or qualities trigger the fields and allow particles or an excitation of these fields to act upon the object. Let’s say that we are talking about the moon. The moon has mass. This quality of the moon excites the field of gravity. Because it has this quality, it has gravitons, which in turn give it gravity. A graviton is the particle or excitation of the Gravity field. Each fundamental force has its own field upon which it acts upon and its own particle or excitation. They are Gravitons (Gravity), Photons (Electromagnetic), and the W- and Z- Bosons (Strong and Weak). These particles are actually energy in a form as an excitation. All particles are some form of condensed energy. Everything is energy.

Please note that the graviton is only hypothetical while the Photon and W- and Z- Bosons are confirmed to exist.