The Number Systems Section 2-2

Natural (counting): 1, 2, 3, ....

Whole: 0, 1, 2, 3, ...

Integers: ...-3, -2, -1, 0, 1, 2, 3, ...

Rational: All numbers that can be written as a fraction or RATIO (decimals-terminating and repeating, fractions, mixed numbers, integers)

Irrational: Decimals that never terminate or repeat - like pi and square root of 2


Real numbers: All the number systems together


Complex or Imaginary Numbers: we don't study these until Algebra II, but they are stated as "
a + bi" where i is equal to the square root of -1



DENSITY PROPERTY: Between any 2 rational numbers is another rational number

Rational number = any number that can be expressed as the ratio (fraction) of two integers
a/b, where a and b are both integers and b cannot be zero
.
"b" cannot be zero because you cannot divide by zero.....IT'S UNDEFINED!

There are positive and negative rational numbers.

One way to put them in order from least to greatest is to simply place them on the REAL number line.
Numbers will be least to greatest if read from the left to the right.

If you have both fractions and decimals, it's often easier to change the fractions into decimals by simply dividing the numerator by the denominator.

REMEMBER: When numbers are NEGATIVE, the closer they are to zero, the bigger they are.
Example: -1/2 is greater than -3/4
-2.3 is greater than -2.5

Algebra Period 4

Using the Distributive Property 2-7
Hanging out with an old friend:
THE DISTRIBUTIVE PROPERTY WITH NEGATIVES
The distributive property works the same when there is subtraction in the ( )
a(b - c) = ab - ac

Inverse of a Sum 2-8
Property of -1:
For any rational number a,
(-1) a = -a
In words: MULTIPLYING BY -1 changes a term to its OPPOSITE SIGN

INVERSE OF A SUM PROPERTY:
DISTRIBUTING THE NEGATIVE SIGN
incognito, it's simply distributing -1
TO EACH ADDEND INSIDE THE PARENTHESES

EXAMPLE: -(3 + x) = -1(3 + x) = (-1)(3) + (-1)(x) = -3 + -x or -3 - x
ALL THAT HAPPENED WAS THAT EACH SIGN CHANGED TO ITS OPPOSITE!
-(a + b) = -a - b

Of course they get MUCH HARDER (but the principle is the same!)
[5(x + 2) - 3y] - [3(y + 2) - 7(x - 3)]

Distribute and simplify inside each [ ] first
[5x + 10 - 3y] - [3y + 6 - 7x + 21]

Now, the subtraction sign between them is really a -1 being distributed!
"Double check" to see this (change the subtraction to adding a negative):
[5x + 10 - 3y] + - 1[3y + 6 - 7x + 21]

Distribute the -1 to all the terms in the 2nd [ ]
[5x + 10 - 3y] + -3y + -6 + 7x + - 21

Simplify by combining like terms:
12x - 6y -17