SYSTEMS

MATHEMATICS

Decimal System

1 . = one

2 .. = two

3 ... = three

4 . . . . = four

5 . . . . . = five

6 . . . . . . = six

7 . . . . . . . = seven

8 . . . . . . . . = eight

9 . . . . . . . . . = nine

10 = 9 + 1 = ten

20 = 10 + 10 = twenty

30 = 20 + 10 = thirty

40 = 20 + 20 = forty

50 = 30 + 20 = fifty

60 = 40 + 20 = sixty

70 = 50 + 20 = seventy

80 = 40 + 40 = eighty

90 = 50 + 40 = ninety

100 = one hundred (90 + 10)

200 = two hundred (100 + 100)

1000 = one thousand (500 + 500)

1,000,000 = one million (1000 * 1000)


Infinitesimal numbers = 0.00000000000...1
                        
0.00000000000...2 etc.
                        0.11111111111...1 etc.
                        
0.99999999999...9

These are small, infinitely repeating numbers.

Rational numbers.

These are numbers with values expressible in fractions
and mathematical relationships.

    X = any number.
    Y = any number, possibly different from X.
    Z= any number, possibly different from X and Y

     10X = 10 of any number.

     X / Y = Any number divided by any number.

     3X / Y = Any number divided by any number
     in which 3X tends to be three times larger than Y.

Equations

1 + 1 = 2

2 + 3 = 5

2 * 10 = 20

1 / 10 = 0.1

1/100 = 0.01

1/1000 = 0.001

10 / 20 = 1/2 = 0.5

10X = Y = Y is exactly 10 * X
That is the same as writing 10X - Y = 0.


Squares and Square Roots

0 ^ 1 = 0 * 1 = 0
1 ^ 0 = 1 * 1 = 1
2 ^ 0 =  1 * 1 * 1 = 1 etc.
1 ^ 2 = 1 * 1 = 1
1 ^ 3 = 1 * 1 * 1 = 1
2 ^ 2 = 2 * 2 = 4
2 ^ 3 = 2 * 2 * 2 = 8

1 root of any number is that number.

The 2 root of any number is the square root of that number.

The square root of 4 is 2, because two 2s multiply to equal 4.

    Number    Sq. Rt.
 -----------------------------
        4               2
        9               3
       16              4
       25              5
       36              6
       49              7
       64              8


Multiplying with Exponents
Add exponents that are multiplied:

(2^2) (2^2) = 2 ^ 4 = 2 * 2 * 2 * 2 = 16
And we know that 2 ^ 2 is 2*2 which is 4, and 4 * 4 is 16.

If a negative exponent is alone, simply turn the multiplied
expression into 1 divided by the original expression raised to the
absolute number or positive value of the negative exponent.

For example,

2 ^ - 2 = 1 / 2 ^ 2 = 1/4

2 ^ - 3 = 1 / 2 ^ 3 = 1/8

Note that these are just fractional powers of 2.

If an expression involving positive and negative numbers is
multiplied, then BOTH RULES APPLY.

(2 ^ 2) (2 ^ -2) = 1 / (2 * 2 ^ 2) = 1 / 16

And indeed, 4 * 1/16 is a 1/4 of a 1/4 which is 1/16.

In the case of multiplying negative exponents, the result is also
multiplication.

In the case of pre-existing exponents in fractions, the advice is to
simplify them by computing values or multiplying or dividing
multiple groups by the opposite of the extracted fraction.

For example,

(4 / 2 ^ 16) + (2 / 3 ^ 16) could reduce to:

(4/2 + 2/3) ^ 16

Now we would either just enter it into our calculator, or multiply
the 2 and 3 or 2/3rds by 2 to equal 4 / 6 and the 4 and the 2 of 4/2
by 3 to equal 12/6 and we get (12/6 + 4/6) ^ 16 = (16/6) ^ 16. At
this point that might be considered irreducible without doing a
further calculation.


Scientific Notation

1 X 10 ^1 = 10
1 X 10 ^2 = 100
1 X 10 ^ 3 = 1000
1 X 10 ^ 4 = 10,000
1 X 10 ^ 5 = 100,000
1 X 10 ^ 6 = 1,000,000 etc.


Trans-Finite Numbers

1/ 0 = Infinity
2/ 0 = 2 * Infinity = Infinity
Infinity * Infinity = Infinity
Infinity / 2 = Infinity



For more advanced material, see
Calculus.



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