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 take the value using a
regular exponent, and then add a minus sign.

For example,

2 ^ - 2 = - (2 ^ 2) =  - 4

2 ^ - 3 =  - (2 ^ 3)=  - 8

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

For example,

(2 ^ 3) (2 ^ -2) = 8 * - 4 = - 32


Fractions cancel with their integer and fractional opposites.

For example,

(2 ^ (1/2)) (2 ^ (-1/2)) = 1, because the 0.5s logically cancel out
and we are left with 2 ^ 0 * 2 ^0, which is just 1*1.

The multiplication of positive and negative exponents is an
exception where you can work across the parentheses.

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

In the case of pre-existing exponents in fractions, the advice is to
simplify them by 1. computing values, and 2. if possible,
extracting any identical exponents.

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 unless we can reduce the fraction it then might be
considered irreducible without doing a further calculation.

However, we can reduce the fraction to 8/3, so now we get
(8/3) ^16, which is less interesting if it is fully calculated.

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


See also:


Percentage to degrees

Knowing geometry

Factors of fractions (?)

Grams to Mols

For more advanced material, see Calculus.



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