SYSTEMSMATHEMATICSDecimal System1 . = one2 .. = two3 ... = three4 . . . . = four5 . . . . . = five6 . . . . . . = six7 . . . . . . . = seven8 . . . . . . . . = eight9 . . . . . . . . . = nine10 = 9 + 1 = ten20 = 10 + 10 = twenty30 = 20 + 10 = thirty40 = 20 + 20 = forty50 = 30 + 20 = fifty60 = 40 + 20 = sixty70 = 50 + 20 = seventy80 = 40 + 40 = eighty90 = 50 + 40 = ninety100 = 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...9These are small, infinitely repeating numbers.Rational numbers.These are numbers with values expressible in fractionsand 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.Equations1 + 1 = 22 + 3 = 52 * 10 = 201 / 10 = 0.11/100 = 0.011/1000 = 0.00110 / 20 = 1/2 = 0.510X = Y = Y is exactly 10 * XThat is the same as writing 10X - Y = 0.Squares and Square Roots0 ^ 1 = 0 * 1 = 01 ^ 0 = 1 * 1 = 12 ^ 0 =  1 * 1 * 1 = 1 etc.1 ^ 2 = 1 * 1 = 11 ^ 3 = 1 * 1 * 1 = 12 ^ 2 = 2 * 2 = 42 ^ 3 = 2 * 2 * 2 = 81 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              8Multiplying with ExponentsAdd exponents that are multiplied:(2^2) (2^2) = 2 ^ 4 = 2 * 2 * 2 * 2 = 16And 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 = 1/42 ^ - 3 = 1 / 2 ^ 3 = 1/8Note 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 / 16And 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) ^ 16Now 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 Notation1 X 10 ^1 = 101 X 10 ^2 = 1001 X 10 ^ 3 = 10001 X 10 ^ 4 = 10,0001 X 10 ^ 5 = 100,0001 X 10 ^ 6 = 1,000,000 etc.Trans-Finite Numbers1/ 0 = Infinity2/ 0 = 2 * Infinity = InfinityInfinity * Infinity = InfinityInfinity / 2 = InfinityFor more advanced material, see Calculus.BACK TO SYSTEMS