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Euclid's Elements WP -
Euclid's Elements Book 1-13 online

Complex number WP -
Imaginary number WP - i^{2} = −1 -
Leonhard Euler and
Carl Friedrich Gauss

Greek alphabet WP - 24 letters from alpha to omega, originally only one letter case, first script with vowels and consonants

Anthony Weaver Department of Mathematics and Computer Science - Bronx Community College

Math is based on + --- y - x = y + (-x) --- x * 3 = x + x + x --- x^{3} = x * x * x --- - reverses + --- : reverses * --- \(\sqrt[n]{x^n} = x\) reverses \(x^n\)

Contents

Ch 1 Whole numbers --- + - * ^{n} : 0 --- order of operations - average - perimeter - area and the Pythagorean theorem - 9

Ch 2 Signed numbers - |-x| include 0 and negative + positive whole numbers = integers - absolute value --- + - * : ^{n} √ - 49

Ch 3 Fractions and mixed numbers - converting - cancelling, precancelling - lowest terms - prime factorization - GCF LCM LCD - reciprocal - 71

Ch 4 - Decimals and Percents - rounding-off - power of 10 - order of magnitude - scientific notation - conversions - 119

Ch 5 - Ratio and Proportion - cross-product property - rates - similar triangles - 149

Ch 6 - Towards algebra - evaluationg expressions - using formulae - functions - linear terms and equations - p169-185 end of book

Ch 1 - Whole numbers

p9 - \(\mathbb{N} \) natural numbers 1, 2, 3, ... - [ \(\mathbb{R} \) { \(\mathbb{Q} \) ( \(\mathbb{Z} \) [ \(\mathbb{N}\) ] ) } ] - real numbers \(\mathbb{R}\) incluce rational numbers \(\mathbb{Q}\) (quotient) (and irrational numbers \(\mathbb{R}\)\\(\mathbb{Q}\) or \(\mathbb{P}\), meaning \(\mathbb{R}\) without \(\mathbb{Q}\), like \(\sqrt{2}\) etc. - an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction), which include integers (Latin meaning "whole") \(\mathbb{Z}\) (German "Zahlen"), which incluce natural \(\mathbb{N}\) - whole numbers incluce 0

place-value system: digit in right-most place for ones, second-from-right tens etc. - example 1: 6040 stands for 6 thousands + 0 hundreds + 4 tens + 0 ones

p10 - property a + b = b + a (commutativity)

p11 - property a + 0 = a (additive identity)

property (x + y) + z = x + (y + z) (associativity)

1.1.2 - Multi-digit addition

line up numbers vertically - ones places (right-most) directly on top of each other - draw line - add digits in each place to obtain sum

p12 - sometimes have to carry a digit to the next higher place - carry on top of next higher column - add including the carried digit etc. - always start with the right-most column and proceed leftward

1.2 - Subtracting Whole Numbers

p14 - Other way to say 5 + 2 = 7 is 5 = 7 − 2, or “5 is the difference of 7 and 2,” or “5 is the result of taking away 2 from 7” - the operation of taking away one number from another, or finding their difference, is called subtraction -

p15 - not commutative (see later when negative numbers are introduced --- 7 - 4 ≠ 4 - 7), not associative --- (7−4)−2 ≠ 7−(4−2) --- 1 ≠ 5 --- x - 0 = x --- 0 is not an subtractive identity, because 0 - x ≠ x (0 - x ≠ 0 ?) if x ≠ 0 --- more about it when negative numbers introduced - Commutative property WP

1.2.2 Multi-digit subtractions

need to distinguish between subtrahend (number doing the diminishing or being taken away) and minuend (number being diminished) - For now subtrahend is no larger than the minuend - line up vertically like in addition - draw line - subtract digits in each column from right to left

1.2.3 - Checking substractions

p16 - Subtraction is opposite of addition - can be restated in terms of addition - if subtraction has been performed correctly, adding difference to subtrahend returns minuend

1.2.4 Borrowing

(example: 85 - 46) --- borrow from the left higher column and set that column -1 --- then check difference + subtrahend = original minuend

p17 - Sometimes we have to go more than one place to the left to borrow from (example: 207 - 69 --- write on top of 207 {1}{9}{17}) --- we see in checking that carrying reverses borrowing :-)

1.3 - Multiplying Whole Numbers

p20 - multiplication is repeated addition --- 4 * 3 = 12 means: start at 0 and add 3 four times over:
0 + 3 + 3 + 3 + 3 = 12 --- result is product

p21 - property of 0: 0 · N = N · 0 = 0 --- property of 1 (multiplicative identity): 1 · N = N · 1 = N --- a * b = b * a (commutative) --- (a * b) * c = a * (b * c) (associative) ---

1.3.2 - Multi-digit multiplications

distinguish multiplicand (the number “being multiplied”) from multiplier (number which is “doing the multiplying") - interchangeable because commutative - saves space if multiplier is number with fewest digits - set numbers up vertically, multiplicand over mutiplier, as in addition (example: 232 * 3)

Sometimes need to carry a digit to next higher place, as in addition (example: 251 * 4)

If multiplier has more than one digit: partial products, one for each digit of the multiplier, then to be added to get total product (example: 24 * 32) - second partial product in tens column, therefore put down 0 in ones place - always first multiply, then, if existing, add carried number etc., then add partial products and get product -

1.4 - Powers of Whole Numbers

start with 1, repeatedly multiply by 3, 4 times over, get 4th power of 3, written 3^{4} = 1 × 3 × 3 × 3 × 3 --- factor of 1 is understood and usually omitted - simply write 3^{4} = 3 × 3 × 3 × 3 --- in expression 3^{4}, 3 is called base, and 4 exponent or power

p27 - definition if exponent is 0: For any nonzero number N, N^{0} =1. (0^{0} is undefined.) - 0^{1} or 0^{2} etc. is 0 or 0 * 0 etc.
1.4.1 - Squares and Cubes

2nd power called square, 3rd cube - 5^{2} read 5 squared, 7^{3} 7 cubed - area of square, x units on a side, is x^{2} square units - and similarly for x^{3} cubic units

x^{0} = 1 for x ≠ 0 --- 0^{0} is undefined --- 0^{x} = 0 for x ≠ 0

exercises

1.4.3 - Square Roots

p28 - ...

...

Ch 2 - ...

...

p53 - now extend ordinary operations of + - * : to all signed numbers ...

2.3 - Adding signed numbers

extended addition operation in terms of motion along the number line:

adding positive number -> move to right --- adding negative number -> move to left

Adding two signed numbers is commutative: -3 + (-4) = -4 + (-3) = 7 --- generally for adding andy two signed numbers: a + b = b + a

p55 - ...

...

2.4 - Subtracting signed numbers

... multiplying ... dividing ... powers ... square roots and signed numbers

Ch 3 - Fractions and Mixed Numbers

p71 - ...

Symbols - Mac keyboard German

√ = alt v --- ∫ = alt b --- ~ = alt n --- º = alt j --- ∆ = alt K --- – = alt - --- ∞ = alt , --- ≈ = alt x --- ª = alt h --- ƒ = alt f --- ∂ = alt d --- ± = alt + --- π = alt p --- ø = alt o --- Ω = alt z --- ∑ = alt w --- « = alt q --- ¿ = alt ? --- ≠ = alt = --- ¡ = alt 1 ---

Sessions

• Session 1 • Mon 2017-2-20 21:05-3:30 Ch 1 p1-20 ♡♡♡

• Session 2 • Tue 2017-2-21 15:50-17:30 Ch 1 p20- ♡♡♡

Bill Evans - Alone (1968 Album)