The document discusses the history and development of the Arabic numeral system. It explains that the numerals originated from the Phoenicians but were popularized by Arabs. It then provides a theory about how the shapes of the numerals may have been derived from representing different numbers of angles, showing examples. The document also briefly outlines some key developments in the history of algebra.

1. “… for a bit of review use the green buttons” INTEGERS

2. Main Menu Decimal (Standard) Form Mixed Number Exponential Form and Roots Fraction Scientific Notation Literal (written) Form Absolute Value Real Number Hierarchy Party in Mathland Parts of Operations Numerals Types of Whole Numbers Venn diagram Comparing Values Percent Conversion Number Properties

3. 013456… The numeral digits used for Numbers This seems to be the most likely theory but counting and writing numbers certainly developed earlier, if nothing more than scratching on a soft rock, bark, etc, 1 2 4 5 3

4. The numbers we write are made up of symbols, (1, 2, 3, 4, etc) called Arabic numerals, to distinguish them from the Roman numerals (I; II; III; IV; etc.). 013456… 1 2 4 5 3

5. 013456… The Arabs popularized these numerals, but their origin goes back to the Phoenician merchants that used them to count and do their commercial accounting. 1 2 4 5 3

6. 013456… Have you ever asked the question why 1 is “one”, 2 is “two”, 3 is “three”…..? 1 2 4 5 3

7. 013456… What is the logic that exists in the Arabic numerals? 1 2 4 5 3

8. 013456… There are angles! 1 2 4 5 3

9. 013456… Look at the decimal numerals written in their primitive form! 1 2 4 5 3

10. 013456… 1 angle 2 angles 3 angles 4 angles 1 2 4 5 3

11. 013456… 5 angles 6 angles 7 angles 8 angles 1 2 4 5 3

12. 013456… 9 angles 1 2 4 5 3

13. 013456… And the most interesting and intelligent of all….. 1 2 4 5 3

14. 013456… No (zero) angles ! This is a theory.. unless there is a few–thousand–year old mathematician. BUT it sounds reasonable. 1 2 4 5 3

15. Known History of Algebra The origins of algebra can be traced to the cultures of the ancient Egyptians and Babylonians who used an early type of algebra to solve linear, quadratic (variable to power of 2), and indeterminate (variable) equations more than 3,000 years ago. Around 300 BC Greek mathematician Euclid in Book 2 of his Elements addresses quadratic equations, although in a strictly geometrical fashion. Around 100 BC Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu (The Nine Chapters of Mathematical Art). Around 150 AD Greek mathematician Hero of Alexandria treats algebraic equations in three volumes of mathematics. Around 200 AD Greek mathematician Diophantus , often referred to as the "father of algebra", writes his famous Arithmetica , a work featuring solutions of algebraic equations and on the theory of numbers. The word algebra itself is derived from the name of the treatise first written by Persian mathematician Al-Khwarizmi in 820 AD titled: Kitab al-mukhtasar fi Hisab Al-Jabr wa-al-Moghabalah meaning ‘The book of summary concerning calculating by transposition and reduction’. The word al-jabr (from which algebra is derived) means "reunion", "connection”, or "completion". Algebra was introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci in 1202.

16. Venn Diagram A Venn diagram is a drawing, in which areas represent groups of items sharing common properties. The drawing consists of two or more shapes (usually circles or ellipses), each representing a specific group. This process of visualizing logical relationships was devised by John Venn (1834-1923). Set C Set C has some elements in both Set A and Set B All elements of Set D are in Set B What is the difference between Set C and Set D? What are the similarities between Set B and C? If elements of Set D are removed, what could this Venn represent? Set B Set A Set D

17. Types of Whole Numbers Primes to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 A Whole Number are positive integers ( 0 to ∞ ) A Prime Number has only 2 factors: “1” and itself. A Composite Number has 3 or more factors. “ 0” and “1” are not composite or prime numbers. WHOLE NUMBERS PRIME COMPOSITE 0 and 1

18. Real Numbers Venn diagram Venn diagram Real Numbers All Numbers (Rational and Irrational) Irrational Numbers PI (3.14….), Square root of a non-perfect square Any number that can be represented by a fraction: Integer Integer Rational Numbers Integers Positive and Negative numbers, and Zero; NO Decimals Whole Numbers Positive non-decimal numbers and Zero Natural (Counting) Numbers Positive non-decimal numbers ; NO Zero or Negative

19. Operation Parts Multiplication Addition Subtraction Division Also called: Multiplicand x Multiplier = Product Addition is the total of groups (sum) of the same and/or different size groups (addend). Subtraction is the amount left (difference) when a total of groups (minuend) is reduced by the same and/or different size groups (subtrahend). Multiplication is adding groups of a same size (multiplicand) so many groups (multiplier) to get the size of all groups (product), Division is subtracting the size of 1 group (divisor) from the total size of all groups (dividend) to get the number of groups (quotient) in the total (dividend). Addend + Addend = Sum 12 + 15 = 27; 2x + x = 3x Factor x Factor = Product 2 x 15 = 30; 3y x 2 = 6y Minuend – Subtrahend = Difference 37 – 15 = 22; 5t – 3t = 2t Dividend ÷ Divisor = Quotient 60 ÷ 15 = 4; 8x ÷ 4 = 2x

20. Operation Basics – Diagrams Division is subtracting groups of a same size ( divisor ) from the total size of all groups ( dividend ) to get the number of groups ( quotient ). Multiplication is adding groups of a same size ( multiplicand ) so many times, or groups ( multiplier ), to get the size of all groups ( product ). Addition is the total of groups ( sum ) of the same and/or different size groups ( addend ). Subtraction is the amount left ( difference ) when a total of groups ( minuend ) is reduced by the same and/or different size group ( subtrahend ). – = ● 3 = + = 3 ÷ = The multiplicand and multiplier can be switched, due to the commutative property, and both are typically referred to as factors .

21. Absolute Value The absolute value is the distance to 0. Absolute value can NEVER be negative ! Negative becomes positive … | –2 | = 2 ; | 2 – 3 | = 1 ; | –2 | 3 = 2 3 = 8 Positives remain positive … | 2 | = 2 ; | 3 – 2 | = 1 ; | 2 | 3 = 2 3 = 8 Examples: The symbol is | a |, where a is any value. -5 5 0 10 -10 7 7

22. Properties of Numbers Additive Identity a + 0 = a Multiplicative Identity a * 1 = a Additive Inverse a + (-a) = 0 Commutative of Addition a + b = b + a Multiplicative Inverse a * (1/a) = a / a = 1  (a ‡ 0) Commutative of Multiplication a * b = b * a Associative of Addition (a + b) + c = a + (b + c) Associative of Multiplication (a * b) * c = a * (b * c) Basis for solving equations and inequalities… isolates the variable by getting an identity number on one side Order of terms CHANGES Term Order does NOT CHANGE.. Grouping DOES = One group of 3 (a=3) = = = ( ) = ( ) + = nothing

23. Properties of Numbers –cont– Definition of Subtraction a - b = a + (-b) Distributive Property a(b + c) = ab + ac Definition of Division a / b = a(1/b) Zero Property of Multiplication a * 0 = 0 Adding a negative number is subtraction , so subtracting is adding a negative number Multiplying by a fraction is dividing by its denominator, so division is dividing by a common factor ex. No ( zero ) piles of 4 crates equals no ( zero ) piles of crates … where “ a ” is a common factor of “ b ” and “ c ” ex. 2( 3 – x ) = 6 – 2x ex. 4x + 2 = 4( x+ ½ ) ex. –x+2 = 2–x ex. x+(–2) = x–2 ex. 3 / 4 = 3● 1 / 4 ex. 3 / 4 = 3● 1 / 4 a - b = a + ( -b ) = a • a a b ( + ) = • + •

24. Fraction A fraction is division of 2 integers but used as one number. There are 2 types of fractions: Proper is < “1” so numerator is smaller than denominator Improper is ≥ “1” so numerator is greater than denominator Any integer can become an improper fraction with “1” as the denominator ex. –⅞, ⅔, ⅓ Ex. 8 / 7 , – 23 / 7 , 3 / 2 Ex. –8 = – 8 / 1 , 23 = 23 / 1 , 3 = 3 / 1 Repeating bar (ignore the “+” / “–” signs for this discussion) This is done because a fraction is more exact in value than a decimal 1 / 3 =0.33

25. Mixed Number The integer and proper fraction parts are added, so addition is implied … 3 ½ = 3 + ½ 10 + ¼ = 10 ¼ A mixed number is an improper fraction reduced to an integer and a proper fraction part, if needed. An improper fraction is in its lowest terms when it is reduced to an integer and its remaining proper fraction part is reduced. Integer part Fraction part

26. Decimal (Standard) Form All numbers have a decimal point. If there is NO decimal portion then the decimal point is implied after (to the right of) the last digit, and is not shown. A decimal (point) separates value greater or equal to “1” and that less than “1” in a number. In the number 12.3  “12” ≥ 1 and .3 < 1 30%= 30 . 0 % 23 = 23 . 0 –123 . 002 0 . 123 22 . 5 % All numbers have a decimal part (after decimal) and an integer part (before decimal) . If it needs to be shown it is followed by a zero(s). 23 = 23.0 = 23.00… This is called “padding” and does not change the value. (ignore the “+” / “–” signs for this discussion) decimal point

27. Exponential Form (Exponent) Exponential form is a short way of show multiplication of the same factor. It has 2 parts:  Base: the only factor to be multiplied  Exponent: the number of times the base is a factor The exponent identifies the number of times the base is used as a factor only!!! b e = 1 x b 1 x b 2 x b 3 x… b e = p where: b = the base which is any term (number) or Grouping symbols contents… this is the factor e = the exponent (power) which is the number of times to multiply the base by itself… this is not a factor p = the product of the exponential form “ p is the e th power of b ” 2 3 = 8 (3●4–1) 2 = 121 – 2 4 = –16 (–2) 4 = 16 4 -3 = ¼ ● ¼ ● ¼ = 3 / 4 “ b to the e th power equals p” A negative exponent means to use the reciprocal of the base as a factor

28. Exponential Form (examples) “ 1” (multiplicative identity) is always implied in multiplication 1; –1; 1 25 6.25 1 / 512 31,000 0.00031 3 8 8 – 8 – 4 – 8 4 2 exponent value calculation Find the value! 3(2+14.3 • 2÷x) 0 = 3(1) 3(2+14.3 • 2÷x) 0 3.10 x ( 1 / 10 x 1 / 10 x 1 / 10 x 1 / 10 ) 3.10 x 10 -4 3.10 x (10 x10 x10 x10) 3.10 x 10 4 (⅛)(⅛)(⅛) (⅛) 3 2.5 * 2.5 2.5 2 (3+1 • 2) 2 = (5) 2 (3+1 • 2) 2 1 ; -1 x 1; 1 2 0 ; -2 0 ; {2x+3 (12-2)} 0 1(2) 2 1 1x(-2) x (-2) (-2) 2 ; exponent is even 1x(-2) x (-2) x (-2) (-2) 3 ; exponent is odd -1(2 * 2) -2 2 ; exponent is even -1(2 * 2 * 2) -2 3 ; exponent is odd (2) x (2) x (2) (2) 3 2 * 2 * 2 2 3

29. Roots A root is the inverse operation of exponent form Exponential form : b e = 1 x b 1 x b 2 x b 3 x… b e = p where: “b” is the base , “e” is the exponent , and “p” is the product Root form : e p = b where: “b” is the base , “e” is the index , and “p” is the radicand If “e” (index) is not shown the root is assumed to be a square root (“e” = 2) Operations with roots and exponents

30. Scientific Notation  very large numbers (a lot of trailing zeroes before decimal) 1,220,000,000,000  very small numbers (a lot of leading zeroes after the decimal) 0.00000000023 (1) The unit digit is always 1-9; AND it is the only digit to the left of the decimal point in the decimal factor. This factor is always ≥ 1 and <10. (2) An explicit multiplication symbol is present. Usually “X”, but also “ • ”, “  ”. Scientific Notation is a short way to show: A value in Scientific Notation form has 3 distinct characteristics (3) The other factor is an exponent with a base of “10”. = 1.22 X 10 12 = 2.3 ● 10 -10 Positive exponent when value ≥ 1 Negative exponent when value < 1 Multiplication of a decimal (>1 and <10) and an exponent

31. Literary (Written) Form Used in speech, thought, and word problems, they must be converted to/from algebraic expressions, inequalities, and equations. Solving math word problems: Translate the wording into a numeric equation, then solve the equation! An expression in Math is like a phrase in Grammar… no subject and verb. A sentence in Math is like a sentence in Grammar. The verb typically includes: is will was equals equal calculate sum estimate subtract can times It is very important to understand the word use in the context of the problem… like determining the meaning of a word when context reading.

32. Literal (Written) Examples There are many others! ( ),{},[] = Results - Reduce = Will be - Diminished Quantity = Equal - Difference % Percent = Was - Subtract 1:4, ¼ 1 of 4 = Is - Less than 3–2 Difference between 3 and two ≈ about - Decreased by 2 x Product of 2 and x /, ÷ Quotient + Greater 2÷4 Quotient of two and 4 /, ÷ Divide + In excess < Less than or equal to /, ÷ Per + Increased < Less than *,•,x Interest on + More than > Greater than or equal to *,•,x Product + In addition > Greater than *,•,x Percent of + Add ≠ Not equal to *,•,x Times + Sum

33. A number increased by 5 n + 5 4 decreased by the quotient of a number and 7 4 - n / 7 7 less than a number n - 7 7 less a number 7 – n The product of ½ and a number is 36 ½ • n = 36 3 more than twice a number is 15 2n + 3 = 15 When you see the words: ‘ less than ’ vs. ‘less in subtraction… switch it around. Literal (Written) Examples Important!

34. The Party in Mathland A dd, S ubtract, M ultiply, and D ivide positive and negative values (integers).

35. Multiply and Divide Party Everyone is happy and having a good time (they are ALL POSITIVE). Suddenly, who should appear but the GROUCH (ONE NEGATIVE)! The grouch goes around complaining to everyone about the food, the music, the room temperature, the other people.... Everyone feels a lot less happy... the party may be “negatized”!! ODD NUMBER OF NEGATIVES MAKES EVERYTHING NEGATIVE I feel odd here.

36. Multiply and Divide Party continues Everyone feels a lot less happy... the party may be doomed! Everyone is so negative! ... is that another guest arriving? Yes, another grouch (A SECOND NEGATIVE) appears? The two negative grouches pair up and gripe and moan to each other about what a horrible party it is and how miserable they are!! But look!! They are starting to smile; they're beginning to have a good time, themselves…is that a POSITIVE attitude!! PAIRS OF NEGATIVES BECOME POSITIVE Now that the two grouches are together the rest of the people (who were really positive all along) become positive again. The party is positive!!

37. The moral of the story Negatives in PAIRS are POSITIVE: Negatives NOT in pairs, they're NEGATIVE: When multiplying or dividing the number of positives doesn't matter … but watch out for those negatives!! To determine whether the outcome will be positive or negative ,  count the number of negatives :  If there are an even number of negatives the answer will be positive  If not ( odd number of negatives )... It will be negative – , + , – , – , + , + , – equals + , – , – , + , + , – equals + – + + +

38. Addition with the Same Signs If the signs are the same; the answer will keep the same sign. – 4 + ( –2 ) = –6 4 + 2 = 6 + = + = Positives Negatives + –

39. Addition with different signs (alias Subtraction) – 32 + 11 32 – 11 Wait a second! ... This is subtraction! What about… ? = – 21 32 – 11 21 32 – 11 21 = + 21 32 + ( –11 ) If the signs are different ; then subtract the absolute value of the small value from the larger value. The sign of the larger value is the answer’s sign. Oh yeah! Subtracting is adding a negative, so adding a negative is subtraction.

40. Subtracting a Negative (adding a positive) – times – = + I’m leaving! I do feel much better… I’ll go back! 2 – ( –1 ) = This operation is based on two properties of multiplication: Multiplicative Identity Property A negative value times a negative value gives a positive value . – 1 = – (1) 1 – 1 ( –1 ) = 1 Ronco’s POSITIZER © Yes! You can change those negatives into a positive in 1 easy step!! That’s one less negative. Welcome back! 2 + 1 = 3 +

41. Scholarly Subtracting a Negative (adding a positive) Subtracting a negative is adding the subtrahend’s absolute value to the minuend Wait a second! ... This is addition! Oh yeah! Multiplying two negatives gives a positive product. 32 – (–11) – 32 – (–11) This is addition with different signs! Is the addition of 2 negatives subtracting a negative? No, when adding 2 negatives, like 2 positives, the sum’s sign is the same as the addends… –2+(–4) = –6 and 2+4=6. So, adding 2 negatives is adding 2 negatives. Subtracting a negative: –2–(–4) = –2+4 = 2 32 + 11 43 = 43 + 32 + 11 = –21 + – 32 + 11 32 – 11 21

42. Your Turn (reduce to decimal form or an expression/equation/inequality ) 23 2 – 9 103,000 0.4 – 8 2.25 – 8  4 – 12 – 72  12 – 4  v ÷ 3 =15  t > 30  ⅔ ● 12 |–23| = |4–2| = – 3 2 = 1.03 X 10 5 = 2 / 5 = -|–2| 3 = (–2) 3 = 12 less than 4 – 3 2 (2 3 ) = 12 less 4 Quotient of a value and 3 is 15. Total is greater than 6 groups of 5 . 2 ¼ = Two–thirds of a dozen

43. Your Turn (reduce to non–decimal form) – 14 0 – 6 1 3 2 40 – 32 – 1 1 / 2 – 24

44. Decimal / Percent Conversion Converting to a percent from a decimal is dividing by 100 (or multiplying by 1 / 100 ). Since decimals are based on 10, we can move the decimal 2 places for conversion… do NOT forget to add / remove the “%”. Move decimal 2 places to the left for conversion from percent to decimal and remove the “%” % Shortcut! Add the percent symbol Move decimal 2 places to the right for conversion from decimal to percent and add the “%” 3.24% = 0.0324 5 ½ % = 0.05 ½ = 0.055 .02% = 0.0002 3.24 = 324% 5 1 / 3 = 5.33 = 533 1 / 3 % .02 = 2%

45. Comparing Values Use this method when not finding a more obvious way. With this method Least–to–Greatest and Greatest–to–Least mistakes are easily remedied. 2) Convert all values to decimal 3) Pad with zeroes all values to the same decimal position. 4) Number the increasing/decreasing values starting with one. 1) Write each value in a different row. Ex. 2.3%, 3 / 25 , 2.31 x 10 2 , 2 1 / 3, , 0.233 greatest–to–least 2.3%, 3 / 25 2.31 x 10 2 2 1 / 3 0.233 2.3 0.12 231 0.233 000 00 0 1 2 .0000 5 4 3 Oh No! I wanted least–to–greatest! Oh Yeah! I can reverse the order. least–to–greatest 2.3333