intro num int asmd

1Intro_num_int_ASMD.ppt

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

2Intro_num_int_ASMD.ppt

Main Menu

Decimal (Standard) Form

Mixed Number

Exponential Form and Roots

Fraction

Scientific NotationLiteral (written) Form

Absolute Value

Real Number Hierarchy

Party in MathlandParty in Mathland

Parts of OperationsNumerals

Types of Whole Numbers

Venn diagramComparing Values

Percent Conversion

Number Properties

3Intro_num_int_ASMD.ppt

013456…

The numeral The numeral digits used for digits used for

NumbersNumbersThis 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,

12455553

4Intro_num_int_ASMD.ppt

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…

12455553

5Intro_num_int_ASMD.ppt

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.

12455553

6Intro_num_int_ASMD.ppt

013456…

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

12455553

7Intro_num_int_ASMD.ppt

013456…

What is the logic that exists in the Arabic numerals? 1245555

3

8Intro_num_int_ASMD.ppt

013456…

There are angles! 1245555

3

9Intro_num_int_ASMD.ppt

013456…

Look at the decimal numerals written in their primitive form! 1245555

3

10Intro_num_int_ASMD.ppt

013456…

1 angle 2 angles

3 angles 4 angles

12455553

11Intro_num_int_ASMD.ppt

013456…

5 angles 6 angles

7 angles8 angles

12455553

12Intro_num_int_ASMD.ppt

013456…9 angles

12455553

13Intro_num_int_ASMD.ppt

013456…

And the most interesting and intelligent of all…..

12455553

14Intro_num_int_ASMD.ppt

013456…

No (zero) angles !

12455553

This is a theory.. unless there is a few–thousand–year old mathematician.BUT it sounds reasonable.

15Intro_num_int_ASMD.ppt

Known History of AlgebraThe 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.

16Intro_num_int_ASMD.ppt

Set B

Set A

Venn DiagramA 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 DSet DSet 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?

17Intro_num_int_ASMD.ppt

WHOLE NUMBERS

Types of Whole Numbers

PRIME

COMPOSITE

0 and 1

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.

18Intro_num_int_ASMD.ppt

Real Numbers

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:IntegerInteger

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

19Intro_num_int_ASMD.ppt

Operation Parts

MultiplicationAddition

Subtraction

Division

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 = 2tDividend ÷ Divisor = Quotient60 ÷ 15 = 4; 8x ÷ 4 = 2x

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).

20Intro_num_int_ASMD.ppt

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.

21Intro_num_int_ASMD.ppt

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 = 23 = 8

Positives remain positive…

|2| = 2; |3–2| = 1; |2|3 = 23 = 8

Examples:

The symbol is |a|, where a is any value.

-5 50 10-10

77

22Intro_num_int_ASMD.ppt

Properties of NumbersAdditive Identity

a + 0 = a Multiplicative Identity a * 1 = a

Additive Inverse Additive Inverse a + (-a) = 0 a + (-a) = 0

Commutative of Addition a + b = b + a

Multiplicative Inverse Multiplicative Inverse a * (1/a) = a * (1/a) = aa//aa = 1   = 1     (a   (a ‡‡  0)  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)

=One group of 3 (a=3)

Basis for solving equations and

inequalities… isolates the variable by

getting an identity number on one

side

= ==

( ) = ( )

+ =

noth

ing

Order of terms CHANGES

Term Order does NOT CHANGE.. Grouping DOES

23Intro_num_int_ASMD.ppt

= a •a ab

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 Multiplicationa * 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 (zerozero) piles of 4 crates equals no (zerozero) piles of crates

…where “a” is a common factor of “b” and “c”

ex. 2( 3 – x ) = 6 – 2xex. 4x + 2 = 4( x+ ½ )

ex. –x+2 = 2–xex. x+(–2) = x–2

ex. 3/4 = 3●1/4ex. 3/4 = 3●1/4

)( 441

4

4

x

x ex.

( + ) = • + •

a - b = a + (-b)

24Intro_num_int_ASMD.ppt

FractionA fraction is division of 2 integers but used as one number.

There are 2 types of fractions:There are 2 types of fractions:

Proper is < “1” so numerator is smaller than

denominator

This is done because a fraction is more exact in value than a decimal 1/3 =0.33

Improper is ≥ “1” so numerator is greater than denominator

Any integer can become an improper fraction with “1” as the denominator

ex. –⅞, ⅔, ⅓ex. –⅞, ⅔, ⅓ Ex. 8Ex. 8//77, –, –2323//77, , 33//22

Ex. –8 =Ex. –8 = ––88//11, 23 = , 23 = 2323//11, 3 = , 3 = 33//11

Repeating bar

(ignore the “+” / “–” signs for this discussion)(ignore the “+” / “–” signs for this discussion)

25Intro_num_int_ASMD.ppt

Mixed Number

4

33

4

15

4

33

2

13

8

43

8

28

The integer and proper fraction parts are added, so addition is implied …

3½ = 3 + ½ 10 + ¼ = 10 ¼

A mixed number is an improper fraction reduced to an integer and a proper fraction part, if needed.

Integer part

Fraction part

An improper fraction is in its lowest terms when it is reduced to an integer and its remaining proper fraction part is reduced.

31

3

26Intro_num_int_ASMD.ppt

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%= 3030.0 % 23 = 2323.0 –123–123.002 00.123 2222.5 %

All numbers have a decimal part (after decimal) and an integer part (before decimal)integer part (before decimal).

decimal point

If it needs to be shown it is followed by a zero(s). 23 = 23.0 = 23.00…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)(ignore the “+” / “–” signs for this discussion)

27Intro_num_int_ASMD.ppt

Exponential Form (Exponent)

Exponential form is a short way of show multiplication of the same factor.

It has 2 parts: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 b1 x b2 x b3 x… be = 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

factorp = the product of the exponential

form

“p is the eth power of b”

23 = 8 (3●4–1)2 = 121 –24 = –16 (–2)4 = 16

4-3 = ¼ ● ¼ ● ¼ = 3/4

“b to the eth power equals p”

A negative exponent means to use the

reciprocal of the base as a factor

28Intro_num_int_ASMD.ppt

Exponential Form (examples)

“1” (multiplicative identity) is always implied in multiplication

23 2*2*2

(2)3 (2) x (2) x (2)

-23 ; exponent is odd -1(2*2*2)

-22 ; exponent is even -1(2*2)

(-2)3 ; exponent is odd 1x(-2) x (-2) x (-2)

(-2)2 ; exponent is even 1x(-2) x (-2)

21 1(2)

20 ; -20 ; {2x+3 (12-2)}0 1 ; -1 x 1; 1

(3+1•2)2 (3+1•2)2 = (5)2

2.52 2.5 * 2.5

(⅛)3 (⅛)(⅛)(⅛)

3.10 x 104 3.10 x (10 x10 x10 x10)

3.10 x 10-4 3.10 x (1/10 x 1/10 x 1/10 x 1/10)

3(2+14.3•2÷x)0 3(2+14.3•2÷x)0 = 3(1)

1; –1; 1

256.251/512

31,000

0.00031

3

8

8–8

–4–842

exponent valuecalculation

Find the value!

29Intro_num_int_ASMD.ppt

Roots

n

nn

b

a

b

a 3

13 66

3

22

3

13 2 xxx

A root is the inverse operation of exponent formExponential form: b e = 1 x b1 x b2 x b3 x… be = p where: “b” is the base, “e” is the exponent, and “p” is the product

Root form: e p = bwhere: “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)

nnn baab 3

2

3

123 2 xxx

Operations with roots and exponents

30Intro_num_int_ASMD.ppt

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

Scientific Notation

(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:Scientific Notation is a short way to show:

A value in Scientific Notation form has 3 distinct characteristicsA value in Scientific Notation form has 3 distinct characteristics

(3) The other factor is an exponent with a base of “10”.

= 1.22= 1.22 XX 10101212

= 2.3= 2.3 ●● 1010-10-10

Positive exponent when

value ≥ 1

Negative exponent when

value < 1

Multiplication of a decimal (>1 and

<10) and an exponent

31Intro_num_int_ASMD.ppt

Literary (Written) Form

Used in speech, thought, and word problems, they must be Used in speech, thought, and word problems, they must be converted to/from algebraic expressions, inequalities, and converted to/from algebraic expressions, inequalities, and equations.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 A sentencesentence in Math is like a in Math is like a sentencesentence in Grammar. The verb typically includes: in Grammar. The verb typically includes:

iswill wa

sequals

equal calculate

sum estimatesubtract

cantimes

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.

32Intro_num_int_ASMD.ppt

Literal (Written) Examples

Sum + Times *,•,x Not equal to ≠

Add + Percent of *,•,x Greater than >

In addition + Product *,•,x Greater than or equal to

>

More than + Interest on *,•,x Less than <

Increased + Per /, ÷ Less than or equal to

<

In excess + Divide /, ÷ Quotient of two and 4

2÷4

Greater + Quotient /, ÷ Product of 2 and x 2x

Decreased by - about ≈ Difference between 3 and two

3–2

Less than - Is = 1 of 4 1:4, ¼

Subtract - Was = Percent %

Difference - Equal = Quantity ( ),{},[]

Diminished - Will be =  

Reduce - Results =  

There are many others!

33Intro_num_int_ASMD.ppt

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

34Intro_num_int_ASMD.ppt

The Party in Mathland

Add, Subtract, Multiply, and Divide positive and negative values (integers).

35Intro_num_int_ASMD.ppt

Multiply and Divide Party

Everyone is happy and having a good time Everyone is happy and having a good time (they are ALL POSITIVE). (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.

36Intro_num_int_ASMD.ppt

Multiply and Divide Party Multiply and Divide Party continuescontinues

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!!

37Intro_num_int_ASMD.ppt

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 negativescount 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

+ +

+

+

–

38Intro_num_int_ASMD.ppt

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

39Intro_num_int_ASMD.ppt

Addition with different signs (alias Subtraction)

–32 + 11 = –21

32– 11 21

32– 11 21

= +21

32 – 11

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.

Wait a second! ... This is subtraction!

Oh yeah!

Subtracting is adding a negative, so adding a negative is subtraction.

Wha

t abo

ut…

?

40Intro_num_int_ASMD.ppt

Subtracting a Negative (adding a positive)

– ti

mes

– =

+

I’m leaving!

I do feel much better…I’ll go back!

2 – (–1) =

2 + 1 = 3

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!

41Intro_num_int_ASMD.ppt

Subtracting a negative is adding the subtrahend’s absolute value to the minuend

Scholarly Subtracting a Negative (adding a positive)

Wait a second! ... This is addition!

Oh yeah! Multiplying two negatives gives a positive product.

32+ 11 43

= 4332 – (–11)

+

32 + 11

= –21–32 – (–11)

+

–32 + 11 32– 11 21

This is addition with different signs!

Is the addition of 2 negatives Is the addition of 2 negatives subtracting a negative?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

42Intro_num_int_ASMD.ppt

Your Turn Your Turn (reduce to decimal form or (reduce to decimal form or an

expression/equation/inequality))

|–23| =

|4–2| =

–32 =

1.03 X 105 =

2/5 =

-|–2|3 =

(–2)3 =

12 less than 4

–32(23) =

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

23

2

–9

103,000

0.4

–8

2.25

–8

4 – 12

–72

12 – 4

v ÷ 3 =15

t > 30

⅔ ● 12

43Intro_num_int_ASMD.ppt

Your Turn Your Turn (reduce to non–decimal (reduce to non–decimal form)form)

912

3

912

37

14

10122

252 )(

1212

8244

33

2

173420

65

2

5)(

)(

–14

0

–6

1

3

2

40

–32

–1 1/2

–24

44Intro_num_int_ASMD.ppt

Decimal / Percent Conversion

Converting to a percent from a decimal is dividing by Converting to a percent from a decimal is dividing by 100 (or multiplying by 100 (or multiplying by 11//100100).).

100

1

1

23

100

2323 %

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 Move decimal 2 places to the leftleft for conversion from for conversion from percent percent to decimalto decimal and and removeremove the “%” the “%”

2301023010001100

100

1..

.

. so %

Add the percent symbol

Move decimal 2 places to the Move decimal 2 places to the rightright for conversion from for conversion from decimal to decimal to percentpercent and and addadd the “%” the “%”

3.24% = 0.03243.24% = 0.0324 5 ½ % = 0.05 ½ = 0.055 .02% 5 ½ % = 0.05 ½ = 0.055 .02% = 0.0002= 0.0002

3.24 = 324%3.24 = 324% 5 5 11//33 = 5.33 = 533 = 5.33 = 533 11//33%% .02 .02 = 2%= 2%

45Intro_num_int_ASMD.ppt

Use this method when not finding a more obvious way.

With this method Least–to–Greatest and Greatest–to–Least mistakes are easily remedied.

Comparing Values

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 102 , 2 1/3,, 0.233 greatest–to–least

2.3%,3/25

2.31 x 102

2 1/3

0.233

2.3

0.12

231

0.233

2.3333

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