Intro Num Int Asmd
description
Transcript of Intro Num Int Asmd
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“…“…for a bit of review use the green buttons”for a bit of review use the green buttons”
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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
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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,
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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.).
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The Arabs popularized these numerals, but their origin goes back to the Phoenician merchants that used them to count and do their commercial accounting.
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Have you ever asked the question why 1 is “one”, 2 is “two”, 3 is “three”…..?
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What is the logic that exists in the Arabic numerals? 1245555
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There are angles! 1245555
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Look at the decimal numerals written in their primitive form! 1245555
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1 angle 2 angles
3 angles 4 angles
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5 angles 6 angles
7 angles8 angles
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And the most interesting and intelligent of all…..
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No (zero) angles !
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This is a theory.. unless there is a few–thousand–year old mathematician.BUT it sounds reasonable.
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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.
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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?
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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.
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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
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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).
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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.
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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
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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
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= 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)
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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)
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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
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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)
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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
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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!
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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
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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
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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.
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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!
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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
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The Party in Mathland
Add, Subtract, Multiply, and Divide positive and negative values (integers).
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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.
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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!!
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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
+ +
+
+
–
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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
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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…
?
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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!
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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
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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
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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
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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%
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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