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85 Cards in this Set
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Natural or counting numbers

1,2,3,4


Whole numbers

0,1,2,3


odd numbers

Whole numbers not divisible by 2: 1,3,5,7


Even numbers

Whole numbers divisible by 2: 0, 2, 3, 6


negative integers

3,2,1


positive integers

the natural numbers 1,2,3,4


rational numbers

fractions, such as 3/5 or 7/8. All intergers ar rational numbers; for ex. the number 5 may be written as 5/1. All rational numbers can be written as fractions a/b, with a being an integer and b being a natural number. Both terminating decimals (such as .5) and repeating decimals (such as .333) are also rational numbers because they can be written as fractions in this form


irrational numbers

numbers that cannot be written as fractions a/b, with a being an integer and b being a natural number. √3 and (pi) are examples of irrational numbers


Axioms of addition
closure 
Closure is when all answers fall into the original set. If you add two even numbers, the answer is still an even number (2 + 4 =6); therefore, the set of even numbers is closed under addition (has closure). If you add two odd numbers, the answer is not an odd numver (3+5=8); therefore, the set of odd numbers is not closed under addtion (no closure).


Axioms of addition
commutative 
means that the order does not made any difference
2+1=1+2 a+b=b+a Note: commutative does not hold for subtraction 32 ≓ 23 

Axioms of addition
Associative 
means that the grouping does not make any difference
(2+3) +4 = 2 + (3+4) (a+b) +c = a +(b+c) note: associative does not hold for subtraction 4(21) ≓ (42) 1 

Identity element of addition

0
5 +0=0 

The additive inverse

is the opposite (negative) of the number. Any number plus its additive inverse equals 0 (the identity)


Axioms of multiplication
Closure 
When all answers fall into the original set. If you multiply two even numbers, the answer is still an even number (2 x 4=8); therefore, the set of even numbers is closed under multiplication (has closure). If you multiply two odd numbers, the answer is an odd number (3 x 5 = 15); therefore, the set of odd numbers is closed under multiplication (has closure)


Axioms of multiplication
Commutative 
Means that the order does not make any difference
4x3=3x4 axb=bxa Note: commutative does not hold for division 12 / 4 ≠ 4 / 12 

Axioms of multiplication
Associative 
Means that grouping does not make any difference
(2x3)x4 = 2 x (3x4) (axb)xc = a x (bxc) the grouping has changed (parantheses moved), but the sides are still equal Note: Associative does not hold for division (8 / 4) / 2 ≠ 8 / (4/2) 

Identity element for multiplication

1
Any number multiplied by 1 gives the original number 5x1=1 ax1=a 

multiplicative inverse

is the reciprocal of the number. Any number multiplied by its reciprocal equals 1
2 x 1/2 =1 a x 1/a =1 

A property of two operations
distributive property 
Is the process of distributing the number on the outside of the parenthesis to each term on the inside
2(3 +4) = 2(3) + 2(4) Note: you cannot use the distributive property w/ only one operation 3(4 x 5 x 6) ≠ 3(4) x 3(5) x 3(6) 

Grouping symbols and order of operation
what order do you put in parenthesis ( ) brackets [] braces {} 
parenthesis first
then brackets then braces 

what is the order of operation?

1. parenthesis
2. Exponents 3. multiplication 4. division (which ever comes first left to right) 5.addition 6.subtraction (which ever comes first left to right) Please excuse my dear aunt sally 

place value system

our number system
each place is assigned a different value. For instance 675 6 hundreds 7 tens 5 ones number system is based on powers of 10 10^0 =1 10^1 =10 10^2 =100 

Expanded notation

523 in expanded notation
523 = 500 +20 + 3 = (5x100) + (2x10) + (3x1) =(5x10^2) + (2x10^1) + (3 x 10 ^0) These last two are the more common forms of expanded notation. one w/ exponents and one w/o 

rounding off

If the number to the right of the number your rounding is 5 or higher round up.


Estimating sums,differences,quotients, products

use rounded numbers to estimate sums
3762 becomes 4000 5021 becomes 5000 about 9000 If both mulitpliers end in 50 or are halfway numbers, then rounding 1 number up and one number down will give you the best estimate 

divisibility rules



factors

numbers that are multiplied together to get a product
ex factor x factor =18 1 x 18 = 18 2 x 9 =18 3 x 6 = 18 Factors of 18 are 1,2,3,6,9, and 18 these numbers are also called Divisors of 18. Factors of a number are also called divisors of that same number. 

Prime number

A number that can be divided by only itself and 1.
Another definition: A prime number is a positive number that has exactly two different factors: itself and 1. Only even number that is prime is 2 0 and 1 are not prime numbers 

Composite numbers

is a number divisible by more than just 1 and itself.
Another definition: A composite number is a postive number that has more than two different factors. Numbers 0 and 1 are not composite numbers (they are neither prime or composite) 

factor tree



Decimal system

The system of numbers we use and is based on powers of ten (BASE TEN SYSTEM)
Every number to the right of the decimal point is called a DECIMAL FRACTION 

Expanded notation
Decimals 
.365
.3 + .06 + .005 (3 x.1) + (6 x .01) + (5 x .001)(3 x 10 ^ 1) + (6 x 10^ 2) +(5 x 10^ 3) 

writing decimals
reading decimals 
to read a decimal or write a decimal in words, you start at the left and end w/ the place value of the last number on the right. Where a whole number is included, use the word "and" to show the position of the decimal point.
.75 would read seventy five hundreths 45.23 would read 45 and 23 hundreths 

comparing decimals
which is greater .37 or .365 
.37


rounding decimals

If the number to the right is 5 or higher round up.


adding and subtracting decimals

To add or subtract decimals, just line up the decimal points and then add or subtract in the same manner you would add or subtract whole numbers


Multiplying decimals

just multiply as usual. Then count the total number of digits above the line which are to the right of all decimal points. Place your decimal point in your answer so there is the same number of digits to the right of it as there was above the line.


Dividing decimals

Same thing as dividing other numbers, execpt that if the divisor (the number you're dividing by) has a decimal, move it to the right as many places as necessary until it is a whole number. Then move the decimal point in the divident ( the number being divided into) the same number or places. Sometimes, you may have to add zeros to the dividend (the number inside the division sign)


terminating decimals

decimals that stop


repeating decimal

is a decimal that continues on indefinitely and reapeats a number or block of numbers in a consistent manner such as .666 or .232323


Vinculum

A horizontal line over the numer or numbers is the standard notation used to show that a number or group of numbers is repeating.


common math symbols



fractional number

used to represent a part of a whole. fractions consist of two numbers; a numerator (which is above the line) and a denominatior (which is below the line).
The denominator tells you the number of equal parts into which something is divided. The numerator tells you how many of these equal parts are being considered 

proper fraction

a fraction where the numerator is smaller that the denominator. <1
ex. 3/4 <1 

improper fraction

numerator is greater than denominator
>1 ex. 12/7 

mixed number

when a term contains both a whole number and a fraction


changing improper fractions

to change an improper fraction to a mixed number, you divide the denominator into the numerator
10/3 = 3 1/3 (remainder becomes numerator) 

changing mixed numbers

to change a mixed number to an improper fraction, you multiply the denominator times the whole number, add in the numerator, and put the total over the original denominator
5 3/4 = 23/4 

equivalent fractions



reducing fractions

when given as a final answer, a fraction should be reduced to lowest terms.
done by dividing both the numerator and denominator by the largest number that will divide evenly into both 

enlarging denominators

The denominator of a fraction may be enlarged by multiplying both the numerator and denominator by the same number


common factors

those factors that are the same for two or more numbers
common factors of 6 and 8 are 1 and 2 

greatest common factor

the greatest common factor (GCF) is the largest factor common to two or more numbers
GCF of 12 and 30 is 6 

multiples

multiples of a number are found by multiplying that number by 1, 2, 3, 4 etc.
fist 3 multiples of 9 are 9, 18, 27 

common multiples

are multiples that are the same for two or more numbers
common multiples of 2 and 3 are 6, 12,18 

Least common multiple

the least common multiple (LCM) is the smallest multiple that is common to two or more numbers
LCM of 2 and 3 is 6 

Add fractions

To add fractions, you must have a common denominator. To add like fractions, simply add the numerators and keep the same denominator ( reduce)
To add unlike fractions, first change all denominators to their lowest common denominator,(lowest common multiple of the denominator), 

Like fractions

Fractions that have a common denominator


Unlike fractions

Fractions that have different denominators


Subtracting fractions

to subtract fractions, the same rule as in adding fractions applies (find the LCD), except that you subtract the numerators.


Add mixed numbers

the same rule as in adding fractions applies (find the LCD), but make sure that you always add the whole numbers to get your final answer.
ex. 2 1/2 + 3 1/4 = 5 3/4 Sometimes, you may end up w/ a mixed number that includes an improper fraction. In that case, you must change the improper fraction to a mixed number and combine it with the sum of the integers. 

subtracting mixed numbers

When you subtract mixed numbers, you sometimes may have to "borrow" from the whole number, just as you sometimes borrow from the next column when subtracting whole numbers.
Note: When you borrow 1 from the whole number, the 1 must be changed to a fraction 

Multiply fractions

simply multiply the numerators; then multiply the denominators. (reduce)
You can cross cancel as well 

multiply mixed numbers

first change any mixed number to an improper fraction. Then multiply the numerators together and the denominators together.


Multiply fractions

simply multiply the numerators; then multiply the denominators. (reduce)
You can cross cancel as well 

multiply mixed numbers

first change any mixed number to an improper fraction. Then multiply the numerators together and the denominators together.


Multiply fractions

simply multiply the numerators; then multiply the denominators. (reduce)
You can cross cancel as well 

multiply mixed numbers

first change any mixed number to an improper fraction. Then multiply the numerators together and the denominators together.


dividing fractions

to divide fractions, invert (turn upside down) the second fraction (the one "divded by) and multiply. reduce
1/6 ÷ 1/5 = 1/6 x 5/1 

dividing complex fractions



dividing mixed numbers



simplifying fractions and complex fractions

If either numerator or denominator consists of several numbers, these numbers must be combined into one numbers.


changing fractions to decimals

fractions may also be written in decimal form (decimal fractions) as either terminating ( for example, .3) or infinite repeating (for example, .66) decimals. To change a fraction to a decimal, simply do what the operation says. In other words, 13/20 means 13 divided by 20. Insert decimal points and zeros accordingly.


Changing terminating decimals to fractions

To change terminating decimals to fractions, simply remember that all numbers to the right of the decimal point are fractions with denominators of only 10, 100, 1000 and so on. Next, use the technique of read it write it , reduce it
ex. .8 would be 8/10 

Changing infinte repeating decimals to fractions

.54444
n=.54 10n=5.44444 100n=54.4444 100n 10n=90n 

change decimal to percent

1. Move the decimal point two places to the right.
2. Insert a percent sign. 

Change percent to decimal

1. Eliminate the percent sign.
2. Move the decimal point tow places to the left. (Sometimes, adding zeros is necessary.) 

Change Fraction to percent

1. Change to a decimal.
2. Change the decimal to a percent 

Changing percents to fractions.

1. Drop the percent sign.
2. Write over one hundred 3. Reduce if necessary 

Finding percent of a number

To determine percent of a number, change the percent to a fraction or decimal (whichever is easier for you) and multiply. Remember: the word of means multiply


Finding what percent one number is of another

One method to find what percent one number is of another is the division method. To use this method, simply take the number after the OF and divide it into the number next to the IS. Then change the answer to a percent
Another method to find what percent one number is of another is the equation method. Simply turn the question word for word into an equation. For WHAT, substitue the leter X, for IS, substitute and equal sign (=); for of, substitute a multiplication sign (X). Change percents to decimals or fractions, whichever you find easier. Then solve. 10 is what percent of 50 10 = X x 50 10= x(50) 10/ 50 = x(50) / 50 10/50 = x 1/5 =5 

Finding a number when a percent of it is known.

You can also use the division method to find a number when a percent of it is known. to use this method, simply take the number of percent, change it into a decimal, and divide that into the other number.
ex A 15 is 50% of what number? 30 

percentproportion method

another simple method commonly used to solve any of the three types of percent problems is the proportion method (also called the is/of method). first set up a blank proportion and then fill in the empty spaces by using the following steps.
1.Whatever is next to the percent (%) is put over 100. ( the word what is the unknown, or x) 2. Whatever comes immediately after the word of goes on the bottom of one side of the proportion. 3. Whatever is left ( comes next to the word is) goes on top, on one side of the proportion. 4. Then solve x/x = x/x 

Finding percent increase or percent decrease

To find percent change ( increase or decrease), use this formula:
change/starting point = percent change ex. what is the percent decrease of a 500 item on sale for $400? change = 500400=100 100/500 =1/5=20% 

addition of intergers

when adding two integers with the same sign, (either both positive or negative), add the numbers and keep the sign.
When adding integers with different signs( one positive and one negative), subtract the numbers and keep the sign from the larger on (that is, the number that is larger if you disregard the positive or negative sign). 