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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

3-2 ≓ 2-3
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-(2-1) ≓ (4-2) -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
percent-proportion 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 = 500-400=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).