First flash cards
Natural Numbers/Whole Numbers/Integers
natural numbers {1,2,3…} .
whole numbers: {0,1,2…} .
integers: {…−2,−1,0,1,2…} . Zero is an integer, but it is neither positive nor negative.
Absolute Value
3 and −3 both have an absolute value of 3.
Rounding Numbers
Multidigit numbers can be rounded to any place.
Denominator
The number on the bottom of a fraction is called the denominator.
Numerator
The number on the top of a fraction is called the numerator. The numerator tells how many of all the parts the fraction represents. For example, the fraction 12 means 1 out of 2 parts.
Simplifying or Reducing Fractions
dividing both the numerator and the denominator by a same number
912=9÷312÷3=34
Adding or Subtracting Fractions with the Same Denominator
To add or subtract fractions with the same denominator, you can simply add the numerators. For example:
38+28=58
Adding or Subtracting Fractions with Different Denominators
Multiplying Fractions
multiply the numerators and multiply the denominators and then simplify or reduce as needed.
Dividing Fractions
To divide fractions, multiply the first fraction by the reciprocal of second fraction. The reciprocal of a number is the number that when multiplied by that number gives a product of 1. For example, the reciprocal of 34 is 43 because 34×43=1212=1 . Notice that for a fraction, the reciprocal is that fraction flipped, or with the numerator and denominator reversed. Since all numbers are equal to themselves over 1, you can also use this rule for integers. For example, the reciprocal of 2 is 12 . Therefore, for example:
23÷2=23×12=26=13
Distributive Property
The distributive property states that when multiplying a number by a sum of two numbers in parentheses, you can distribute the outside number to both numbers in the parentheses. For example:
12(13+14)=12(13)+12(14)=16+18=424+324=724
Percentages
A percentage represents an amount out of 100, so decimals can also be converted to percentages by simply multiplying the decimal by 100%. For example:
0.5 = 0.5 × 100% = 50%
Exponents
An exponent or a number raised to a power indicates the number of factors of that number that need to be multiplied together, and squaring a number means raising it to a power of two. For example:
42=4×4=16
Squaring 4 means multiplying two factors of 4 together, which gives you 16.
Converting a Fraction to a Percentage
To convert a fraction to a percentage, convert the fraction to a decimal first, and then multiply the decimal by 100 and add the % symbol to the answer. To convert a percentage to a fraction, we reverse the process. First, convert the percentage to a decimal by dividing it by 100 and removing the % symbol. Then convert the decimal to a fraction as shown previously.
Order of Operations
When performing operations on real numbers of any kind, it is important to remember to use the order of operations. We often use the acronym PEMDAS to help remember the order of operations. It stands for “parentheses, exponents, multiplication and division, addition and subtraction,” and that is the order in which operations in an equation should be performed.
Commutative Property
Both addition and multiplication have a commutative property, meaning you can add or multiply two numbers in either order and the sum or product will be the same:
5+73×6==7+5=126×3=18
This does NOT apply to subtraction or division, however. If you subtract numbers in the opposite order, you get the additive inverse of the difference, and if you divide numbers in the opposite order, you get the multiplicative inverse or reciprocal of the quotient. For example:
7−55−76÷33÷6====2−2212
Associative Property
Addition and multiplication have an associative property, meaning you can group additions or multiplications any way and the final sums or products come out the same. This property does not apply to subtraction or division. For example:
(7+8)+3(3×5)×6==(7+3)+8=18(3×6)×5=90
Identity Property
Addition and multiplication each have their own identity property. For addition, any number plus zero equals itself, and for multiplication, any number times 1 equals itself. Multiplication also has its own zero property, which means that any number times 0 equals 0.
a+0a×1a×0===aa0
Inverse Property
Multiplication has an inverse property, which means that any number times its multiplicative inverse or reciprocal always equals one. Note in the first equation that follows that when a fraction has the same numerator and denominator, it always equals 1.
5×15=5523×32ab×ba===111
Distributive Property of Multiplication and Addition
The distributive property of multiplication and addition means that when multiplying a number by a sum of two or more numbers in parentheses, you can distribute the outside number to the numbers in the parentheses:
5(4+3)a(b+c)==5×4+5×3=20+15=35a×b+a×c
Divisibility
We say a number is divisible by a second number if dividing the first by the second gives you an integer quotient. For example, 10 is divisible by 5 because 10 divided by 5 is 2, which is an integer, but 10 is not divisible by 4 because 10 divided by 4 is 2.5, which is not an integer. We call the set of numbers by which a number is divisible the factors of that number.
Roman Numerals
There are some number systems that don’t use place or position value at all. For example, Roman numerals use different characters or letters to represent different values. The values of Roman numerals are shown in the following table:
Roman Numeral Value
I 1
V 5
X 10
L 50
C 100
D 500
M 1,000
Prime and Composite Numbers
All numbers can be classified as either prime or composite. Prime numbers are divisible only by themselves and 1, while composite numbers are divisible by other numbers, all of which can be written as products of prime numbers. All the factors of a composite number can be broken down into prime numbers, and that set of prime numbers is called the prime factorization of that number.
Rules of Exponents: Product Rule
The product rule says that to multiply two exponents with the same base, you can simply keep the base the same and add the exponents.
43×45ax×ay==43+5=48ax+y
Rules of Exponents: Quotient Rule
Similarly, the quotient rule says that to divide two exponents with the same base, you can simply keep the base the same and subtract the exponents.
86÷84ax÷ay==86−4=82ax−y
Rules of Exponents: Power to a Power Rule
The power to a power rule says that to raise an exponent to a power, you can keep the base the same and multiply the exponents.
(53)4(ax)y==53×4=512axy
Binomials
A binomial is an expression with two terms that is in parentheses and raised to a power such as:
(x4+x5)2
Scientific Notation
One of the most important applications of exponents is using scientific notation to express very large or very small numbers. Scientific notation is written in the form a×10n , where 1≤|a|<10 and n is an integer.
To express a large number such as 30,000,000,000 (30 billion) in scientific notation, we move the decimal point to the left until the number is between 1 and 10 and then multiply by 10 to the power of the number of spaces the decimal was moved. In this case, we would need to move the decimal point 10 places to the left to get 3, so this number in scientific notation would be 3×1010 .
To express a small number such as 0.000005 (5 millionths) in scientific notation, we move the decimal point to the right until the number is between 1 and 10 and then multiply by 10 to the negative power of the number of spaces the decimal was moved. In this case, you would need to move the decimal point 6 places to the right to get 5, so this number in scientific notation would be 5×10−6 .
The Nature of Sets
A set can be any grouping of elements or objects that is well defined. For example, if set S is defined as “the first five letters of the alphabet,” then it would contain the elements a,b,c,d, and e . The notation a∈S means “a is an element of S,” while the notation z∉S means “z is NOT an element of S.”
When working with sets, we call the largest set that contains all of the elements we are using the universal set, which is represented by the letter U. A set that contains no elements at all is called the empty set, also known as the null set or the void set. It is represented by the Greek letter phi (φ), by a circle with a line through it ( ⊘ ), or by a set of curly braces with nothing inside: { }. Two sets that share all of the same elements are called equal sets.
Writing Sets: Roster Form
One way to write sets is called roster form, and it involves simply listing the elements of the set separated by commas inside curly brackets. For example, the set S containing the factors of 10 would be written as S={1,2,5,10} . If there are too many elements to list them all, you can establish a pattern and then use an ellipsis (…) to indicate that the pattern continues. For example, the set E of even numbers would be written in roster form as E={2,4,6,8,10…}
