Numbers Flashcards
Real Numbers (R)
Any number that can be identified on the number line. Includes negative numbers, positive numbers, square roots, pi, and almost all other numbers. Non-real numbers cannot be identified on a number line such as the square roots of negative numbers and 0/0.
Properties of Real Numbers
- Commutative Property
- Associative Property
- Distributive Property
- Identity Property
- Inverse Property
Note: The properties of real numbers deal only with addition and multiplication. The properties do not apply to subtraction and division
Commutative Property
A binary operation (equation consisting of two numbers) of addition or multiplication can be rearranged in order and the value will not change.
Addition: 2+3 = 3+2
Multiplication: 2x3 = 3x2
Associative Property
The numbers in equations made up of only addition OR only multiplication can be grouped in any order and the value will not change.
Addition: 2+(3+4) = (2+3)+4
Multiplication: 2x(3x4) = (2x3)x4
Distributive Property
When multiplying a number by a group of numbers being added together in parentheses, the number can be multiplied individually to each of the numbers in parentheses, and the resulting products are added together. The same is true when multiplying a number by a group of numbers being subtracted together in parentheses.
2x(3+4) = (2x3)+(2x4) 2x(3-4) = (2x3)-(2x4)
Identity Property
Adding zero or multiplying one by a number does not change the original value.
Addition: 5+0=5
Multiplication: 5x1=5
Inverse Property
Adding a number by its additive inverse equals zero. Multiplying a number by its multiplicative inverse equals 1.
Addition: 5+(-5) = 0
Multiplication: 5x(1/5)=1
Note: the one exception is zero. Zero is a real number, but there is no additive inverse or multiplicative inverse for the number zero.
Components of Real Numbers
WINQS
- Whole Numbers (W)
- Integers (I)
- Natural Numbers (N)
- Rational Numbers (Q)
- Irrational Numbers (S)
Integers (I)
A number with no fractional part or decimal.
…., -3, -2, -1, 0, 1, 2, 3, ….
Whole Numbers (W)
Non-negative integers that are greater than or equal to zero.
0, 1, 2, 3, 4, …..
Natural Numbers (N)
Non-negative integers greater than or equal to 1 (AKA counting numbers).
1, 2, 3, 4, ….
Rational Numbers (Q)
All fractions consisting of two integers in which the bottom integer is not equal to zero.
Examples: 1/2, 3/4, -2, -2/5, 3
Note: This includes all numbers with repeating decimals such as .22222…. (show notation for repeating decimals), because all such numbers can be re-written as a fraction. .2222… re-written as a fraction is 2/9.
Irrational Numbers (S)
All numbers that are not rational are irrational. This includes numbers with fractional parts in which the decimals do not terminate and do not repeat.
Examples: pi, square root of 2
Proper and Improper Functions
Proper fractions - Fractions in which the numerator is less than the denominator and the value is between -1 and 1. Examples: 1/2, 3/4, 150/160
Improper fractions - Fractions in which the numerator is greater than the denominator (all other fractions besides proper fractions). Examples: 8/5, 3/1, 60/20.
Note: Improper fractions can be re-written as mixed numbers, i.e. 8/5 can be re-written as 1 3/5
Odd and Even Numbers
Even numbers are easily divisible by two (includes 0). Odd numbers are not easily divisible by 2.
Even+Even=Even
Odd+Odd = Even
Even + Odd = Odd
Even x Even = Even
Odd x Odd = Odd
Even x Odd = Even
Factor
Any natural number that that divides into a another number without a remainder.
Example: The factors of 18 are 1, 2, 3, 6, 9, and 18
Multiple
The value that results when multiplying a given number by any natural number.
Example: the multiples of 3 include 3, 6, 9, 12, 15….
Absolute Value
The magnitude of a number without regard to its sign. The distance between zero and any number on a number line is the absolute value of that number (always positive).
Example: |4| = 4 and |-4| = 4
The Rules of Absolute Value
- |-a| = |a|
- |a|>=0 (the only situation in which |A|=0 is when A=0)
- |a/b| = |a| / |b| (b cannot equal 0)
- |ab| = |a| x |b|
- |a^n| = |a|^n
The Order of Operations
The order in which mathematical equations are solved is the following: parentheses, exponents, multiplication and division (left to right), addition and subtraction (left to right).
AKA PEMDAS (Please Excuse My Dear Aunt Sally)
Prime Numbers
Positive integers that are greater than 1 and are only the only factors include 1 and the prime number itself.
Examples: 2, 3, 5, 7, 11…..
Composite Numbers
All integers greater than 1 that are not prime numbers.
Examples: 4, 6, 8, 9, 10…..
Order Property of Real Numbers
If x and y are both real numbers, then only one of the following statements can be true:
- x>y
- x=y
- x
Transitive Property of Inequalities
If x, y, and z are real numbers, then both of the following statements are true:
1. If xy and y>z, then x>z
Addition Property of Inequalities
If x, y, and z are real numbers AND x>y, then both of the following statements are true:
- (x+ z) > (y+z)
- (x-z) > (y-z)
Set-Builder Notation
A way of writing subsets (i.e. intervals) of numbers on a number line.
Using set builder notation (also called set notation), the set of numbers on a number line are written in the format {x | inequality} and are read in the following format: “the set of values x such that x….”
Example: {x | x>3} is read as “the set of values of x such that x is greater than 3”
Scientific Notation
A form of notation used to represent very large or very small numbers. The form is: a x 10^n where 1 <= |a| < 10 and n is an integer.
Scientific notation is converted to standard notation by moving the decimal place of a n number of times, and inserting zero for any values not designated in a. If n is positive, the decimal place is moved to the right. If n is negative, the decimal place is moved to the left.
Example: 2.4 x 10^5 = 240,000
Example: 3.6 x 10^-8 = .000000036
Example: -7.2326 x 10^3 = -7,232.6
Converting from Standard Notation to Scientific Notation
For numbers greater than 1, a = the first non-zero digit followed by a decimal place and the rest of the non-zero digits, and n = the number of digits to the right of the first non-zero digit (n). Example: 446,900 = 4.469 x 10^5
For numbers less than 1, a = first non-zero digit followed by a decimal place and the rest of the non-zero digits, and n = the number of zeroes to the left of the first non-zero digit and before the decimal place + 1. Example: 0.0000052 = 52 x 10^6
Multiplication of Numbers in Scientific Notation
(a x 10^n) (b x 10^m) = ab x 10^(n+m). In other words, multiply the main numbers and add the exponents, then solve. If a x b >=10, express ab in scientific notation as well, which will require adding the number of digits in ab in excess of 1 to the exponent.
Example: (2 x 10^3) (4 x 10^-6) = 2(4) x 10^(3-6) = 8 x 10^-3
Example: (8 x 10^2) (3 x 10^4) = 8(3) x 10^(2+4) = 24 x 10^6 = 2.4 x 10^7
Example: (-3 x 10^4) (-5 x 10^10) = -3(-5) x 10^(4-10) = 15 x 10^-6 = 1.5 x 10^-5
Division of Numbers in Scientific Notation
(a x 10^n) / (b x 10^m) = a/b x 10^(n-m). In other words, divide the main numbers and subtract the exponents, then solve. If a/b < 1, express ab in scientific notation as well, which will require subtracting 1 from the exponent.
Example: (8 x 10^3) / (2 x 10^4) = 8/2 x 10^(3-4) = 4 x 10^-1
Example: (2 x 10^3) / (4 x 10^-6) = 2/4 x 10^(3+6) = .5 x 10^9 = 5 x 10^8
Imperial System Units of Length
- 1 foot (ft) = 12 inches (in)
- 1 yard (yd) = 3 feet (ft)
- 1 mile (mi) = 5,280 feet (ft)
Metric System Units of Length
- 1 meter (m) = 100 centimeters (cm)
- 1 meter (m) = 1,000 millimeters (mm)
- 1 kilometer (km) = 1,000 meters (m)
Imperial System Units of Weight
- 1 pound (lb) = 16 ounces (oz)
* 1 ton = 2,000 pounds (lb)
Metric System Units of Weight
- 1 gram (g) = 1,000 milligrams (mg)
* 1 kilogram (kg) = 1,000 grams (g)
Imperial System Units of Volume
- 1 pint (pt) = 16 ounces (oz)
- 1 quart (qt) = 2 pints (pt)
- 1 gallon (gal) = 4 quarts (qt)
Metric System Units of Volume
• 1 liter (L) = 1,000 milliliters (mL)
Units of Time
- 1 minute (min) = 60 seconds (s)
- 1 hour (h) = 60 minutes (min)
- 1 day = 24 hours (h)
- 1 year (yr) = 365 days
Converting Units
Multiply the original unit by the conversion factor for the new unit.
Example: 5 feet converted to inches = 5 feet x (12 inches / 1 foot) = 60 inches
Example: 6 pints converted to gallons = 6 pints x (1 quart / 2 pints) = 3 quarts x (1 gallon / 4 quarts) = 3/4 gallon
Interval Notation
A way of writing subsets of numbers (i.e. intervals) on a number line
Written in the format (minimum, maximum). Soft parentheses are used for open endpoints and hard brackets are used for closed endpoints.
Example: -2 <= x < 5 is written in interval notation as [-2,5)
Disjoint intervals are joined with the U symbol. Example: x =/0 is written in interval notation as (-inf,0) U (0,inf)
Solving Negative Exponents
a^-n = 1/a^n
Note: This rule is true as long as the base number is not equal to zero. If zero is raised to a negative power, the result is undefined because we cannot divide by zero.
Solving Fractional Exponents
a^(m/n) = n root of a^m
Note: The order of taking the power and the root is flexible. The base number is raised to the power of the numerator and then the denominator determines what root take of the result.
You will get the same result if you take the n root of the base number first and then raise the result to the m power.
General Laws of Exponents
- a^m(a^n) = a^(m+n)
- (a^m)^n = a^mn
- a^m/a^n = a^(m-n)
- ab^m = a^m(b^m)
- (a/b)^m = a^m/b^m, b=x0
Imaginary Numbers
The square roots of negative numbers. The imaginary number i is defined as i = √-1.
Note: i^2 = -1
Complex Numbers
The sum of a real number and an imaginary number. The standard form for a complex number is a + bi, where a and b are real numbers and i=√-1.
The set of all complex numbers is represented as C = {a + bi | a and b are all real numbers}.
All real numbers are complex numbers but not all complex numbers are real numbers.
Adding Complex Numbers
(a + bi) + (c + di) = (a + c) + (b + d)i
Combine like terms
Subtracting Complex Numbers
(a + bi) - (c + di) = (a - c) + (c - d)i
Combine like terms
Multiplying Complex Numbers
When multiplying with a constant by a complex number, the distributive property is used. When multiplying two complex numbers, the FOIL method is used.
(a + bi)(c + di) = a(c) + a(di) + bi(c) + bi(di)
Dividing Complex Numbers
To solve for the division of two complex numbers, we multiply both the numerator and the denominator by the complex conjugate in order to put the equation in a form that can be solved.
a + bi / (c + di) = [(a + bi) x (c - di) ] / [ (c + di) x (c - di)}
