Chapter 10 (Math Foundations)- Arithmetic and Number Properties
Rules of Operation (PEMDAS)
-Parenthesis -Exponent -Multiplication (simultaneously from left to right) -Division -Addition (simultaneously from left to right) -Subtraction
Fraction
division of a part by a whole (i.e. part/whole). Example- 3/5
Product
the result of multiplication
The product (multiplication) of zero and any number is...
zero Example- (3)(0)= 0
Greatest common factor
(of a group of integers) is the largest factor that they share. Example- GCF of 12 and 14? 12= 1, 2, 3, 4, 6, 12 14= 1, 2, 7, and 14 GCF= 2
Scientific notation
Example- 123,000,000,000 to 1.23x10^11 (scientific notation) (move decimal to right 11 places) -0.000000009 to 9x10^-9 (multiplying by 10-^9 is the same as dividing by 10^9) -5.6x10^6 = 5600000. (move decimal to right 6 places) Note: use rules for multiplying and dividing by powers of 10, to translate a number from scientific notation. to ordinary notation of 10.
The square root of any non-negative number (x)
a number that when multiplied by itself yields x.
Integer
a number without fractional or decimal parts, can be positive or negative whole numbers, and zero. All integers are multiples of 1. Examples- -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...
Subtracting a number from zero...
changes the numbers sign Example: 0-5= -5 (can change to addition by (0) + (-5)= -5
Adding to or subtracting zero from a number...
does not change the number Examples- x+0= x and x-0= x
Properties of 1 and -1 (multiplying or dividing a number by 1)...
doesn't change the number. Examples- 3x1= 3 and 3/1= 3
Rules of odds/evens
odd + odd= even even + even= even odd + even= odd odd x odd= odd Multiplying an even number by any integer always produces another even number: even x even= even odd x even= even Note: pick any two numbers to test (only exception is when dividing) Example- integer x is evenly divisible by 2. Is x/2 even? 6/2= 3 (odd) and 4/2= 2 (even) Answer: x/2 is BOTH.
Digit
one of the numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). A number can have several digits. Example- 542= 3 digits vs. 321,321,000= 9 digits (only 4 distinct digits)
to DIVIDE two powers with the same base...
keep the base and SUBTRACT the exponents of the denominator from the exponent of the numerator. Example- 4^5/4^2 = 4^3
to divide by power of 10...
move the decimal point the corresponding number of places to the left. Note: insert zeros (0s) as placeholders if necessary. Example- (416.03)/(10^4)= 0.041603 (move decimal four places to the left)
to multiply by power of 10...
move the decimal point to the corresponding number of places to the right. Example (0.029)x(10^3)= 29 (move decimal 3 places to the right) = 0.029 x 1,000= 29
to multiply two powers with different base, but the same power
multiply the bases together and raise to the same power. Example- 3^2 x 5^2= 15^2
to raise a power to another power...
multiply the exponents. Example- (3^2)^4 = 3^8
Powers of 10
when 10 is raised to an exponent, that has a positive integer exponent, it tells us how many zeros (0s) the number would contain if written out. Example- (10)^6= 1,000,000 (6 factors)
Rules of Divisibility: An integer is divisible by...
2- if last digit is divisible by 2 3- if digits add up to a multiple of 3 4- if last two digits are a multiple of 4 5- if last digit is 0 or 5 6- if divisible by both 2 and 3 9- if digits add up to multiple of 9
Number line
A straight line that extends indefinitely in either direction (i.e. positive or negative), on which numbers are represented as points. -decimals, fractions, and irrational numbers can be depicted. -values get larger as move to the right along the number line (negative (-), 0 (neither positive or negative), positive (+)) -any positive (+) number is larger than any negative (-) number
Every positive number has a positive and negative square root
Example- 25: positive square root = 5 (5^2 = 25) negative square root = -5 (-5^2 = 25) Note: square root stands for the positive square root only (square root of 9 = 3 only (even though both 3^2 and -3^2= 9) -When applying the four basic arithmetic operations, radicals (roots written with radical symbol (square root)) are treated similar to variables.
Simplifying radicals...
If the number inside the radical is a multiple of a perfect square, the expression can be simplified by factoring out the perfect square.
Numerator and Denominator
Numerator- quantity in top of a fraction Denominator- quantity in bottom of a fraction
(Exponents) to MULTIPLY two powers with the same base...
keep the base and ADD the exponents together. Example- 2^3 x 2^4 = 2^7
Decimal
a fraction written in decimal system format Example- divide numerator by the denominator of a fraction = decimal (3/5 = 0.6)
Operation
a function or process performed on one or more numbers. The four basic arithmetic operations are addition (+), subtraction (-), multiplication (x), and division (/).
Divisibility
a number is said to be evenly divisible by another number if result of the division is an integer with no remainder. Note: a number that is evenly divisible by a second number is also a multiple of the second number. Example- 52/4= 13 (integer) 52 is evenly divisible (no remainder such as 38/14= 2 R10) by 4 and is also a multiple of 4.
Cube
a number raised to the third (3rd) power Example- 2^3= (2)(2)(2)= 8 (8 is the cube of 2)
Multiples
a product of a specified number and an integer. Multiples don't have to be integers but must be the product of a specific number and an integer. (i.e. 2.4, 12, 132 multiples/product of (1.2)(x) (x= 2, 10, and 110) Example- 3, 12, and 90 are all multiples of 3 (i.e. 3x1, 3x4 (4 is not multiple of 3 because there is no integer that. an be multiplied by 3 to yield 4), 3x30)
Part
a specified number of the equal sections that compose a whole.
Set
a well-defined collection of items, numbers, objects, or events. (curly brackets) Example- {2, 4, 6, 8...}
Adding two numbers with the same sign...
add the number parts and keep the sign
The Distributive Law of X
allows to distribute (multiply) a factor over numbers that are added or subtracted. Multiply the factor by each number in group. Example- a (b+c)= ab + ac Also works for NUMERATOR in division (when sum or difference is in the numerator). Example- (a+b)/c = a/c + b/c Note: when the sum or difference is in the DENOMINATOR, no distribution is possible. Example- a/(b+c) does not equal ab + ac
Properties of Odd and Even Numbers
apply only to integers (either positive or negative) Odd- integers that are not evenly divisible by 2 or whose last digit is 1, 3, 5, or 9. Even- integers that are evenly divisible by 2 or whose last digit is 2, 4, 6, or 8.
Factors
are the numbers you multiply together to get another number. Example- 2x3=6 (2 and 3 are factors) also known as divisors, of an integrate positive and negative integers by which it is evenly divisible (no remainder). Example- positive factor of 36? 1, 2, 3, 4, 6, 9, 12, 18, and 36 (regrouped into pairs and multiplied)
Consecutive numbers
ascending (+) or descending (-) order of numbers of a certain type, following one another without interruption. Examples- consecutive integers: -2, -1, 0, 1, 2 consecutive multiples: -3, 0, 3, 6, 9 consecutive prime numbers: 2, 3, 5, 7, 11
To subtract a positive number from a negative number (or from smaller positive number)...
change the sign of the numbers subtracting from positive to negative and follow the rules for addition of signed numbers. Example- (-4)-1= (-4) + (-1)= -5 4-5= (4) + (-5)= -1
Multiplying or dividing a non-zero number by -1...
changes the sign of the number. Examples- (3)(-1)= -3 and 3/-1= -3 or (-3)(-1)= 3
Symbols
equal to, not equal to, less than, greater than, less than or equal to, greater than or equal to, divided by, pi, plus or minus, square root, angle.
raising any nonzero number to an exponent of zero...
equals 1 Example- 3^0= 1
Adding two numbers with different signs...
find the difference between the number parts and keep the sign of the number whose number part is larger. Example- (-7)+4= -3
Prime factorization
finding which prime numbers multiply together to make the original number. Method: figure out one pair of factors and determine their factors, continuing the process until left with only prime numbers. (prime numbers will be the prime factorization) Example: Prime factorization of 1,050? -10 -> 5 x 2 -105-> 5 x 21-> 7x 3 Prime factors (in increasing order): 2, ,3, 5, 5, and 7 or (2)(3)(5^2)(7)
Multiplying or dividing two numbers with opposite signs...
gives a negative result Example- (-3)(4)= -12 or (-3)/(4)= -3/4
Multiplying or dividing two numbers with the same sign...
gives a positive result Example- (-3)(-4)= 12 and (3)(4)= 12 or (3)/(4)= 3/4 and (-3)(-4)= 3/4
raising an even number to ANY positive integer exponent...
gives an even number. Example- (2)^5= 32
raising an odd number to any integer greater than or equal to zero...
gives an odd number. Example- (3)^5= 243
a base with a negative exponent...
indicates the reciprocal of that base to the positive value of the exponent Example- 5^-3 = 1/5^3= 1/125
Prime number
is a whole number greater than 1 that can not be made by multiplying other whole numbers. Has only two factors 1 and itself. (1 is not prime as it is divisible only by itself) Examples- 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 (If we can make it by multiplying other whole numbers it is a Composite Number. Example- 2x3=6) *2 is the only even prime number
Exponent
number denotes power to which another variable or number is raised (i.e. superscript). Exponents can be positive or negative integers, or fractions, and may include variables. Example- 5^3= (5)(5)(5)= 125 (superscript= 3) (125 is the third power of 5)
Decimal system
numbering system based on the power of 10, each place is 10X larger than the place to its right. (and only number system used in the GRE) Example- 315.246 (hundreds, thousands, units, tenths, hundredths, thousandths) (315= integer portion or whole numbers, "."= decimal point, 246= decimal part or decimal fractions)
Number Properties (Addition and Subtraction)
numbers can be treated as though they have two parts, a positive and a negative sign and a number.
Element
one of the members of a set. Example- {2, 3, 4}= 2 is an element
Addition and subtraction of radicals...
only like radicals can be added to or subtracted from one another.
Sequence
ordered list of terms, terms are indicated by a letter with a subscript to indicate the position of the number in the sequence. Example- an (n= subscript denotes the nth term in a sequence)
raising a negative number to an odd power...
produces a negative result Example- (-2)^3= -8
raising a negative number to an even power...
produces a positive result Example- (-2)^2= 4
(Powers) raising a fraction between 0-1 to a power...
produces a smaller result Example- (1/2)= 0.5 vs. (1/2)^2= 0.25
Square
product of a number multiplied by itself, raised to the second power. Example- 4^2= (4)(4)= 16 (16 is the square of 4)
Whole
quantity that is regarded as a complete unit
Pi
ratio of circumference of a circle to the diameter
Associative Laws of Addition and Multiplication (regrouping)
regrouping the numbers does not affect the result. Example- Addition: (a+b)+c= a+(b+c) Multiplication: (ab)(c)= (a)(bc)
To subtract a negative number...
rephrase as an addition problem and follow the rules for addition of signed numbers. Example- 9-(-10)= 9+10= 19
Difference
result of a subtraction
Sum
result of addition
Variable
special type of amount or quantity of unknown value
Commutative Laws of Addition and Multiplication (order)
switching the order of any two numbers being added or multiplied does not affect the result. Division and subtraction are not commutative. Examples- Addition: 5+6=6+5 Multiplication: (3)(2)=(2)(3) Subtraction: 3-2= 1 and 2-3= -1 Division: 6/2= 3 and 2/6= 0.33
Remainder
the amount left over when one number is divided by another. Note: remainder is always smaller than the number one is dividing by Example: What is the remainder of 17/3? 17/3= 5R2 (3X5= 15 and 17-15= 2) Note: if numerator unknown, or solving for numerator, figure out number that will subtract by the denominator to give the remainder (if final answer is 1R__). Example- numbers that leave a R2 when divided by five are two greater than multiple of 5 (e.g. 7 (5+2), 12 (10+2), 17 (15+2))
Least common multiple
the smallest multiple (other than zero) that two or more numbers have in common. Begin by finding the prime factors. Example- LCM of 8 and 6 6= 3x2 (3 appears 1 time in prime factorization of 6) 8= 2x4-> 2x2 (2 appears 3 times in prime factorization of 8) LCM: (3)(2)(2)(2)= 24 Note: LCM is always smaller than their product, if have any factors in common. 8x6= 48 (greater than LCM 24)
Absolute value
the value of a number WITHOUT its sign, the number's distance from zero on a number line. Examples- I-3I= I3I= 3 (each are 3 units from zero and have an absolute value of 3) IxI= 5 (x could equal -5 or 5) Note: "absolute value" means to remove any negative sign in front of a number, and to think of all numbers as positive (or zero). (No negatives!)
Multiplication and division of radicals...
to multiply or divide one radical by another 1. Multiply or divide the numbers outside the radical signs 2. (Then) the numbers inside the radical signs
Division by zero is...
undefined Example- 3/0= undefined/error Note: fractions are divisions, any fraction with zero in the denominator is undefined. Assure that a fraction with algebraic expression in denominator is not equal to zero.