Arithmetic Flashcards
Integers
Positive and negative numbers, including 0
Positive and Negative integers
Positive Integers are greater than 0
Negative Integers are smaller than 0
0 is neither positive or negative
Multiplication of Integers (positive and negative)
The product of two positive integers is as positive integer
The product of two negative integers is a positive integer
The product of a positive and negative integer is a negative integer
Factors (or Divisors)
Integers which are multiplied with other integers are factors of the resulting integer.
Multiples and Divisible
An integer is the multiple of all of its factors and divisible by all of its divisors
Each nonzero integer has infinite multiples
Least common multiple
The LCM of two nonzero integers c and d is the least positive integer which is a multiple of both c and d
The greatest common divisor or greatest common factor
The GCD(F) of two integers c and d is the greatest positive integer which is a divisor of both c and d
Quotient and remainder
When integer c is divided by positive integer d the greatest multiple of d that is less than c can be multiplied by q.
q is the quotient and c - qd is the remainder (always > 0 and < d)
When dividing an integer c by an integer d which is a divisor of c then the result is always a divisor of c.
If d is not a divisor then the result can be viewed as a fraction or decimal or a quotient with a remainder.
Even and Odd integers
If an integer is divisible by 2 then it is an even integer, otherwise it is an odd integer
Even integers contain 0
Even and odd integers facts
E + E = E O + O = E E + O = O E * E = E O * O = O E * O = E
Prime Number
Integer grater than 1 which has only two possible divisors, 1 and itself
Prime divisors
Every integer grate than 1 is either a prime number or van be uniquely expressed as a product of factors or prime divisors
Prime factorization
Expression of an integer in the form of products of prime divisors
Composite Number
Integer grater than 1 which is not a prime number
Fractions
number in the form c/d where c and d are integers and d is different from 0
c is the numerator and d is the denominator
Rational numbers
Numbers in the form of c/d
Every integer n is a rational number n is equal to n/1
Equivalent fractions
if a fraction c/d is multiplied at the nominator and denominator by an integer n the resulting fraction is equivalent
-c/d and c/-d and -(c/d) are all equivalent
If a numerator and denominator have common factor they can be factored to reduce the fraction to an equivalent fraction
Adding and subtracting fractions
If the denom is equal then just add the numerator
If the denoms are different then find a common denom (a multiple of both denoms) and then convert both fractions to equivalent fractions with the same denom, then add the numerators and keep the common denom
Multiplying and Dividing fractions
Multiplications of fractions is done by multiplying num with num and denom with denom
Division of fractions is done by finding reciprocal of second fraction and then multiplying the first with the reciprocal od the second
Mixed numbers
An expression which consists of an integer part and a fraction part n * (c/d) where the fraction part has a value between 0 and 1
Converting mixed number to fraction
Convert integer part to a equivalent fraction with same denominator of the fraction and then add it to the fraction part
Fractional expressions
numbers in the form of c/d where either c or d is not an integer and d is different from 0
Exponents
Used to denote the repeated multiplication of a number by itself
n^e where n is the base and e is the exponent
For all nonzero numbers n^0 = 1
0^0 is undefined
n^(-e) = 1/(n^e)
n*n^(-1) = 1
Squaring
A expression with an exponent of value 2
Negative and positive numbers raised to an exponent
A negative number raised to an even power is always positive
A negative number raised to a odd exponent is always negative
Square root
The square root of a number n is a number r such that r^2 = n
All positive numbers have two square roots, one positive and one negative
The square root of 0 is 0
Square roots of negative numbers are not defined in the real number system
4 rules for square roots operations
(√a)^2 = a
√(a^2) = a
√a√b = √(ab)
√a/√b = √(a/b)
Odd and even order roots
Odd order roots there is exactly one root for every number n even when n is negative
Even order roots there are exactly two roots for every positive number n and no roots for any negative number n
Decimals
A decimal is number which has nonzero values after the decimal point (or that has nonzero values of 10^-x)
A decimal can terminate or repeat without end.
Every fraction with integers in the num and denom is equivalent to a decimal which either terminates or repeats
Every rational number can be expressed as a terminating or repeating decimal and every decimal as a rational numbers
Irrational numbers
A decimal which does not terminate or repeat as √2
Real numbers
Set that consists of both rational and irrational numbers (it includes all integers, fractions and decimals)
The real numbers can be represented on a line where each point corresponds to a real number (continuous)
Absolute value
the distance between a number x and 0
written as |x|
|-x| = x and |x| = x
The absolute value of any nonzero number is positive
12 real numbers properties
r+s = s+r and rs = sr
(r+s)+t = r+(s+t) and (rs)t = r(st)
r(s+t) = rs + rt
r + 0 = r and r0 = 0 and r1 = r
if rs = 0 then either r = 0 or s = 0 or both
Division by 0 is undefined
If r and s are positive then r + s and rs is positve
If r and s are negative then r + s is negative and rs is positive
If r is positive and s is negative then rs is negative
|r + s| ≤ |r| + |s| triangle inequality
|r|*|s| = |rs|
if r > 1 then r^2 > r if 0 < s < 1 then s^2 < s
Ratio
Proportion between two quantities which expresses their relative sizes
The first quantity is the num and the second is the denom
apples to oranges is expressed as apples/oranges or apples:oranges
Ratios can be reduced to equivalent ratios
Proportion
An equation which relates two ratios which can be solved by cross multiplication
Percent
means per hundred or hundreths
Ratios which are used to represent parts of a whole considered to have 100 parts
Percents to decimals and fractions
1% is equal to 1/100 or 0.01
Converting part and whole to percent
Divide part by whole and then multiply by 100
Find part that is a certain percent of a whole
Multiply the whole by the decimal equivalent of the percent
Set up a proportion to find a part
Given the percent and the part calculate whole
Use decimal equivalent of percent
- whole = part/percent
Use proportions
- whole = (part * 100)/perc
Percent grater than 100%
If the numerator of a percent is greater than the denominator then the percent is greater than 100
Percent change
Amount of change between two values in percent of the initial value
Percent increase
(new - previous)/previous
percent decrease
(previous - new)/previous
