Algebra Flashcards
Numbers which allow us to count the objects or ideas in a given collection
Cardinal Numbers
States the position of the individual objects in a sequence
Ordinal Numbers
Symbols or combination of symbols which describe a number
Numerals
Most widely used numerals
Arabic and Roman Numerals
What is the Arabic Number equivalent of the following:
I V X L C D M
I-1 V-5 X-10 L-50 C-100 D-500 M-1000
How does the romans indicate large numbers?
Bracket-multiply 100 times
|V|=500
Vinculum(bar above the number):multiply 1000 times
Doorframe=multiply 100,000 times
A specific symbol used alone or in combination to denote a number
Digit
Numbers which are considered as the “Counting Numbers”
Natural Numbers
Are all natural numbers, negative of natural numbers and the number zero
Integers
Are numbers which can be expressed as a quotient (ratio of two integers). The term rational comes from the word ratio
Rational Numbers
Are numbers which cannot be expressed as a quotient of two integers
Irrational Numbers e.g. sqrt(2), pi, Euler’s number
The numerical value of the number neglecting the sign
Absolute Value
Defined as part of the whole
Fractions
A fraction in which the numerator and denominator are both integers. Also known as common fraction
Simple Fraction
Is a fraction with unity for its numerator and positive integer for the denominator
Unit Fraction
A fraction in which the numerator or denominator, or both are fractions
Complex Fraction
Two or more simple fraction where the denominators are similar
Similar Fraction
Positive integers that have more than two positive whole number factors
Composite Numbers
An integer greater than 1 hat is divisible only by 1 and itself
Prime number
What is the fundamental theorem of arithmetic
Every positive integer greater than 1 is a prime or can be expressed as a unique product of primes and powers of primes
What is the only prime number that is greater than 1 and even?
number 2
numbers that have only two factors: 1 and itself
Natural Prime numbers
Set of two consecutive odd prime that differs by two
Twin Primes
Pair of prime numbers that are the same distance from a given number in a number line
Symmetric Primes/Euler Prime
Prime numbers that remains a prime when its digits are reversed
Emirp
Numbers whose greatest common factor is 1
Relatively Prime numbers
A number whose factors are prime numbers raised to a certain power
Unique Product of power prime
eg: 360=2^3 * 3^2 5^1
An integer that is equal to the sum of all its possible divisors, except the number itself
Perfect number
Perfect Number formula
2^(p-1)*(2^p-1)
If the sum of the possible divisors is greater than the number
abundant number
An integer with the sum of all possible divisor is less than the number itself
Deficient number
Refers to two integers where each is the sum of all the possible divisors of the other
Amicable/Friendly numbers
Represents the product of all positive integers from 1 to n, inclusive
factorial
eg: n!=n(n-1)(n-2),,,3,2,1
The factorial symbol was introduced by who?
Christian Kramp
Digits that define the numerical value of a number
Significant figure/digit
Forms of approximation
Rounding and Truncating
It means replacing the number with another number having fewer significant decimal digits
Rounding
Refers to the dropping of the next digits in order to obtain the degree of accuracy beyond the need of practical calculation/aka rounding down
Truncation
Closure Property of addition
a+b=integer
commutative property of addition
a+b=b+a
Associative Property of addition
(a+b)+c=a+(b+c)
Identity property of addition
a+0=a
additive identity
0
multiplicative identity
1
transitive property
if a=b and b=c then a=c
What is Zero-Factor property
if ab=0, then a=0 or b=0
Who first used the symbol of radical
Christoff Rudolff, 1525 in Die Coss
A radical expressing an irrational number
Surd
contains no rational number and all its terms are surds
Pure surd eg sqrt(3)=sqrt(2)
a surd that contains at least one rational number
Mixed Surd eg 5sqrt(2)
an expression of two terms with at least one surd
Binomial Surd
eg 5+sqrt(2)
Difference of two squares x^2-y^2
(x+y)(x-y)
Cube of a binomial (x+y)^3
x^3+3x^y+3xy^2+y^3
difference of two cubes x^3-y^3
(x-y)(x^2+xy+y^2)
sum of two cubes x^3+y^3
(x-y)(x^2-xy+y^2)
square of a trinomial (x+y+z)^3
x^2+y^2+z^2+2xy+2xz+2yz
In proportion, what do you call the first and second term respectively
antecedent/consequent
Refers to the product of several prime numbers each taken with its greatest multiplicity
Least common donominator
A number that two other numbers will divide into evenly
common multiple
Lowest multiple of two numbers
least common multiple
A factor that divides into a larger number evenly
Greatest common factor
Logarith comes from the greek words _____ and _____
logus=ratio
arithmus =number
who invented logarithm?
John Napier, 1614 using e=2.718 for its base
What do you call a logarithm with base e?
Natural logarithm/Napierian Logarithm
who improved the logarithm using base 10?
Henry Briggs. Its now called common logarithm/Brigssian Logarithm
Euler’s number formula
e=lim as n->infinity (1+1/n)^n
if a polynomial p(x) is divided by the binomial(x-a), the remainder is p(a)
Remainder Theorem
What is the factor theorem?
if the polynomial is divided by x-a, and the reminder is zero, then x-a is a factor of the polynomial
who suggested the remainder and factor theorem?
Etienne Bezout
Sum of two roots formula
r1+r2=-B/A
Product of two roots formula
r1*r2=C/A
rth term of the binomial expansion of (x+y)^n
rth=nC(r-1)x^(n-r+1)y^(r-1)
What is the degree of the polynomial 3x^4y-2x^3z^4=7yz^5?
- second term, 3+4
Work Problems tips
solve in man-hours
what is the equivalent of the following coins?
Penny, Nickel, Dime
Quarter, Half dollar?
Penny 1c Nickel 5c Dime 10c Quarter 25c halfdollar 50c
magic clock formula
degrees=11/2m-30H
arithmetic progression formula
an=a1+(n-1)d
s=n/2(a1+an)
Geometric Progression Formula
an=a1r^(n-1)
S= a1(r^n-1)/(r-1)
Infinite Geometric Progression Formula
S=a1/(1-r)
Fibonacci Numbers
1 1 2 3 5 18 13
Lucas Sequence
1 3 4 7 11 1 29
A set of ordered pairs
Relation
A relation where X has one and only y value
Function
Who introduced matrices?
James Joseph Sylvester
what is the main/principal diagonal?
in square matrices, the diagonal from the upper left to the lower right is the main diagonal. The entries are called diagonal entries
What is the trace of a matrix?
It is the sum of all diagonal entries of a square matrix
Define square matrix
has the same number of rows and columns
Define Diagonal Matrix
Square matrix that has values only in its diagonal
Define scalar matrix
diagonal matrix where all the entries are equal
Define identity/unit matrix
a type of scalar matrix whose non-zero elements are equal to 1
Define zero/null marix
matrix that does not contain any non-zero element
Define Triangular Matrix
Lower Triangular matrix: diagonal matrix whose entries above the main diagonal are all zero
Upper triangular matrix: diagonal matrix whose entries below the main diagonal are all zero
Define Symmetric Matrix
The transpose is equal to the original matrix
Skew-Symmetric Matrix
Transpose is equal to the negative of the original matrix
Steps required to get the inverse of a matrix
- Form the co-factor
- Transpose the cofactor
- multiply the determinant of the original matrix to the transpose of the cofactor
Dot product
A*B=|A||B|cos(theta)
Cross Product
AxB=|A||B|sin(theta)
What refers to the operation of root extraction
Evolution
To compute the vaue of n factorial, in symbolic form (n!), where n is a large number, a formula called ______ is used
Stirling’s Approximation
*Caltech
in simplifying complex fraction/simplification and rationalization
substitute the given using a number and compare the choices using the same number
Curvature Formula
1/p=(d^2y/dx^2)/(1+(dy/dx)^2)^(3/2)
