ALGEBRA Flashcards
Review
is an expression involving a combination of real and imaginary numbers. They are written in the form: a + bi
Complex Numbers
are the rational and irrational numbers taken together.
Real Numbers
are the square roots of negative numbers.
Imaginary Numbers
Are numbers which can be expressed in the form m/n, where m and n are integers and 𝑛 ≠ 0 .
Rational Numbers
are numbers, which cannot be expressed in the form m/n.
Irrational Numbers
are the natural numbers, along with their negatives, and
zero (0).
Integers
a number that is not a whole number, a negative whole
number, or zero.
Non-Integers
are numbers that have a value less than zero. They do not include fractions or decimals. For example, -7, -10 are negative integers.
Negative Number
are numbers that are positive and zero.
Whole Number
number representing an empty quantity
Zero
a whole number not including zero.
Natural Number
Types of Natural Number
- Even Number
- Odd number
- Composite Number
Are natural numbers that are neither 1 nor a prime number.
Composite Numbers
Are natural numbers that are divisible by 1 and itself only.
{2,3,5,7,11,etc.}
Prime Number
Types of Prime Numbers
- Euler primes or Symmetric primes
- Twin primes
- Emirp
- Mersenne primes
are pairs of prime numbers that are equidistant from a given number on a number line.
Euler primes or Symmetric primes
are pairs of two consecutive odd prime numbers that differ by 2.
Twin primes
are prime numbers that remain a prime when its digits are reversed.
Emirp
are prime numbers can be made from the Expression 2𝑛 − 1. This method for generating prime numbers works only when n itself is prime, but not always. For example, it works when n = 2, 3, 5 or 7 but not when n is 11, and not when n = 23 as well as several other prime values. (3, 7, and 31, etc.)
Mersenne primes
PROPERTIES OF REAL NUMBERS
A.) Closure Property
B.) Commutative Property
C.) Associative Property of Addition
D.) Distributive Property
E.) Identity Property
F.) Inverse Property
The set of real numbers is closed under addition, subtraction and multiplication. This means that adding, subtracting or multiplying two or more real numbers always results to another number that belongs to the same set of real numbers.
Closure Property
The order of adding two or more numbers of a sum or multiplying two or more
factors of a product does not affect the result.
Commutative Property
When two or more real numbers are added or multiplied together, no matter how the numbers are grouped, or associated, when performing the operation the result is not affected.
Associative Property of Addition
The product of a number a by the sum of two or more numbers (b +c +d +…) is equal to the sum of the products ab, ac, ad, …
Distributive Property
- Additive Identity Property
When zero (0) is added to a real number, the sum is the real number itself. - Multiplicative Identity Property When one (1) is multiplied to a real number, the product is the real number itself.
Identity Property
Additive Inverse
The additive inverse of a real number is its opposite, so that the sum of that number and its additive inverse is 0
Multiplicative Inverse
The multiplicative inverse of a real number is its reciprocal, so that the product of that number and its multiplicative inverse is 1.
Inverse Property
largest number identified in
the list of common factors is known as
GCF
defined as the smallest multiple that two or more
numbers have in common
LCM
THEORY OF EQUATIONS
- The Fundamental Theorem of Algebra
- The Remainder Theorem
- The Factor Theorem
States that every polynomial equation of degree n with complex number coefficients has n roots, or solutions, in the complex numbers. The roots can have a multiplicity greater than zero. It also states that every single-variable polynomial with complex coefficients has at least one complex root.
The Fundamental Theorem of Algebra
If a polynomial f(x) is divided by (x-k) , the remainder is f(k).
The Remainder Theorem
If (x-k) is a factor of a polynomial f(x), then, the remainder f(k)=0.
The Factor Theorem
Determines the maximum number of positive and negative real roots that a polynomial will have by counting the number of sign variations in the polynomial. The polynomial must have real coefficients and be arranged in terms of descending powers of x.
Descartes’ Rule of Signs
In the quadratic formula, the quantity under radical sign b2 - 4ac is called
discriminant
the roots are real, rational and unequal
Perfect Square
the roots are real, irrational and unequal
Not a perfect square
b2-4ac < 0, then the roots are
complex conjugate
a set of numbers in a definite or specific order and formed according to a definite rule
a SEQUENCE of numbers
The numbers of the sequence are called
terms
It is a sequence of numbers in which each term is obtained from the preceding term in the same way
Progression
It is a sequence in which there is a common difference “d” between any two consecutive terms
Arithmetic Progression
It is a sequence in which there is a common ratio of each term to its receding term.
Geometric Progression
A sequence of terms in which each term is the reciprocal of the corresponding term of a series in arithmetic progression
Harmonic Progression
It is any well-defined collection of symbols or objects.
A set
The objects comprising the set are called
elements or members
It is the set of elements which belong to A or to B or to both A and B
Union
It is the set of elements which belong to both A and B
Intersection
If A and B do not have any element in common, it is said to be
disjoint
It is the set of elements which belong to A but not to B
Difference
denoted by A raised to c, is the set of elements, which belong to the universal set but not to the set A
Complement
Coin Value and total value of penny(p)
1 cent, p
Coin Value and total value of nickel(n)
5 cents, 5n
Coin Value and total value of dime(d)
10 cents, 10d
Coin Value and total value of quarter (q)
25 cents, 25q
Coin Value and total value of half(h)
50 cents, 50h
If equals are added to equals, the results are equal
Axiom
A mathematical argument that appears to prove something that we know is incorrect.
Fallacy
It is an algebraic expression consisting of two terms.
Binomial
“Googol” is one of the smallest large numbers. What does it stands for?
1 followed by hundred 0s or 10 raised to 100
Irrational numbers are also known as?
transcendental numbers
A number which is divisible by the sum of its own digits is called
Harshad Number
Who introduced the multiplication symbol “X” in mathematics?
William Oughtred
Who introduced the symbol “=” for equality?
Robert Recorde
Who invented the symbol “n!” for factorial of n?
Christian Kramp
Who gave the symbol “i” for √-1?
Leonard Euler
The number o.123123123… is a/an
Rational Number
MCMXCIV is the Roman Numeral equivalent to
1994
Any combination of symbols and numbers related by the fundamental operation of
algebra is called a/an
Algebraic Expression
What is the identity element for addition?
0
What is the identity element for multiplication?
1
If a = b =a. This illustrates which axiom in Algebra?
Symmetric Axiom
In algebra, the operation of root extraction is called
Evolution
