Algebra and Complex Numbers

Objective and formulas

______ is an item that describe a magnitude or a position

Number

______ and _______ are the two types of number.

Cardinal numbers and ordinal numbers

_______ are numbers which allow us to count the object or ideas in a given collection.

Cardinal number

______ state the position of the individual object sequence.

Ordinal numbers

______ are symbols or combination of symbols which describes a number.

Numerals

The most widely used numerals are ______ and the ______

Arabic numerals and roman numerals

______ were simply the modification of the hindu-arabic number sign and atr written in Arabic digits.

Arabic numerals

_______ are numbers which are written in latin alphabet.

Roman numerals

Roman numerals and their equivalent arabic number

I = ______

1

Roman numerals and their equivalent arabic number

V = ______

5

Roman numerals and their equivalent arabic number

X = ______

10

Roman numerals and their equivalent arabic number

L = ______

50

Roman numerals and their equivalent arabic number

C = ______

100

Roman numerals and their equivalent arabic number

D = ______

500

Roman numerals and their equivalent arabic number

M = ______

1000

_____ to multiply it by 100

Bracket

_____ to multiply the number by 1000 times

Vinculum (bar above number)

______ to multiply the number by 1000000 times

Door frame

______ is a specific symbol or symbols used alone or in combination to denote a number.

Digit

Two categories for number system: ______ and ______

Real Numbers and Imaginary Numbers

What are the Real numbers?

1. _______

2. _______

3. _______

4. _______

1. Natural numbers

2. Integers

3. Rational numbers

4. irrational numbers

______ are numbers considered as counting numbers

example: 1, 2, 3, ….

Natural numbers

_____ are all the natural numbers, the negative of the natural numbers and the number zero.

Integers

______ are numbers which can be expressed as a quotient (ratio) of two integers.

example: 0.5, 2/3 , -3, 0.333….

Rational numberes

The term “rational” comes from the word _______

ratio

_____ are numbers which cannot be expressed as a quotient of two integers.

example : sqrt of 2 , pi , e , …..

Irrational Numbers

An _____ is denoted as i .

Imaginary number

Imaginary number

i = ________

Imaginary number

Sqrt of -1

Imaginary number

i² = ________

Imaginary number

-1

Imaginary number

i³ = ________

Imaginary number

= -i = - sqrt of -1

Imaginary number

i^4 = ________

Imaginary number

= 1

_______ is an expression of both real and imaginary number combined.

Complex numbers

what is the form of complex number?

Form of Complex Numbers

a + bi

Complex Numbers

a + bi , if a = 0 , then ______

Pure imaginary number

Complex Numbers

a + bi , if b = 0 , then ______

Pure real number

_______ of a real number is the numerical value of the number neglecting the sign.

example. I -5 I = 5

absolute value

_______ are numbers which are in the form of a/b ,

where a = numerator ; b = denominator

Common fraction

If the numerator is smaller than the denominator , it is called _______

Proper fraction

when the numerator is greater than the denominator called ______

Improper fraction

________ are common fraction with unity for numerator and positive integer for the denominator.

example : 1/5 , 1/25

Unit fraction

_______ is a number that can be written as product of two or more integers, each greater than 1.

Example: 60 = 2 × 2 × 3 × 5

Composite number

______ is an integer greater than 1 that is divisible only by 1 and itself. According to 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 power of primes

example of prime number : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …….

example of unique product of primes : 360 = 2³ . 3² . 5^1

Prime number

_____ are prime that appear in pair and differ by 2

example : 3 and 5, 11 and 13, 17 and 19 …..

Twin primes

_______ is an integer number that is equal to the sum of all its possible divisors, except the number itself.

Example: 6, 28, 496 …..

6 = 1 + 2 + 3

Perfect number

________is an integer number, the sum of all its possible divisor is less than the number itself.

Defective or deficient number

if the sum of the possible divisors is greater than the number, it is referred to as _______

Abundant number

________ refers to two integer numbers where each is the sum of all the possible divisors of the other.

Amicable number or friendly numbers

What is the smallest known amicable number ?

220 and 284

______ denoted as n!, represents the product of all positive integers from 1 to n.

Factorial

(n!)(n + 1) = (n +1)!

This is known as ________

Recursion formula

_________ are digits that define the numerical value of a number.

Significant figures or digits

What are the two forms of approximations ?

Rounding and truncation

_______ means replacing the number with another number having the fewer significant decimal .

Example :

3.14159 shall be rounded up to 3.1416

3.12354 shall be rounded down to 3.1235

Rounding

_____ refers to the dropping of the next digits in order to obtain the degree of accuracy beyond the need.

example :

3.14159 is truncated to 4 decimal as 3.1415

Truncation

Revolution and its equivalent in units of angle

1 Rev = ________ deg.

360

Revolution and its equivalent in units of angle

1 Rev = ________ radians

2pi

Revolution and its equivalent in units of angle

1 Rev = ________ grad

400

Revolution and its equivalent in units of angle

1 Rev = ________ mils

6400

Revolution and its equivalent in units of angle

1 Rev = ________ centissimal degree

6400

Revolution and its equivalent in units of angle

1 Rev = ________ gons

6400

Temperature

Relation between Celsius and Fahrenheit

deg. Celsius = ________

= 5/9 ( deg F - 32 )

Temperature

Relation between Celsius and Fahrenheit

deg. Fahrenheit = ________

= 9/5 ( deg. C + 32 )

Temperature

Relation between Celsius and Kelvin

deg. Kelvin = ________

= deg. C + 273

Temperature

Relation between Rankine and Fahrenheit

deg. Rankine = ________

= deg. F + 460

Density of water = ________ kg/ cubic meter

1000

Density of water = ________ lb / cu.ft

62.4

Density of water = ________ N/ cubic meter

9810

Density of water = ________ gram / cc

1

the symbol ________, which is the ration of the circumference of a circle to its diameter was introduced by william jones in 1706 after the initial letter of the greek word meaning “periphery”.

pi

_______ is a radical expressing an irrational number. The ____ is described after the index of the radical.

Surd

_______ contains no rational number and all its terms are surd.

Example: sqrt of 3 + sqrt of 2

Pure surd or entire surd.

_______contains at least one rational number.

Example: 5 sqrt of 3

MIxed surd.

_____ is an expression of two terms with at least one term is a surd.

Example : 5 + sqrt of 2

Binomial surd

_________ is an expression of three terms with at least two of them are surds and cannot be expressed as a single surd, otherwise it will become a binomial surd.

Trinomial surd

a:x = y:d

a & d = ________

x & y = _________

a & d = extremes

x & y = means

a:x = a/x

a & a = ________

x & x = _________

a & a = antecedent

x & x = consequent

_________ refers to the product of several prime numbers occurring in the denominators, each taken with its greatest multiplicity

Least common denominator (LCD)

______ refers to the smallest integer that is a multiple of each given numbers.

Least common multiple (LCM)

________ refers to the largest integer which is a factor of each of the given number

Greatest common factor

_______ is an expression or an equation that contains the variable squared, but not raised to any higher power

Quadratic

Ax² + Bx + C = 0 , what equation?

Quadratic equation

Ax² + Bx + C = 0

When B= 0 is known as ______

Pure quadratic equation

A quadratic equation in x is also known as a _______

Second degree polynomial equation

X = (-B (+ or -)sqrt of (B² -4AC) ) / 2A

_______ formula

Quadratic formula

Sqrt of (B² -4AC) is known as the ______

Discriminant

Sqrt of (B² -4AC) = 0

Therefore : _______

Only one root

Sqrt of (B² -4AC) = >0

Therefore: ________

Real and unequal

Sqrt of (B² -4AC) = <0

Therefore:_______

Imaginary and unequal

Sum of the roots

r1 + r2 = ________

- B/A

Product of the roots

r1 . r2 =_______

C/A

______ is an expression containing two terms joined by either + or -

Binomial

________Gives tne results of raising a binomial expression to a certain power

Binomial theorem

_________. The coefficient of a binomial expansion can also be conveniently obtained by arranging them in a triangular array or pattern.

Pascal's Triangle

Coefficient of any term

C= _______

C = (( coeff. Of preceding term(Pt)) ( exponent of x of Pt ) ) / (exponent of y of Pt - 1 )

r^th term of the binomial expansion

r^th = ________

r^th = nCb (C)^a (D)^b

_______ or equation with only one variable refers to the exponent of the variable.

Degree of polynomial

______ a number or variable x to base b , log b x ; is the exponent or b.needed to give x.

Logarithm

Logarithm comes from greek word “ _____” meaning ratio and ______ meaning number.

Logus

Arithmus

Logarithm with base e ( log e or ln ) is called ______

Natural or the napierean logarithm

The logarithm with base 10 is known as _____

Common logarithm or the Briggsian logarithm