Algebra 1

Algebra

It is branch of mathematics which deals with the study of formal manipulations of equations involving numbers and numerals (signs or symbols for graphic representation of numbers).

Al-jabr - Restoration

The word Algebra is taken from the Arabic word "___", which means _____.

Algebra

One of the earliest mathematical concepts was to represent a number by a symbol and to represent rules for manipulating numbers in symbolic form as equations.

Algebra

Science of restoration, completion, reduction, or balancing.

Number

It is an entity describing the quantity or position of a mathematical object or extensions of these concepts.

Number

An item that describes magnitude and position.

Cardinal and Ordinal Numbers

Two distinct type of numbers.

Cardinal Numbers

It refers to the magnitude (size) and quantity.

Ordinal Numbers

It refers to the position relative to an ordering.

Numerals

Signs or symbols for graphic representation of numbers.

Roman Numerals

Consist of seven symbols written in Latin Alphabet.

Number System

It is a simple tree diagram showing the classifications of numbers.

Positive Numbers

Natural or counting numbers

Integers

Negative, zero, and positive numbers

Rational Numbers

Integers and non-integers

Rational Number

Any number that can be expressed in the ratio of 2 integers a to b or (a/b).

Non-Integers

Fractions, non-repeating and terminating decimal, repeating and non-terminating decimal

Real Numbers

Irrational and rational

• Axiom

• Theorems

The properties of real numbers are simply the rules where many of which are ____ (assumptions), and ____ (consequences of these assumptions) that are to be followed in working with mathematical expressions or equations.

Transcendental Numbers

Other term for irrational numbers

Irrational or Transcendeantal Numbers

Constants like pi and e

√2

What is the first known irrational number?

Christian Kramp

Who introduced the factorial sign “!”?

Robert Recorde

Who introduced the equal sign “=”?

Neither

Is one (1) a composite number or prime number?

Twin Prime

A prime number that is either 2 less or 2 more than another prime number. For example, (41, 43) or (11, 13).

Emirp

A prime number that results in a different prime when its decimal digits are reversed.

Palindromic Prime (Pal Prime)

Both prime numbers that reads the same forwards and backwards that only divisible by 1 and itself.

Surd

It is under irrational numbers; represents with the radical symbol like √3, 1+√6, √3+√6

Complex Numbers

Real and Imaginary numbers

Gaussian Integer (Number Theory)

The complex numbers are generally called?

Argand Diagram

Complex Numbers are plotted in _____.

Complex Numbers

It is in the form of a+bi (i.e. 1+6i)

Leonhard Euler

Who introduced the i with the value of √-1?

Composite Numbers

Numbers with more than 2 factors (i.e. 4, 6, 9, 12)

Prime Numbers

Numbers having only 2 factors (unity and itself)

2²¹⁶⁰⁹¹-1

What is the highest known prime?

2

What is the only prime even number?

Because it only has 1 factor

Why does 1 is not a prime number?

Complex Number

The sum of real and imaginary number.

a = real part

b = imaginary part

i = √-1

What are a, b, and i in complex numbers?

Pure Imaginary

In a+bi, when a=0, the number is called?

Real

In a+bi, when b=0, the number is called?

Irrational Numbers

Numbers that cannot be expressed as a ratio of two integers.

Happy Number

A number which will yield 1 when it is replaced by the sum of the square of its digits repeatedly.

3 = index

1 = radicand

√ = radical

Parts of ∛1

1. ⁿ√a^m = (ⁿ√a)^m

2. ⁿ√a * ⁿ√b = ⁿ√ab

3. (ⁿ√a)ⁿ = a

4. ⁿ√a/ⁿ√b = ⁿ√a/b

5. ^m√(ⁿ√a) = ^mn√a

Laws of Radicals (5)

Common Factor

A ____ of two or more counting numbers which is a factor of each of the given number. The set of _____ of the two numbers is the instersection of these two sets.

Greatest Common Factor (GCF)

The largest counting number which is a factor of each of the given number.

Greatest Common Factor (GCF)

The product of the smallest prime factors common to both.

Multiple

The _____ of a number is the product that the number gives when multiplied by a counting or natural number.

Least Common Multiple (LCM)

The smallest counting number which is a multiple of each of the given number.

Least Common Multiple (LCM)

The product of the prime factors with the highest power in the fatorization.

Lucky Number

The number that is generated by a certain “sieve”.

Perfect Number

A number that is equal to the sum of its proper divisors.

Proper Divisors

Positive factors of a number other than the number itself.

Proper Divisors

6 = 1 × 6

6 = 2 × 3

6 = (1, 2, 3)

1 + 2 + 3 = 6

Abundant Number

A number for which that sum of its proper divisors is greater than the number itself.

Abundant Number

12 = (1, 2, 3, 4, 6)

1 + 2 + 3 + 4 + 6 = 16

Abundance Value

The diffenrence of the sum of its proper divisor to the original number.

Untouchable Number

A number that cannot be expressed as the sum of all the proper divisors of any positive integer including the number itself.

5

What is the only known odd untouchable number?

Most Beautiful Number

The value of the golden ratio.

Amicable Numbers

Two different numbers related in such a way that the sum of the proper divisors of each is equal to the other number.

Amicable Numbers

220 = (1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110)

1+2+4+5+10+11+20+22+44+55+110 = 284

284 = (1, 2, 4, 71, 142)

1+2+4+71+142 = 220

Friendly Numbers

Two or more numbers with a common abundancy index, the ratio between the sum of divisors and the number itself.

Friendly Numbers

Two numbers with the same “abundany” form a friendly pair: (140, 30)

Algebraic Expression

It is made up of variabes and constants along with algebraic operation and made up of terms.

Algebraic Equation

It is indicated by the sign “=”.

3 = numerical coefficient

x² = term

- = algebraic operation

x = variable

9 = constant

3x² - x + 9

1. a (x + y) = ax + ay

2. (x + y)² = x² + 2xy + y²

3. (x +y) (x - y) = x² - y²

4. (x - y)² = x² - 2xy + y²

5. (x + y)³ = x³ + 3x²y + 3xy² + y³

6. x³ + y³ = (x + y) (x² - xy + y²)

7. x³ - y³ = (x - y) (x² + xy + y²)

Special Products (7)

1. a^maⁿ = a^m+n

2. a^m/aⁿ = a^m-n

3. (a^m)ⁿ = a^mn

4. (ab)^m = a^mb^m

5. (a/b)^m = a^m/b^m

6. a⁰ = 1

7. a^-1 = 1/aⁿ

Exponential Properties (7)

Exponential Notation

An easier way to write a number as a product of many factors.

Logarithm

The inverse function of exponentiation.

Brigssian Logarithm or Common Logarithm

A logarithm with base equal to 10.

Ex.: log₁₀(x) = 4 → log(x) = 4

Naperian Logarithm or Natural Logarithm

A logarithm with base equal to e.

Ex.: log_e(x) = 4 → ln(x) = 4

• Logus

• Arithmus

Logarithm was derived from the word _____ (ratio) and _____ (number).

John Napier

Who introduced logarithm?

Henry Briggs

Who is the mathematician notable for changing the original logarithms into common logarithms.

Antilogarithm or Antilog

The inverse function of logarithm, used to find the original number (x) from a given logarithm value (y).

1. log_a(AB) = log_aA + log_aB

2. log_a(A/B) = log_aA - log_aB

3. log_a(A^c) = clog_aA

4. log_bx = log_ax/log_ab

5. logx = 0.4343lnx

6. logx = log₁₀x

7. lnx = log_ex

8. lne = 1

9. e^lnx = x

10. lnx = 2.302logx

Properties of Logarithm (10)

Characteristics

The integral part (whole number) of a common logarithm.

Mantissa

The non-negative decimal part.

Logarithm of a Number

Characterisitics + Mantissa

colog (x) = log (1/x) = -log x

Cologarithm

Fundamental Theorem of Algebra

States that every nonzero, single variable, degree n polynomial with complex coeffients has, counted with multiplicity, exactly n complex roots.

Quadratic Equation

Any equation that can be rearrange in the standard form.

• Ax² + Bx + C = 0, where A is not equal to 0.

x₁x₂ = C/A

Products of Roots formula

x₁ + x₂ = - B/A

Sum of Roots formula

Quadratic Formula

The mathematical formula in solving the zeros.

x = (-B ∓ √B²-4AC)/2A

Quadratic Formula

Discriminant

A parameter of an object or system calculated as an aid to its classification or solution.

Real and Equal Roots

If the discriminant is B²-4AC = 0?

Real and Distinct Roots

If the discriminant is B²-4AC > 0?

Imaginary Roots

If the discriminant is B²-4AC < 0?

Descartes’ Rule of Signs

A technique for determining an upper bound (maximum value) on the number of positive or negative real roots of a polynomial. It is not a compete criterion, because it does not provide the exact number of positive or negative roots (approximate).

Descartes’ Rule of Signs

The rule is applied by counting the number of sign changes in the sequence formed by the polynomial’s coefficients.