Basic concepts

Álgebra: variables y constantes

In Algebra we use symbols to represent quantities. These symbols can be divided into two groups:

1. Constants: These are quantities that don't change in value, like the number π that will always equal 3.14 no matter where we're, or like the number 100 will always be 100.

2. Variables: They are unknown quantities and are represented by the last letters of the alphabet: u, v, w, x, y, z. Although we can use the letter or the symbol that we prefer, it won't modify the results. These amounts may vary and we are the ones who give them values It's important that we know that although Mathematics is exact, there're many ways to reach the same result and Algebra gives us the freedom to explore different paths and rules to solve problems

Laws of signs (arithmetic and relation)

In Algebra we use the same signs as in Arithmetic. It is important to review how each of these is read and the different ways to express and write them

The sign is like having a first name and the last name; so let's not see the sign before the variable or constant, it will always be there

Sum: The sign for the sum is the "+", which is read more. It can also be called "addition" a + b is read “a plus b” Subtraction: The sign of the subtraction is “-”, which is read minus. It can also be called "subtraction" a - b is read “a minus b” Multiplication: The sign of multiplication is "x" which is read multiplied by. It can also be called "product" a x b is read “a multiplied by b” Another way to represent it is: a.b equals “a x b” (a)(b) equals “a x b” ab equals “a x b” Division: The division sign is “÷” which is read divided by. It can also be called "quotient" a ÷ b is read “a divided by b” Another way to represent it is: a/b equals “a ÷ b”

Relationship signs: We will use these signs to establish the relationship between two quantities:

=, then a=b is read “a equals b” ≠, then a≠b is read “a different than b” , then a>b is read “a greater than b” <, so a<b is read “a less than” ≥, so a≥b is read “a greater than or equal to” ≤, so a≤b is read “a less than or equal to”

Grouping Signs

The grouping signs will help us a lot and will be very helpful to avoid making mistakes and maintain order. The signs we are familiar with are parentheses “()”, brackets “[]” and braces “{}”.

They've got a hierarchical order, meaning that everything enclosed in parentheses must be resolved first, followed by everything enclosed in square brackets, and finally, everything enclosed in curly braces.

Laws of exponents: Multiplication and division

• When we've any expression or element raised to any power, we call the element the base, and the small number that appears above the base is called the exponent

• The function of an exponent is to tell us how many times our base is multiplied by itself

• In a term that has a variable and a constant being multiplied, we've to be very careful which exponent is affecting, because if it's accompanying only the constant, it's the one that will be multiplied by itself; if the entire expression is enclosed in parentheses and the power is outside, then the entire term must be raised to the indicated power. On the other hand, if the power accompanies the variable, only it will be affected

• Any expression that is raised to zero power will result in "1"

Laws of exponents:

• Multiplication: When we've got the same base and equal or different exponents. The rule is to add the exponents

• Division: It's easier to express division as a fraction because we identify our base and its exponent. The rule is that from the exponent in the numerator (the one on top) we subtract the exponent that is in the denominator (the one on the bottom)

Laws of Exponents: Power of a power and radiation

Laws of exponents:

• Power of power: We will have a base raised to a power and at the same time all of this will be enclosed in parentheses and raised to another power. The rule is to multiply both exponents

• Radiation: We change from a root to a fraction and vice versa. It is important to know that whenever we find an expression in the form of a root, it is better to convert it to a fraction so that it becomes easier for us to operate it. The rule is that the exponent of the base will be our numerator and the radical of our root will be the denominator

Algebraic language and algebraic expressions

Algebraic language: Set of symbols and rules used for the transmission of mathematical ideas

Algebraic language rules

• Each multi-stage combined operation must be preceded by the equals symbol “=”

• If the symbol = is followed by a fraction dash, it must appear midway between the two equals dashes

• The number 1 can be omitted when it is multiplying another number or when it is acting as an exponent

• The multiplication symbol can be omitted when parentheses appear after it, or when the product of two variables (letters) is indicated

Algebraic expression: Combination of numbers, letters and symbols of mathematical operations, which respects the rules of the algebraic language

n+1 is read as “the successor of n” n-1 is read as “the ancestor of n” 2n reads "integer always EVEN" 2n+1 and 2n-1 are read "integer always ODD" 2n and 2n+2 is read “2 consecutive EVEN" 2n+1 and 2n+3 is read "2 consecutive ODD" n² is read “perfect square of n”

Material: Basic concepts

Basic concepts

Algebraic expression: It's a combination of numbers, letters, and symbols of mathematical operations, which respects the rules of the algebraic language

Examples

Expression	Algebraic expression
addition of two numbers	$x + y$
subtraction of two numbers	$x - y$
product of two numbers	$xy$
quotient of two numbers	$\frac{x}{y}$
number increased by three units	$x + 3$
number reduced by five tenths	$x - 5(0.1)$
double of a number	$2x$
triple of a number	$3x$
fourth part of a number	$\frac{x}{4}$
third part of a number	$\frac{x}{3}$
cube of a number	$x^3$
square of a number	$x^2$
half of a number plus a quarter of another number	$\frac{x}{2}+\frac{x}{4}=x+\frac{x}{2}$
double of the sum of two numbers	$2(x+y)$
square of the sum of two numbers	$(x+y)^2$
sum of the squares of two numbers	$x^2+y^2$
sum of two consecutive numbers	$x+(x+1)$
product of three consecutive numbers	$x(x+1)(x+2)$
half of the sum of two numbers	$\frac{(x+y)}{2}$
double of the subtraction of two numbers	$2(x-y)$
subtraction of the doubles of two numbers	$2x-2y$
Carlos's age three years ago compared to his current age	$a-3$
perimeter of a square of unknown side	$4s$
two numbers that differ in thirteen units	$x=y+13; x-y=13$
a son is twenty-two years younger than his father	$s=f-22; f-s=22$
area of ​​a rectangle whose base is six meters longer than its height	$A=s(s+6)=s^2+6s$
triple of the square of a number	$3x^2$
Pedro is five years younger than Andrés	$p=a-5; a-p=5$
area of ​​a triangle	$A=\frac{bh}{2}$

Uncle Paco's farm is on a rectangular plot of 'l' meters long and 'a' wide. In it live 'c' pigs, 'v' cows, and 'g' chickens. It is requested:

Expression	Algebraic expression
area of ​​the plot	$A=la$
length of the fence surrounding the plot	$L=2(l+a)$
number of animals on the farm	$N=c+v+g$
number of legs of all the pigs	$P=4c$
number of legs of all the cows	$C=4v$
number of legs of all the chickens	$H=2g$
number of legs of all animals	$N=4c+4v+2g= 2(2c+2v+g)$

Challenges: Convert the following expressions to algebraic language:

Expression	Algebraic expression
subtract 5 from a number	$x-5$
twice a number	$2x$
square of a number	$x^2$
area of ​​a square of side r	$A=r^2$
price of a pair of pants increased by 15%	$P=p(1+0.15)=1.15p$
five times a number	$5n$
sum of a number and its square	$n+n^2$
perimeter of a square of side r	$4r$
17% of a number	$n(0.17)=0.17n$

Given two numbers, the first 'a' and the second 'b', it is requested:

Expression	Algebraic expression
sum of the two numbers	$a+b$
difference of the two numbers	$a-b$
product of the two numbers	$ab$
ratio of the first to the second	$\frac{a}{b}$
quotient of the second between the first	$\frac{b}{a}$
double the first	$2a$
double the first by the second	$2ab$
triple of the second by the first	$3ba$
double the sum of the two numbers	$2(a+b)$
triple the difference of the two numbers	$3(a-b)$