Expressing in algebraic form is the beginning of the mathematical basis, as a mathematician is not a calculator.

The mathematician works with general and non-specific ideas. For example, when we express “two more than five”, it is a specific thought, as we are adding two to five.

In mathematics, thinking is more general, in the example mentioned above, a mathematician says “two more than $x$”.

$$x+2$$

In this case, we add “two to any number $x$”, where this number can be included in the set of natural numbers

$$x+2,\quad x\text{ pertence ao conjunto dos números naturais } x\in \mathbb{N} = \\{0,1,2,3,\dots\\}$$

The bottom line

The origins of the morphology of a variable $x,$ that originated from a lexical paranomasia (word similarity) syntactic and semantic phonos of that letter.

This letter has origins from the linguistic metaphorical plane and was heavily used in mathematics.

First Reason

Around 166 algebra authors spelled terms $x_1, x_2, x_3 ...$ belonging to

$$\cancel{1}, \cancel{2}, \cancel{3}, \dots$$

Mathematician, philosopher and writer Pietro Antonio Cataldi wrote $x$ in two distinct and curious ways:

  • 1 cut by inclined hairline
  • letter x cut by perpendicular thin line.
Grafiko's Galileo.

Second Reason

In the Low German 11th century, which gave rise to Old English 13th century, we find the word Shei transcribed by Xei, which designated something unknown, enigmatic, ignored, …, unknown.

Grafiko's Galileo.

Third Reason

In the same semantic phonetic linguistic trunk that occurred in Europe, Low German, Old English (14th century) the facts, dates or references to Jesus Christ They were written in two basic ways:

  • Christmas
  • ✝but
Grafiko's Galileo.

Algebraic expressions in different operations

Next, we present algebraic expressions in different mathematical operations. In the previous section, we talked about the algebraic variable used in mathematics, which is $x.$

It is worth remembering that in mathematics, we cannot be stuck with this letter. In this case, we can use $x$, $y$, $z$, $k$, among others. Just note that it is a variable or we call it an unknown (something we don’t know the value of).

Mathematical Operation Description Algebraic expression
$+$ Two more, Added to two, two more, plus two $x+2$
$-$ Minus one, Subtracted from one, one less, reduced from one $y-1$
$\times $ multiplied by 5, product of $k$ by 5 $5\times k$
$\div $ divided by 3, quotient of $b$ by 3 $b\div 3$

Factors

To define the term factor, let’s look at the example:

$$9x$$

In this case, $9$ and $x$ are factors. In mathematics, there is a popular saying that:

  • The order of the factors does not change the product.
  • $8\times 5 = 5\times 8 = 40$
  • $8$ and $5$ are the factors of the product of these two numbers.

Coefficients

In the previous section we said that from $9x$ we have the factors $9$ and $x$. Now, let’s define a coefficient. If we have a constant multiplied by a variable factor. This constant is called the coefficient.

In the example $9x$, the value 9 is a coefficient.

Algebraic expressions with more than one variable

Next, we will define some concepts of one or more variables.

A variable

When we have an unknown, for example $x$ or any other letter, we say that the algebraic expression has a variable.

Example:

  • $x+1$
  • $y+5$
  • $z+15$

Two variables

When we have two unknowns $x, y$ or any other letter, we say that the algebraic expression has two variables.

Example:

  • $x+y+18$
  • $3xy+5$
  • $z+15y$
  • $\dfrac{x}{y}+9$

Three variables

When we have three unknowns $x, y, z$ or any other letter, we say that the algebraic expression has three variables.

Example:

  • $x+y+18+2z$
  • $8xyz+20$
  • $\dfrac{xz}{y}+23$

n variables

Like all mathematicians, let’s generalize our thinking. When we have $n$ unknowns $x_1, x_2, \dots, x_n$, we say that the algebraic expression has $n$ variables.

Example:

  • $x_1+x_2+\dots+x_n$
  • $x_1~.~x_2~.~\dots~.~x_n$
  • $\dfrac{x_1~.~x_2}{15} + (x_3-\dots - x_n)$

Numeric expressions

An important point in mathematics is following some orders. In operations with numerical expressions, we have to follow orders of mathematical operations. Below we will show the orders of operations:

  1. Multiplication or division
  2. Addition or subtraction

Multiplication

Next we will present a numerical example to apply the expression with multiplication.

$$\underbrace{5\times 3}_{\text{escolha primeiro} } + 8 = 15 + 8 = 23$$

What if we make the wrong choice?

See what happens:

$$5\times \underbrace{3 + 8}_{\text{nunca faça isso} } = 5\times 11 = 55$$

My god 😂

Division

Next we will present a numerical example to apply the expression with division.

$$\underbrace{15\div 3}_{\text{escolha primeiro} } - 2 = 5 - 2 = 3$$

What if we make the wrong choice?

See what happens:

$$5\div \underbrace{3 - 2}_{\text{nunca faça isso} } = 5\div 1 = 5$$

Oh shit!!!! 💩

Solving numeric expressions

In lagebraic expressions, the order of execution of the parts of the expression is fundamental. To make numerical expressions easier, we will follow the following order:

  • Parentheses.
  • Brackets.
  • Keys.

And the order of operations given in the previous section.

This way, let’s solve the example:

$$ \begin{aligned} ?? &= \{[(\underbrace{8 ~.~ 4}_{=32} + 3) \div 7 + (3 + \underbrace{15 \div 5}_{=3}) ~.~ 3] ~.~ 2 – (19 – 7) \div 6 \} ~.~ 2 + 12 = \\ &= \{[(\underbrace{32 + 3}_{=35}) \div 7 + (\underbrace{3 + 3}_{=6}) ~.~ 3] ~.~ 2 – (\underbrace{19 – 7}_{=12}) \div 6 \} ~.~ 2 + 12 = \\ &= \{[\underbrace{35 \div 7}_{=5} + \underbrace{6 ~.~ 3}_{=18}] ~.~ 2 – \underbrace{12 \div 6}_{=2} \} ~.~ 2 + 12 = \\ &= \{\underbrace{[5 + 18]}_{=23} ~.~ 2 – 2 \} ~.~ 2 + 12 = \\ &=\{\underbrace{23 ~.~ 2}_{=46} – 2 \} ~.~ 2 + 12 = \\ &= \underbrace{\{46 – 2 \} }_{=44} ~.~ 2 + 12 = \\ &= \underbrace{44 ~.~ 2}_{=88} + 12 = \\ &= 88 + 12 =\\ &= 100 \end{aligned} $$

That’s it for here, wait for the next posts.