**What are the basic operations?**

The **basic operations** in mathematics they are addition, subtraction, multiplication and division. Some authors additionally consider three more operations: power, root and logarithm. These basic operations apply to both numbers and algebraic expressions.

When basic operations are carried out with numbers, it is called arithmetic. When carried out with algebraic expressions it is algebra. In both, mastery of basic operations is essential, as well as in the field of more advanced mathematics and its applications to other sciences.

In this sense, electronic calculators are of great help, despite this, it is highly recommended to experience the resolution «by hand» of these operations to fully understand the meaning and usefulness of each one.

Let’s look at the 7 main types of basic operations:

**sum or addition**

The addition consists of adding or joining elements of a similar nature. Let the values “a” and “b”, which when added give the number c as a result:

**a+b=c**

The quantities a and b are called *addends*and the result c is called *addition*. For example:

5 + 3 = 8

**Examples of sums**

1 + 3 = 4 4 + 4 = 8 8 + 5 = 13 13 + 6 = 19

**Addition Properties**

**commutativity**

The order of the addends does not alter the sum, that is:

**a + b = b + a**

5 + 3 = 3 + 5 = 8

**associativity**

The order in which the addends are grouped does not change the result. For example, if there are three addends, you can add the first two and add the last to the result. Or you can add the last two and to which is added the first, like this:

**(a + b) + c = a + (b + c)**

(10 + 4) + 25 = 10 + (4 + 25) = 39

**neutral element**

It is the element that when added to another results in this second element. That value is 0, since:

**0 + a = 0**

0 + 5 = 5

**Opposite**

The opposite of a number is one that when added to it gives 0 as a result. If the number is “a”, its opposite is “−a”, so that:

**a + (−a) = 0**

12 + (−12) = 0

**subtraction or subtraction**

Let be a number «a», which receives the name of *minuendo*because its value will decrease according to another number “b”, called *subtrahend*. The subtraction consists of taking away from «a» the quantity «b», to give rise to the new quantity «c», called *subtraction*, *subtraction* either *difference*:

**a − b = c**

If the subtraction is carried out with natural numbers, the minuend is always greater than the subtrahend.

7 − 3 = 4

But subtraction can also be carried out with whole, fractional, real or complex numbers, if it is defined as *the sum of the opposite* and the law of signs is conveniently applied:

**a − b = a + (− b)**

Where (− b) is the opposite of b. For example, suppose you want to do the subtraction:

3−14

Then it is expressed as the sum of the opposite of 14, which is − 14:

3 + (−14)

And the law of signs says that when adding two numbers with different signs, the greater and the lesser are subtracted, and the sign of the greater is placed on the result:

3 + (− 14) = − 11

It is important to note that subtraction is not commutative, that is to say that in general:

**a − b ≠ b − a**

**subtraction examples**

10 − 3 = 7 20 − 7 = 13 13 − 8 = 5 30 − 20 = 10

**multiplication or product**

Between two quantities “a” and “b”, called *factors*, its product consists of adding b, as many times as the value of a indicates. The multiplication is denoted with the symbol “×” or with the point at half height “∙”:

**a × b = a ∙ b = c**

For example, the product 4 × 6 means that 6 must be added four times:

4 × 6 = 6 + 6 + 6 + 6 = 24

Or alternatively, the 4 can be added six times, to obtain the same result, since the order of the factors does not change the product:

4 × 6 = 4 + 4 + 4 + 4 + 4 + 4 = 24

**multiplication examples**

7 × 3 = 21 8 × 6 = 48 9 × 3 = 27 5 × 5 = 25

**multiplication properties**

**commutativity**

The order of the factors does not alter the product, as stated before:

**a × b = b × a**

3×5 = 5×3=15

**associativity**

When you have the product of three or more factors, you can group it in the most convenient way:

**(a × b) × c = a × (b × c)**

(4 × 3) × 7 = 4 × (3 × 7)=84

**neutral element**

When multiplying any value by the neutral element, the value is not altered, so said neutral element is 1:

**a × 1 = a**

5 × 1 = 5

**reciprocal or inverse**

The multiplicative inverse of an element is another value such that the product of both is 1. Let the element be “a”, then its reciprocal is:

Given that:

For example, the reciprocal of 2 is:

#### **Distributive property with respect to addition**

If a number “a” is multiplied by the sum (b + c), the multiplication can be distributed among the addends like this:

a × (b + c ) = a × b + a × c

As an example:

3 × (10 + 12) = 3 × 10 + 3 × 12 = 30 + 36 = 66

**Division**

It consists of distributing an amount called *dividend* among others, which is *divider*Being the *quotient* the result of the operation. To denote it, the symbols are used interchangeably: “÷”, “:” and “/”, with the dividend to the left of the symbol and the divisor to the right.

The division can be exact if the divisor is contained exactly in the dividend a certain number of times, but if not, there is a part left over, called the dividend. *residue*.

Let «a» be the dividend, «b» the divisor, «c» the quotient and «r» the remainder, then:

Equivalent to:

a = (b × c) + r

For example:

7 ∟3

1 2

In this example, a = 7, b = 3, c = 2 and r = 1, and indeed it is verified that:

7 = (3×2) + 1 = 6 + 1

Regarding the division, it is important to note that:

In general a ÷ b ≠ b ÷ a, therefore the division is not commutative. The dividend can be any number including 0, but 0 divided by any value is always 0: 0 ÷ b = 0 Division by 0 is undefined, so the divisor can have any value except 0.

**Division Examples**

9 ÷ 3 = 3 21 ÷ 3 = 7 40 ÷ 2 = 20 100 ÷ 4 = 25

**empowerment**

Potentiation consists of multiplying an expression, called *base,* by itself a certain number of times, given by a value *no* called *exponent*. If the base is «a», then:

**an = a × a × a… × a**

Examples of powers are:

23 = 2 × 2 × 2 = 8

(−3)4 =(−3) × (−3) × (−3)× (−3)= 81

It must be taken into account that both the base a and the exponent n can be real numbers including 0. Powers follow these laws:

an × am = an + m an ÷ am = an − m (an)m = an∙m a0 = 1 a1 = a an∙bn = (a∙b)n an ÷ bn = (a ÷ b)n

If the exponent is negative, it can be rewritten like this:

For example:

And if it is a fraction, it can be written as a root, as will be seen in the next section.

**filing**

It is the inverse operation of potentiation. For example, if a certain number x raised to the exponent n is a:

xn = a

So the value of x is:

Where «a» is the subradical quantity and «n» is the root index. For example:

Since 33 = 27

The general way to write a root as a fractional exponent is:

The root index is the denominator of the fraction in the exponent, and the numerator is the power of the subradical quantity. For example:

**logarithms**

To find out how much «n» is worth in the expression bn = c, the operation called *logarithm*. A logarithm is therefore an exponent:

n = logb c

The value of «b» is called the base of the logarithm.

For example, it is known that 23 = 8, therefore it is written:

3 = log2 8

Which reads like this “The base 2 logarithm of 8 is equal to 3”, which means that the logarithm is the exponent to which the base must be raised to obtain the number.

Another example:

81 = 34

Therefore 4 is the exponent to which 3 must be raised to obtain 81:

log3 81 = 4

It is important to highlight the following aspects:

There are no logarithms of negative numbers nor of 0. The base is always positive

**Properties of logarithms**

**logarithm of the base**: logb b = 1, since b1 = b

**The logarithm of 1 is 0**since any number raised to the power of 0 is equal to 1: logb 1 = 0.

**Product**: logb (a∙b) = logb a + logb b

**Quotient:** logb (a÷b) = logb a − logb b

**Power**: logb(an) = n∙logba

An example of applying the logarithm of the product is the following:

log10 (2∙4) = log10 2 + log10 4 = 0.30103 + 0.60206 = 0.90309

The base 10 logarithm or decimal logarithm is one of the most used. On any scientific calculator it appears simply as “log”. The reader can check the result with a scientific calculator or with any online calculator.

**References**

Baldor, A. 2007. Practical Theoretical Arithmetic. Grupo Editorial Patria SA de CV Math is Fun. Basic Math definitions. Recovered from: mathisfun.com. Math e Mania. Basic Math operations. Retrieved from: mathemania.com Superprof. Operations in Mathematics. Recovered from: superprof.es. Universal Class. The four basic mathematical operations. Retrieved from: universalclass.com.