The invariant property holds for subtraction and division among the four fundamental arithmetic operations but not for addition or multiplication.

The invariant property of subtraction states that if we add or subtract the same number to the minuend and the subtrahend, *the difference does not change*. In symbols, given three numbers 𝑎, 𝑏 and 𝑐,

The distributive property applies to **binary operations** such as multiplication and, partially, division.

In the case of multiplication, the distributive property states that, given three numbers 𝑎, 𝑏 and 𝑐,

The associative property applies to some **binary operations**, including addition and multiplication. However, it does not apply to subtraction and division nor exponentiation.

This property establishes that the order in which a **sequence** of two or more operations of the same type is performed does not change the result, as long as the order of the **operands** remains the same.

Since the use of parentheses in an expression determines the **order** in which calculations are done, it follows from the associative property that the use of parentheses in specific sequences of operations is *irrelevant*. Thus, we can eliminate those parentheses…

Division is a **binary operation** defined as the **inverse** of multiplication. More precisely, dividing a number *a* by a number *b* **other than zero** means finding a third number *c* which, when multiplied by *b*, yields *a*. In symbols, 𝑎 ÷ *b = c *if and only if *b* × *c* = *a*. For example, 18÷6 = 3 because 6×3 = 18.

The three numbers have specific names. The **dividend** is the quantity to be divided; the **divisor** is the number by which it is divided; the **quotient** is the result of the operation. …

Subtraction is a **binary operation** that consists of decreasing a given number, called the **minuend**, by an amount equal to that indicated by another number, called the **subtrahend**. The minus sign (−), which is the subtraction **operator**, connects the two numbers. The result of the operation is called the **difference**.

Subtraction is defined as the **inverse** of addition. More precisely, the difference of two numbers is that number which, added to the subtrahend, returns the minuend. For instance, if 12 is the minuend and 5 the subtrahend, the difference is that number which, added to 5, gives 12, i.e., 7…

An operation is called **binary** when it combines two elements called **operands** to produce a third element: the result of the operation.

The **commutative property** applies to some binary operations. It states that the result of these operations does not change if we *reverse *the order in which the two operands are combined.

It applies to both addition and multiplication.

Given two numbers 𝑥 and 𝑦, the commutative property of addition states that 𝑥 +* y = y + x*. Its application is pretty simple. For example, 5 + 8 = 13; symmetrically, 8 + 5 = 13.

Mathematics has enormously broadened its horizons in the last centuries, and multiplication has followed its path, finding application in gradually different fields. Multiplication is possible between integers, fractions, real numbers, complex numbers, polynomials, sets, vectors, matrices, and more. But the most elementary and ancient form of multiplication is and remains that between natural numbers.

This fundamental type of multiplication is based upon the idea of a **recursive addition**, i.e., an addition repeated as many times as required by the factor by which a number is multiplied. For instance, what does it mean to multiply 4×6? It means *repeating* the addition…

Of the four, addition is undoubtedly the most **basic** operation, considering that the other three are defined through a logic chain that starts from addition: subtraction is defined as the inverse of addition, multiplication as a recursive addition, division as the inverse of multiplication.

Although it is a basic operation, performing an addition requires sophisticated mental skills. To understand why let’s imagine a concrete case. In a primitive society, thousands of years ago, a farmer collected the eggs laid by her hens and placed them in some containers of different sizes. One contains four eggs, another six, another ten, and…

The numbers form ordered sets, within which precise ratios and relationships exist. For example, within the natural numbers, 4 is the next of 3 and the previous of 5; 9 is less than 27; 11 is greater than 10. These relations are fixed and immutable and can be expressed in a purely symbolic way (e.g., 9 < 27; 11 > 10).

However, our understanding of numerical proportions and properties can be improved and facilitated if the mathematical symbols that express them can be associated with a **spatial representation** of those connections. …

Precisely because of their close connection with material things that can be counted (goods, animals, people, etc.), some mathematicians hold that the series of natural numbers must begin with 1. Others think, however, that it must start with 0, partly because this choice allows a rigorous formalization of the concept of natural numbers, such as that provided by the so-called Peano axioms.

Because of this ambiguity, the symbol used to denote the set of natural numbers is often accompanied by a clarifying character in subscript or superscript:

- ℕ₀ indicates that the series starts from 0; in this case, the set…

Science writer with a lifelong passion for astronomy and comparisons between different scales of magnitude.