In the field of mathematics, the symbology given to numbers, variables and terms within an equation is very important. This makes a linear or non-linear sequence more understandable and solvable. This is one of the reasons why academic teachers make sure that students numbers have their corresponding positive or negative sign.
In the case of negative numbers, it’s an aspect of mathematics that often causes a lot of confusion for some reason. So today we will explain what negative numbers are, as well as how you can differentiate it from a positive number. In addition, you will know the most fundamental characteristics of them, without leaving aside that you will know what is a negative number in mathematics.
Meaning of a negative number in mathematics
They are all those integers, natural, in the form of a fraction or power that are identified with the negative mathematical sign (-). This is the easiest and most practical way to identify if a number is positive or not. On the other hand, if we take it to the field of geometry and algebra, it is possible to do the identification of the number graphically. To achieve this, you need to do the following:
- Draw a two-dimensional Cartesian plane.
- Locate the X axis and the Y axis.
- Take the numerical coordinates and translate them into the drawn plane.
All points in two-dimensional space have two values. These can be completely positive, negative, or a combination of both. That is why you have 4 quadrants in a Cartesian plane in 2D. To make it easier to identify, on the left side you have the negative numbers and on the right side, the positive numbers. This as long as you take the point of origin or “zero” value as a reference.
Of course, it is not necessary to draw a plane in two dimensions to identify the positive integers from the negative ones. A number line is more than enough. However, another way of identifying them is taking into account that everything Numbers greater than zero are positive, and any number less than zero is negative..
How are negative numbers different from positive numbers?
The first noticeable and visual difference is that the negatives have a sign in front of the number. On the other hand, positives may or may not have said sign. When it does not exist, it is intuited that there is a number greater than zero, although this will depend on the position of the positive number within an equation or algebraic expression. This applies perfectly for large numbers, to a certain power, radical, irrational or rational numbers, as well as independent variables and terms or in square binomials.
In the field of analytic geometry and algebra, the difference becomes noticeable when we express an ordered pair on a Cartesian plane. Bearing in mind that a 2-dimensional plane has 4 quadrants, two will be for positive numbers and the other two for negative numbers. Once such a combination has been drawn on the plan, it is possible to identify the position and whether it is a positive or negative value.
It must be taken into account that by the simple fact that a number has a minus sign in front, does not make it completely negative. That is, we can have the following mathematical expression: (-5)^2. This number, being raised to an even power, automatically makes it a positive expression. Like this one, there are many variants of negative numbers that are not completely negative.
What is the first negative number?
If we use a number line or a two-dimensional plane, we will notice that the The first number that appears to us is (-1). As we move away from zero as the central or reference value, the values decrease. Keep in mind that, unlike positive integers, in negative integers, as you move away from zero, the value has less value or quantity. That is to say, that (-1) is greater than (-20), contrary to everything that happens with numbers greater than zero.
This can be applied for both the X axis and the Y axis of a 2 or 3 dimensional Cartesian plane, there is no difference. In fact, the same principle applies to each of the scientific fields in which one works with numbers and quantifiable values.
What was the origin of negative numbers?
Few people or almost no one knows the origin of negative numbers. Until now, I myself thought that it was something taken out of pure logic. It turns out that this numerical set has its origin in the sixth century after Christ. It was thanks to an astronomer of Indian origin that he considered negative numbers as a way of expressing an idea. This was based on nothing or the debts that people had at that time.
However, this idea did not have much relevance until the era of the renaissance began. It was here that the use of the negative sign began to be part of the world of mathematics. It is in this period of our history that it was established that any number less than zero or nothing had to be negative. These, in general, were expressed with a kind of black rod, while positive numbers were expressed with a red rod.
What are the characteristics of negative numbers?
The main characteristic of this numerical set is that they are all real numbers less than 0. This implies that both integers, as well as rational and irrational numbers, have this same characteristic. An interesting aspect is that every negative number that is inside an absolute value will return a positive numberalthough this is quite a complicated operation that requires understanding.
On the other hand, every negative number has the peculiarity of becoming positive as long as they are raised to an even power or a multiplication is made by another number of the same nature. Let’s see the following examples:
- If we have (-1)*(1), the result will be a negative number
- In situations where we have a product, but with the following structure: (-1)*(-1), the result will become positive.
This simple change is due to the property or law of signs, which tells us that if two negative numbers are within a product or division, the result will be positive. Now, in terms of “sizes” or magnitudes of the values, the greater between a positive and negative number will be the one with the “+” sign. Now, if a comparison is made in the same negative quadrant, the largest will be the one that is closest to zero.
Examples of Negative Numbers in Mathematics
Negative numbers can be found in equations with independent variables and terms with variables. They can also be found reflected in fundamental theorems of calculus, as well as mathematical operations. For example, if we have a binomial like the following: Y = 2x – 10x, we have that the result will be a negative value, since the greater term has a negative sign. On the other hand, if we have the same equation, but with the following difference: Y = 2x – 10y, here is the result cannot be minimized to its bare minimumsince they are terms with different variables.
However, it must be taken into account that not always it is possible to have a negative number that exists and can be solved. That is, there are certain conditions in which one can work mathematically with these numbers. One of them is that it is not inside a root, otherwise it would no longer be an ordinary number, but instead it becomes a complex or imaginary number. In these cases, its resolution becomes more complicated and the methods vary depending on the matter and the field in which they are used.