 # Early Operation Tools on Arithmetic System Section 02 ## Early Operation Tools on Arithmetic System Section 02

### Fraction

A number like 1, 2, 3, 5, 10, 120, and 3000 addressed as an integer number. If we do a multiplication between two integers, then we will always get an integer number. Like 5 times 6 or 5 multiplied by 6 will be giving us a result of 30, or 7 times 9 is 63. However, we may not always get an integer number in a result if we try to divide between one number of integers with another integer number. This is can be true once we divide 8 apples into 4 plates, then each place will have 2 apples. On the other hand, if we divided 8 apples into 3 plates, then we will not have an integer number. For example, 8 divided by 3, will give us `8/3` or 2.6667, which is what we will be called as a fraction for `8/3` , and 2.6667 will be addressed as decimal numbers and/or irrational numbers. This `8/3` doesn’t mean we need to divide 8 apples into three selections or groups and giving `8/3` to each place. We know and understand about `3/3` equals one apple, and `6/3` equals two apples. Therefore, to divide 8 apples into 3 equal plates, we will put 2 apples into each plate. Then we will divide the remaining 2 by 3 and we will put `2/3` apples into each plate. In mathematics, we can write those like `8 : 3 = 2 2/3` . The term `2 2/3` shows a group of numbers between integer and fraction which will be addressed as a group number of mixtures number.

In a fraction like `2/5` , the number at the top line position will be called the dividend, meanwhile, the number at the below line position will be called the divisor, and their result will be addressed as quotient. Mathematician never addressed `2/5` as two numbers but always addressing it as a number that has its own value. If we write this fraction `2/5` into another form of numbers, like a decimal number, which is 0.4 we will see about this fraction truly similar to a single number.

Each fraction indeed a form of quotient that has two integer numbers. Each fraction perhaps has been written under this kind of format or idea. First, we could write those fraction by written the divisor of 1, then by a divisor of 2, then by a divisor of 3, etc. like, this example:

Divisor 1, then … `1/1 , 2/1 , 3/1 , 4/1 , 5/1 ,` … etc.

Divisor 2, then … `1/2 , 2/2 , 3/2 , 4/2 , 5/2 ,` … etc.

Divisor 3, then … `1/3 , 2/3, 3/3 , 4/3 , 5/3 ,` … etc.

It surely clears to us if we wrote it in such a form then it will take an incredibly unpredictable amount of time. However, if we making such a list of fraction in a perfect form, then those any form of a fraction will always have a placed in the proper position.

Fraction numbers can also get influence by the fundamentals and derive arithmetic’s operational tools like addition, subtraction, multiplication, and even division. Then the answer that can be performed will be the integers number or the fraction itself, or a number of mixtures (a number that has integers and a fraction). During the process of division of fractions, then we have a new property, which is the divisor between two fractions that get divided will become a transverse version of it. By mean, if we have `2/5:7/9` , then we also can write it as `2/5 * 9/7` to lighten our calculation, now with the new form, we can easily calculate to find the result, which given us the result `18/35` for `2/5 : 7/9` .

In the form of a fraction like `18/35` , the divisor which is 35, in reality, is the value that we can divide since 18/35 also means `18 : 35` , because of that also the value of 0 will never become a divisor yet. This is happening to help us on avoiding the value that we haven’t yet been discovered or inventing. For example, if we allowing the divisor of 0, we may be able to prove about the value of 6 is 1. How? The logic terms like, we know about 6 multiplied by 0 are 0. For each section form of those format calculation divided by 0, we will get `( 6 * 0 ) / 0` which is become `0/0` , which also can be written as `6 * 0/0 = 0/0` . Now, we know for each number that divided by that number-self will always giving the result is 1, then we can change that calculation from `6 * 0/0 = 6 * 1` , which is giving us a result of 6, or 6 = 1. In the current concepts, once we need to divide by 0 then we may have to postpone it for a while, even though we have to postpone each calculation that has a divisor of 0, we will proving almost everything that currently exists on Earth.

### A Negative Integer

Let’s back to our integer numbers, if we see it we will know if we make a subtraction between two integers, then we will find out the result may not always positive integers. Even though, we can see on most of the examples about 7 subtract by 4 is positive 3. Meanwhile, if we want to do the opposite like 4 subtract by 7, then to make things possible we will need to be designing a new method for defining a number, which is addressed as a negative integer, and symbolic by a minus sign in front of those integers itself. Therefore, we could have a result from subtraction between 4 and 7 is negative 3 or written as `-3` .

A mechanism for solving the problems like 12 subtract by 6, 9 subtract by 7, six subtract by thirteen, etc. is quite simple. In the idea we just have to look at the bigger value then we take the differences between the bigger value and the smaller value, if there is a need on using a negative sign then we use it, based on the first number that gets subtracted by the second number. For example, twelve subtract by six is positive six, nine subtract by seven is positive two, four subtract by one is positive three, but six subtract by twelve is negative six, seven subtract by nine is negative two, and one subtract by four is negative three.

At this point, the possibility of total negative integers could be equals to the total positive integers. Started by the value of zero (0) then we can make a list of numbers, by going to the left side from our point of view to our screen or paper will be a negative integer meanwhile from the value of zero (0) by going to the right side from our point of view to our screen or paper will be a positive integer, at the end we will have a list like:

`..., -3, -2, -1, 0, 1, 2, 3, ...`

We also can widen those lists until a value that we want it. Also, with the idea of these negative numbers, in general, we will have a negative value for each existence of fraction, each decimal number, each irrational number, and many more. For the example like each positive value of fraction `3/5` , then this value will also have the opposite value, that is a negative fraction of  `3/5` or symbolical identification written as `-3/5` .

All these types for groups of numbers that we have identified up to this point like integers number, negative integers number, fractions, negative value, and zero, often get addresses as rational numbers. This means rational numbers like zero, or an integers numbers in format negative or positive value, or a fraction in format either negative or positive, in which the dividend and the divisor can be an integer. A value of 5 is a rational number, which means negative 8, `1/2` , `- 3/4` also integers numbers.

Some sets of properties have been made for the addition, subtraction, multiplication, and division of negative and positive rational numbers; therefore, such illogical value won’t be happening. Some examples from this property are product numbers between two factors of equals sign symbol in integers will give results as positive integers, like four multiplied by four is sixteen, as well as negative four multiplied by negative four, is positive sixteen. Some other property like a product number between two different sign symbols in factors of integers will giving a result as negative integers, in this case, will be like a positive four multiplied by a negative four then we will have a negative sixteen.

### Irrational Number

Like positive value have the opposite value called negative value, then the rational number also have the opposite number, called irrational numbers. Unlike positive and negative integers that only different in writing by sign symbols, then irrational and rational numbers have different in their value terms. Like a square root of twenty-five from a group of numbers, we know it as products from two similar factors. So, the square root of twenty-five is five, because five multiplied by five is twenty-five. Meanwhile, there is a group of numbers that exist and can’t be defined by the rational numbers, for example like the square root of two. We may able to identify the location of the square root of two is between 1.4 and 1.5 because the product of 1.4 to the power of 2 is 1.96 and the product of 1.5 to the power of 2 is 2.25. Then we could identify the result for the square root of two by adding more decimal points. Therefore like, 1.41 to the power of two which is 1.9881, and 1.414 to the power of two is 1.999396. However, no matter how many decimal points that we added, we wouldn’t be able to find a rational number that forming a square root of two, but we only can reach a number that closed enough to the value of the square root of two itself.

Moreover, we will meet more variants of irrational numbers that may become some standard and/or some new form of some numbers during our progress on dealing with mathematics. For example, one theorem from the theory of geometry that describing each side from the 90 degrees angle of a triangle, will be like a square of their two sides, or the hypotenuse is the squared root of the opposite and the adjacent. Let’s say in one right angle of a triangle, with the side adjacent is 1 and side opposite is 1, then the hypotenuse of that triangle will be hypotenuse = square root of adjacent power up by two + opposite power up by two, and because each value of adjacent and opposite is 1, then we will know about the hypotenuse of this right angle is the sqrt root of 2, written as `sqrt(2)` . Therefore, we know about this irrational number of the square root of 2 is a real existence, that can be found through mathematical arithmetic calculation even though our finding shows that number is not the rational number. Hereby, we called this type of number an irrational number.

There is quite a number that can be categorized as irrational numbers. For example, the square root of 3 is an irrational number. As well as the square root of 5, or `sqrt(5)` . Or the cube root of seven, or `root(3)(7)` . Those are a real number, even though at the current discovery of Mathematics in number, and arithmetic’s unable us on showing it into integers number or even the use of fractions that perform by the integers.

If we combine all the irrational numbers, then we will have a list of mathematical involutions of numbers in form of exponents that is likely such a real number. All of those systems of group numbering can be performed by an unending straight line. For example, one dot identifies as zero (0), then to the right side from zero of our point of view to the screen or paper will become a group of positive numbers, and to the left side from zero of our point of view to the screen or paper will become a group of negative numbers. If the space between an existence integers number we can fill it with any smaller number between them, then we will able to perform a subsection on each rational number. Meanwhile, in whatever numbers of fractional quantities that we can perform by keep on dividing it, but that emptiness between two values of existing integer numbers will remain there. Even though we try to fill the gap between two values of existing integer numbers by a possible irrational number like the square root of 2, square root of 5, and many more, then those straight lines can be filled with only more detail. This straight line can be addressed as a continuum.

### Imaginary Number (`i=sqrt(-1)` )

Likely positive number that has an opposite called a negative number, and a rational number has an opposite called an irrational number. Then there is also an existing that called imaginary number as an opposition form of a real number. The existence of this imaginary number likely a combination of existence between a negative number and the roots. Meanwhile, during the operational tools called subtraction form, we are realizing about a possibility that exists on a negative value, which we addressed as a negative number or negative integers, with also during the operational tools called multiplication form, we can see about a possibility to the existence of negative value, once we have two factors with different sign symbol. Lastly, we know about there is no limitation boundary on when and where we can do those operation tools, in terms we can do it anywhere, anytime, and anyhow by conveniently. Therefore, once we perform such a subtraction or a multiplication that giving us a negative value under the evolutions of the mathematical form called roots, then we will be finding a number like the square root of negative one, with a note, square root of negative one does not equal to the negative of the square root of one, `sqrt(-1)!=-sqrt(1)` .

This combination geometry in the format of the square root of negative one, somehow showing us to against the principal in squared power as we know on finding the real value from roots. Like the square root of twenty-five is either positive five or a negative five, the square root of two is an irrational number, but how should we have addressed the square root of negative one. Meanwhile, if we take a look into these roots, we will always find a factor inside those roots always get perform through two identical numbers with a singular sign of symbols, like two factors of positive six’s, or two factors of negative six’s, or else, and/or two similar numbers in composition, like two factors of one point four one four two, or two factors of one point seven three two, etc. Therefore, again in current mathematical discovery and invention, we know it on here, about there is no such perform that can give us a negative value from two multiplication of identical factors in a singular sign of symbols, and seemingly impossible to find roots from such a perform. However, a Mathematician have using the square root of negative one on designing a fundamental performance from such a group of numbers that can be called imaginary numbers with the close value of the square root of negative one, that often shown by the symbol “`i`

Some cultures were using this imaginary number to recognizing it as directions. Let assume if we pull two giant straight lines that perpendicular to each other, then we can take one straight line that along with our hand as positive integer section to the right hand and negative integer section to the left hand, by means the zero (0) is the center or the connection between two giant straight lines. This also, identifies us for the other straight line that perpendicular with our hands can become like a frontside and backside, which we can be using the positive imaginary number for the front side direction and negative imaginary number for the backside direction. Now, we can have our understanding through analogy in a direction in terms of that straight line along with hand as for +1 means a positive integer, which can be identified as one step unit to the right-hand side, as well as the -1 means a negative integer, which can be identified as one step unit to the left-hand side, then those not along with our hand, but perpendicular to our hand with a center is our body as zero(0), will have a meaning like `+i` means a positive imaginary number that identifies as one step unit to the front side, and `-i`  means a negative imaginary number that identifies as one step unit to the backside. Therefore, in general for that culture, once the community has seen a symbol that written as `5+3i` which means you can walk to the right for five steps then go to the front for three steps, as well as other symbols like `-2i-1` means you walk to the backside of you for two steps then go to your left for one step. One note we have to be known and realize about this system of direction is we need to decide beforehand on which direction will be our front, and we need to committing and consistent it with that direction.

At this moment there are several more groups of numbers, but in this article, I only show you some general information about this arithmetic’s that commonly to be found and seen during the process in whatever we do through Mathematics. In the next article, we will continue with information about the general algebra series.

Create or log in to your registered account on the alberttls website — a free research-reading-based learning platform that delivers differentiated information, to accommodate reader trouble statements, related information, and learning speeds. Aligned with curricula across the English-speaking world, it’s likely by variants of readers, learners, and educators, that been provided in our paid development course.

Therefore, without holding anymore, thank you for your reading and if you think this information is helpful, then feel free to let us know and if you don’t mind giving us support then feel free to send your donation to our PayPal, as a compliment, we will give you a digitally registered hard copy of this document with some additional information in it.

Reference: