# The best online calculators in 2020

When studying subjects that require complicated calculations, there is nothing better than having a good calculator. Although having a calculator in physical is very comfortable, it is not always accessible, which is why we must resort to online options, either surfing the web or as applications for the phone.

My recommendation for my students is to always study accompanied by a calculator, so they can check if they are doing the necessary calculations correctly.

Let’s see then, a list without a particular order (just kidding, they are ordered from the best to the worst) of the best calculators that we can get browsing the internet or in the store of applications of different operating systems.

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## Wolfram Alpha

With no doubt, Wolfran Alpha is the king of online calculators because its calculations are not only based on algorithms as traditional calculators do, but on innovative algorithms, knowledge base and artificial intelligence technology.

Nota: According to Atlassian, A knowledge base is a self-serve online library of information about a product, service, department, or topic. The data in your knowledge base can come from anywhere. Typically, contributors who are well versed in the relevant subjects add to and expand the knowledge base.

Performing a calculation in Wolfram Alpha not only provides the solution to the calculation, but also provides additional information usually needed when performing calculations. You can see the entire development step by step to reach the final result by paying a subscription, but it is not mandatory if you only want results.

Wolfram Alpha is available for free at wolframalpha.com and for a fee at iOS, Android and Microsoft.

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## Calculator N+

My daily use calculator is the Android Calculator N+ application. It is an open source calculator, developed by Trần Lê Duy who, according to his github profile, is a student at Nguyen Binh Khiem High School who loves to study algorithms.

Note: open source commonly refers to software that uses an open development process and is licensed to include source code.

This calculator provides results only, without procedures, but the amount of functions that can be applied is immense. I think the only defect it has (for now), is that it does not have a function finder in the home screen calculator.

In addition to the home screen calculator, this application has specific calculators to work with Equations, Derivatives, Integrals and Matrices, among others; this is what extends its versatility and comfort.

Calculator N+ is only available for Android, however, being open source, it can be built from your code following the instructions in GitHub.

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## GeoGebra

GeoGebra is a multifunctional platform of didactic support that deserves a whole article to be able to expose everything it offers, however, this time we will only focus on the calculator it provides.

The strength of GeoGebra lies in the graphical representations of Functions, Equations and Inequations, or generally, the interaction between two variables (although its application for 3D graphics generalizes these aspects), however, it also allows the calculation of derivatives and integrals.

Graphic representations can be customized to clearly illustrate which elements are involved in the calculations being performed.

The full range of applications provided by GeoGebra is available at GeoGebra.com, iOS and Android.

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## Mathway

Mathway is a calculator with a simple but very versatile interface when making calculations, because as Wolfram Alpha, it is based on innovative algorithms and artificial intelligence.

Although you can use the buttons of the application to make the calculations, you can set the instructions and obtain the results.

Like Wolfram Alpha, you can see the complete development step by step to reach the final result paying a subscription, but it is not mandatory if we only want results.

Mathway is available at Mathway.com, iOS and Android.

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## Symbolab

Symbolab is the son of Wolfram Alpha and Mathway, haha, because it provides similar functionalities to both calculators and its interface also a mixture of both (but with more ads), however, it is equally comfortable to use.

Like Wolfram Alpha, you can see the complete development step by step to reach the final result paying a subscription, but it is not mandatory if we only want results.

Symbolab is available at symbolab.com, iOS and Android.

# What is 6÷2(1+2)?

In 2019 a debate went viral, discussing on what is the result of the operation 8÷2(2+2), I thought that it had been forgotten and that the situation had already been clarified. However, it was reborn on the infamous 2020 as 6÷2(1+2).

It is necessary to understand that when considering mixed operations, there is an established order among the operations. First all the products must be made, then all the divisions, then all the additions and finally all the subtractions. Also consider that if there are signs of grouping you must first solve the contents between parentheses ( ), then brackets [ ] and then braces { }; you must make the operations that are within them considering the original hierarchy between operations. This is what people call BODMAS.

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## Does calculators lie?

When calculating this operation in a calculator, the results will differ depending on how they have been programmed because some have been programmed to prioritize the order of operations and others have been programmed to prioritize the order of appearance of the operations.

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## Write it better…

In my opinion, the problem with that specific case is that the person who originally raised it does not have the slightest idea of how to use the grouping signs because when operations between numbers are proposed, they always come from a real problem, so that kind of problems will always be well proposed if they are written correctly. Ambiguity in mathematics should have no place.

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## How to propose the problem?

### Case 1

Suppose that you work for a party agency and at a party you have been asked to distribute six pieces of cake to a pair of children, this situation is described with the operation 6÷2. Suppose furthermore that you have to do this twice more, then this situation is described with the following operation (6÷2)×2. If again you are told to do this one more times, then at the end you will describe this with the following operation

(6÷2)×(2+1)
= 3×3
= 9

This means that in the end you will have to distribute 9 pieces of cake.

### Case 2

Suppose again that you work at a party agency and you have been given six pieces of cake to distribute to a pair children, this situation is described with operation 6÷2. However, you are being told that now it is not a pair of children but rather two pairs of children, this situation is described with the operation 6÷[2×2]. Finally, you are told that one more pair of children have arrived, so in the end you will describe this with the following operation

6÷[2×(2+1)]
= 6÷[2×3]
= 6÷6
= 1

This means that at the end you will have to give a piece of cake to each child.

## In conclusion…

Considering these two cases, we notice that each one has its own approach and interpretation. Always specifying which operations have to be grouped together and always specifying which operations should be carried out first.

# What is 8÷2(2+2)?

In 2019 a debate went viral, discussing on what is the result of the operation 8÷2(2+2), I thought that it had been forgotten and that the situation had already been clarified. However, I was asked what the result of this operation was, quoting me on a tweet, and even today, the people who responded are still deciding between 1 and 16.

It is necessary to understand that when considering mixed operations, there is an established order among the operations. First all the products must be made, then all the divisions, then all the additions and finally all the subtractions. Also consider that if there are signs of grouping you must first solve the contents between parentheses (), then brackets [] and then braces {}; you must make the operations that are within them considering the original hierarchy between operations. This is what people call BODMAS.

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## Does calculators lie?

When calculating this operation in a calculator, the results will differ depending on how they have been programmed because some have been programmed to prioritize the hierarchy between operations and others have been programmed to prioritize the order of appearance of the operations.

Anuncios

## Write it better…

In my opinion, the problem with that specific case is that the person who originally raised it does not have the slightest idea of how to use the grouping signs because when operations between numbers are proposed, they always come from a real problem, so that kind of problems will always be well proposed if they are written correctly. Ambiguity in mathematics should have no place.

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## How to propose the problem?

### Case 1

Suppose that you work for a party agency and at a party you have been asked to distribute eight pieces of cake to a couple of children, this situation is described with the operation 8÷2. Suppose furthermore that you have to do this twice more, then this situation is described with the following operation (8÷2)×2. If again you are told to do this two more times, then at the end you will describe this with the following operation

(8÷2)×(2+2)
= 4×4
= 16

This means that in the end you will have to distribute 16 pieces of cake.

### Case 2

Suppose again that you work at a party agency and you have been given eight pieces of cake to distribute to a couple of children, this situation is described with operation 8÷2. However, you are being told that now it is not a pair of children but rather two pairs of children, this situation is described with the operation 8÷(2×2). Finally, you are told that two more pairs of children have arrived, so in the end you will describe this with the following operation

8÷[2×(2+2)]
= 8÷[2×4]
= 8÷8
= 1

This means that at the end you will have to give a piece of cake to each child.

## In conclusion…

Considering these two cases, we notice that each one has its own approach and interpretation. Always specifying which operations have to be grouped together and always specifying which operations should be carried out first.

# Fractions

Fractions are an alternative way to denote division between two numbers and are generally used to express proportions, for example, to express three-quarters of a quantity we write $\frac{3}{4}$ or to denote half of a cake we simply write $\frac{1}{2}$. It is possible to represent the fractions graphically to make them easier to understand.

Formally, if we consider two integers $a$ and $b neq 0$, then we will say that $a$ is the numerator of the fraction and $b$ is the denominator of the fraction, and so, the division $a div b$ will be represented by the following expression

$\frac{a}{b}$

The number above is called numerator and the number below is called denominator.

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## Fraction properties

When we work with fractions, we will find very particular expressions that we can identify when we want to simplify mathematical operations. Let’s consider $a$ an non-zero integer and see below what these fractions are.

One divided by one equals one. In general, if we consider any non-zero real, the division of this number by itself, is equal to one, then,

$\dfrac{1}{1} = 1$.

$\dfrac{a}{a} = 1$.

Any integer number can be expressed as the division of itself with one, this information will be useful when we are presented with operations between numbers expressed in fractions and integers.

$\dfrac{a}{1} = a$.

When dividing zero by any non-zero real number, the result will always be the same, zero.

$\dfrac{0}{a} = 0$

On the contrary, if we take any real number, it cannot be divided by zero because this operation is not defined, that is, division by zero is not defined.

$\dfrac{a}{0}$

not defined.

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## The Law of Signs for Fractions

Since fractions represent divisions, we can also establish the law of signs for division, if $a$ and $b$ are integers such that $b$ is non-zero, then

$\dfrac{a}{b} = \dfrac{a}{b}$.

$\dfrac{-a}{b} = -\dfrac{a}{b}$.

$\dfrac{a}{-b} = -\dfrac{a}{b}$.

$\dfrac{-a}{-b} = \dfrac{a}{b}$.

The advantage in the use of fractions is that they provide rigidity in the results and thus avoid approximation or rounding errors when making divisions, which is why it is necessary to master the operations of sum, subtraction, multiplication and division between the fractions.

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## Proper and improper fractions

One way to classify fractions is by considering the size of their numerator and denominator, because these will determine the portion they really represent. If $a$ and $b$ are two integers such that $b neq 0$, we have to

• If $a < b$, we will say that the fraction is $\frac{a}{b}$ is proper, that is, if the numerator is smaller than the denominator.
• If $a \geq b$, we will say that the fraction is $\frac{a}{b}$ is improper, that is, if the numerator is greater or equal than the denominator.

To clarify this idea, let’s see some examples.

### Examples

#### Example 1

The fraction $\frac{1}{2}$, is a proper fraction, because its numerator is less than its denominator.

#### Example 2

The fraction $\frac{7}{15}$, is an proper fraction, because its numerator is smaller than its denominator.

#### Example 3

The fraction $\frac{4}{9}$, is an proper fraction, because its numerator is smaller than its denominator.

#### Example 4

The fraction $\frac{6}{20}$, is an proper fraction, because its numerator is smaller than its denominator.

#### Example 5

The fraction $\frac{5}{3}$, is an improper fraction, because its numerator is greater than its denominator.

#### Example 6

The fraction $\frac{10}{4}$, is an improper fraction, because its numerator is bigger than its denominator.

#### Example 7

The fraction $\frac{20}{12}$, is an improper fraction, because its numerator is bigger than its denominator.

#### Example 8

The fraction $\frac{75}{44}$, is an improper fraction, because its numerator is bigger than its denominator.

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## Mixed fractions

When reading a recipe it is common to find measurements for ingredients such as one and a half cup of sugar or, that is why we can find containers with measurements of $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$ or $\frac{1}{8}$. This also occurs when buying foods that must be weighed, such as one kilograms and a quarter cheese or three and a half kilograms of meat.

Fractions are ideal to express this type of measurements, they are designed to measure portions, for example, to write a cup and a half you can write $1 + \frac{1}{2}$ which in turn is equal to $\frac{3}{2}$. However, the way they are written may not present comfort or clarity in practice, that is why the mixed fractions (or mixed numbers) are defined, then, that instead of writing $1 + \frac{1}{2}$, one writes

$1 \tfrac{1}{2}$

In this way, we define mixed fractions to separate the whole part from its non-integer part, the latter usually represented with a fraction of its own. Any mixed fraction can be rewritten as an improper fraction, because if $a$, $b$ and $c$ are positive integers, then the following mixed fraction

$a \tfrac{b}{c}$

is rewritten as an improper fraction adding $a$ with $\frac{b}{c}$, that is,

$a + \frac{b}{c} = \frac{a \cdot c + b}{c}$

Let’s see some examples of how to rewrite mixed fractions.

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### Examples

#### Example 9

Rewrite the mixed fraction $1 \tfrac{1}{2}$ as an improper fraction.

$1 \tfrac{1}{2} = 1 + \frac{1}{2} = \frac{1 \cdot 2 + 1}{2} = \frac{3}{2}$

#### Example 10

Rewrite the mixed fraction $1 \tfrac{1}{8}$ as an improper fraction.

$1 \tfrac{1}{8} = 1 + \frac{1}{8} = \frac{1 \cdot 8 + 1}{2} = \frac{9}{2}$

#### Example 11

Rewrite the mixed fraction $2 \tfrac{3}{4}$ as an improper fraction.

$2 \tfrac{3}{4} = 2 + \frac{3}{4} = \frac{2 \cdot 4 + 3}{4} = \frac{11}{4}$

#### Example 12

Rewrite the mixed fraction $2 \tfrac{3}{4}$ as an improper fraction.

$3 \tfrac{1}{2} = 3 + \frac{1}{2} = \frac{3 \cdot 2 + 1}{2} = \frac{7}{2}$

#### Example 13

Rewrite the mixed fraction $5 \tfrac{9}{16}$ as an improper fraction.

$5 \tfrac{9}{16} = 5 + \frac{9}{16} = \frac{5 \cdot 16 + 9}{16} = \frac{89}{16}$

# Hermitcraft – How much Mycelium is worth the Diamond Throne?!

## What is Hermitcraft?

Hermitcraft it’s a youtube gaming series based on the popular sandbox game Minecraft with notorious players from the platform usually called hermitcrafters or just hermits. There are diverse locations in the server where the hermits converge, but most of the action develop in the Shopping District, since that is the place where players set up stores to sell and buy all in-game items.

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Elections were made to designate a mayor for the Shopping District, and since it was built on a mushroom biome, the new elected mayor renewed the appearance of the land by changing mycelium for grass but not everyone was happy about, thus the Mycelium Resistance was born.

While the Mycelium vs. Grass conflict started to heat up, the mayor started a “buy back!” program to motivate neutral hermits to eradicate mycelium from the district. The deal was: 2 stacks of 64 blocks of Mycelium for 5 diamonds.

## The Grian math problem

The Mycelium Resistance “mother spore”, Grian, saw this exchange program as the perfect plant to buy the diamond throne in exchange for all the mycelium harvested by resistance, so in his episode Hermitcraft 7: Episode 48 – BUYING THE THRONE he bumped into a curious math problem.

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We know roughly that there are seven a half stacks of diamonds (in the Diamond Throne)… So if we do some math very quickly, I need my calculator… This is like a math paper examination of school…

After saying that, Grian, was amazed that he needed actual math from high school to solve this problem and after thinking a bit, he stated an actual math problem like those in math books:

If you have nine diamonds per diamond block, and there are sixty four blocks in a stack, and there are seven and a half stacks, how many mycelium do you need to farm in order to take the diamond throne?

What we are going to do is to translate this question into math equations to solve the problem.

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## The math translation

Considering the Mycelium Resistance already have 34 shulker boxes full of mycelium, let’s state what are the equalities we have:

• $9$ Diamonds is equal to $1$ Diamond Block.
• $64$ Diamond Blocks are equal to $1$ Diamond Blocks Stack.
• $2$ Mycelium Stack is equal to $5$ Diamonds.
• $1$ Shulker Box full of Mycelium is equal to $27$ Mycelium Stacks.
• $34$ Shulker Box full of Mycelium is equal to $918$ Mycelium Stacks.
• $7 \frac{1}{2}$ diamond block stacks is equal to how many mycelium blocks?

First thing we need to do, is check how many diamonds are in $7 \frac{1}{2}$ diamond block stacks. Since seven and a half is a mixed number represented by $7\frac{1}{2}$, we have to rewrite it as a fraction, that is

$7\frac{1}{2} = 7 + \frac{1}{2} = \frac{15}{2}$

Then, we have $\frac{15}{2} \cdot 64$ diamond blocks, that is $480$ diamond blocks, and that is $480 \cdot 9 = 4 320$ diamonds.

We know that $2$ Mycelium Stacks are equal to $5$ Diamonds, in this way, we can state that if $M$ denotes a Mycelium Stack and $d$ denotes a diamond, then $\frac{M}{2} = \frac{d}{5}$. So, if we solve this equation for $M$, we have

$M = \frac{5}{2} \cdot d$

Considering this last equality, we can now replace $d$ with $4 320$, because that is the total of diamonds that the Diamond Throne, then, $M = \frac{2}{5} \cdot 4 320$ and that is $1 728$.

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So, the Diamond Throne is worth $1 728$ mycelium stacks. However, the question is how many mycelium do you need to farm in order to take the diamond throne? We have to remember that the Mycelium Resistance already have 34 shulker boxes full of mycelium, that is $918$ Mycelium Stacks.

Then, the Mycelium Resistance needs to farm needs to farm $810$ mycelium stacks in order to take the diamond throne, and that is $810 \div 27 = 30$ shulker boxes full of mycelium.