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Arithmetic is the ABC of math. Addition, subtraction, multiplication, and division are the basics of math and every math operation known to humankind. In one way or another, every equation, graph, and an enormous amount of other things can be broken down into the ABC’s of math, the four basic operations. As people say, math is a language, and addition subtraction, multiplication, and division are its alphabet, along with the number line as well. The properties are basically, proven ways to apply the mechanics of arithmetic into certain situations.
You can usually find these situations within an equation. Some properties are much more basic and common then others; consequently, they pop up more often then others. For example, the reflexive property of equality, probably the most basic property there is. This property simply states that a number equals itself (a=a). This property is so fundamental that every time you do anything with a number, you see this property come up.
When solving a multiple step equation, the only way we can really know that a number, if left alone from one step to the next has stayed the same is because of the reflexive property.
This property allows us to establish that 1=1 and anywhere that that specific number comes up, its value is equal to every other number written the same way, “1.” This is connected to the mechanics of arithmetic in the sense that all arithmetic operations assume that every number labeled “1” has a value of one. If you multiply 1 with 2 the “1” has the same value, of one, as it does in any other operation done with that specific number, “1”.
This is the most simple, broken down, and basic concept of math, this property allows us to always make the assumption that a “1” we use in one operation has the same value of a “1” we use in another, so that we can know the set value; otherwise, every number would be a variable. Now, there are more sophisticated properties such as the Distributive Property of Multiplication with respect to Addition, which states that a(b+c)=ab+ac. This property actually helps us deal with a certain situation that we may find in an equation. If in an equation we see that a set of numbers are grouped with parenthesis, the order of operations tells us that we must take care of that grouped set before we do anything else. This is a flat out given that is stated for us in the order of operations (Parenthesis, exponents, multiplication and division, addition and subtraction).
Therefore, if the rules tell us we must solve something in parenthesis but it contains a variable or unknown value that keeps us from solving it, we must find a different approach. For this we look to the distributive property. It tells us we can open the parenthesis by multiplying or dividing what is outside of them with every thing inside. This connects with the mechanics of arithmetic because if you would have solved what was in the parenthesis first then multiplied or divided that with what was outside you would have been doing the same thing if you simply multiply both parts of the inside of the parenthesis separately with what is outside. This works with multiplication and division directly outside the parenthesis and addition or subtraction inside. However, in reality, all a property really does is analyze and explain something we do and take for granted. Although when I actually analyze something to understand how it works, it improves my mathematical ability.
This is because it makes things easier for me since it changes the way I look at things. Instead of seeing an equation as a bunch of properties, I see it as a bunch of numbers doing different operations. If I were to relate this to a language, I could say I fully understand the uses of every letter and the sounds it makes, once I have understood that, I can not only sound out any word, but create my own as well. Instead of memorizing an incredible amount of properties, I can just understand the works behind all of them and only after time and repetition can I pick up and second nature them. Not only that, this approach brings logic behind all of math so that it is not memorization, its understanding and being able to apply what you learned to a number of slightly altered situations. This maximizes the ability to use the properties in every way possible, making them most useful.
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