Arithmetic Properties a Calculus journey

This is a series of articles of my journey once again into Math … building up to Multivariable Calculus.
Today we cover basics that will be used later when working through Algebra.
Place value
In the real world, we are used to numbers like 2400 in fact this is a group of numbers i.e 2 thousands and 4 hundreds the Place Value is kind of the contract on which we understand number by a number’s position in a group of other numbers we can derive that digit’s value.
so 3456 becomes 3 thousands, 4 hundreds, 5 tens, 6 ones this, in essence, tells us the complete value of a complex sequence of numbers. Read more on positional notation to explore this.
Rounding Whole Numbers
This is a method of approximation we actually do this in day to day life quite a bit. Think of a scenario where you have to pay a bill in a supermarket you will usually see something priced as 9.99 and this will read to you as less than 10 or to most of us just as 10 to some careless ones like me 9. Now, what happened here? We looked at a detailed number and to save time we assigned with a shorter, simpler and more explicit representation. In math, we round numbers up or down depending on what place value we are round to 9545 can be rounded a number of ways
- To the nearest
thousand9545will become10000this is because it is closer to10000than it is to9000 - To the nearest
hundred9545becomes9600
Order of Operations
When evaluating a mathematical expressions e.g
y = m + (6 — (2 * 3))+5^5
In order for us to all get the same result aka the correct result we have to follow a convention on which procedures to apply first in order to evaluate this expression.
There are a few Mnemonics we can use to remember the order in which to apply the operations.
In the US the acronym PEMDAS is used.
In the UK the acronym BODMAS is used
these all come down to the same thing
Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction.
Brackets, Of, Division and Multiplication, Addition and Subtraction
The conventions above let us evaluate
y = m + (6 — (2 * 3))+5^5 as follows
- Apply Parenthesis/Brackets first then
Since we have brackets we can solve for those first, the first set we solved are the innermost ones then expand outwards, The whole rule applies when in brackets
y = m + (6 — (6))+5^5 here we solved for the brackets inside the brackets
y = m + 0 +5^5 here we solved for all the brackets in the expression
- Apply Power Of / Exponentials then
y = m + 0 + 3125 here we solved for the exponents in the expressions
- Apply Multiplication and Division
since we don’t have much multiplication or division to apply we can skip this
- Apply Addition and Subtraction
since we have only addition we can evaluate this left to right
Please note is is Addition and Subtraction the both of these fall on the same order hence when encountered should simply be evaluated left to right the same applies for division and multiplication
y = m + 3125
Commutative Property
This is a mathematical law that applies to multiplication and addition. This law simply states that when evaluating expressions with the above operations. The order of operands doesn’t matter.
examples
5 + 7 + 9 = 7 + 5 + 9
try it!
30 * 4 * 6 = 6 * 30 * 4
try it!
If you like pictures more I find the following one explains this well.

Associative Property
This law in mathematics states that in a row of two or more associative operators the order in which the operations are performed does not matter.
2 * (6 * 7) = (2 * 6) * 7
(2 + 3) + 4 = 2 + (3 + 4)
Note:
Addition and Multiplication are associative operators while subtraction and division are non-associative.
Distributive Property
Generally known as the distributive law states that in multiplication expression when one of the factors is rewritten as a sum of two numbers the product of the expression does not change. Let’s use this in an example
y = 4 * 60
is the same as
y = 4 *( 20 + 40)
For those who are more comfortable with this you can further simplify this by finding the greatest common factor
try it!
Note — This law is useful when multiplying large numbers since one can use it to make the factors smaller and easier to manage.
Break your problems down to simpler problems they become easier to solve
Types of Numbers
Finally, let’s cover the last of the arithmetic properties. These rules are used to classify numbers in math.
Whole Numbers — Numbers that can be represented without a fraction or a decimal.
1, 2,3,5000, 67
Integers — These are whole numbers and their opposites i.e their negatives 1,-1, 2,-2
Rational Numbers — These are numbers that can be expressed as a fraction of 2 integers
17/2, 19/5, 18/3
Irrational Numbers — These are numbers that cannot be represented as a fraction of integers these numbers usually occur between two rational numbers and they appear everywhere

