Basic arithmetic/calculus.
In the series: Note 1.

Subject: Basic Arithmetic

Date : 06 Februari, 2016
Version: 0.4
By: Albert van der Sel
Doc. Number: Note 1.
Remark: Please refresh the page to see any updates.
Status: Ready.


Maybe you need to learn basic "arithmetic" rather quickly.
So really..., my emphasis here is on "rather quickly".

So..., I am really not sure of it, but I hope that this note can be of use.
Ofcourse, I hope you like my "style" and try the note anyway.


I assume you have a certain basis, like that you are able to add and subtract numbers.

1. Use of parenthesis "(" and ")" in arithmetic.

Arithmetic is in a way based on (sort of) natural order and observations, but also on a set of rules.
That's really true. People all around the world have agreed to use certain "practices".
One of them is the use of parenthesis, that is the "(" and ")" symbols, in formula's and calculations.

For example, how would you compute the following calculation:

3 + (4 + 5)

It goes like 3 + (4 + 5) = 3 + 9 = 12

So, as a rule, calculate what is in parenthesis first.

True, when the calculation only has "additions" (like above) and/or subtractions, the order is not really relevant.
That is 3 + (4 + 5) is the same as (3 + 4) + 5, since (3 + 4) + 5 = 7 + 5 = 12. So, again, we find "12" as the solution.

However, in a moment we will see some more complex calculations. Take a look at this one:

3 x 2 + (7 + 3) / 2.

You don't need to know how this is done right now, but for now, just follow the same rule with respect to parenthesis.

I mean, write the calculation as follows:

3 x 2 + (7 + 3) /2 = 3 x 2 + 10 / 2 = 6 + 5 = 11

Now, if you wonder about the "order" on how to handle the subterms, then that's good thinking.

Let's look again at "3 x 2 + (7 + 3) /2". Here, we see a multiplication, namely (3 x 2), and we see numbers
enclosed in "(" and ")" (namely 7 + 3), and we see that we need to "divide" something by 2.

So, what's the order of calculation in equations like that? Let's go to section 2.

2. The main order in arithmetic.

If you would see this:

3 x 4 + 8 =

Then, what's the order of calculating? Do we multiply first and then perform the addition, or
do we add first and then multiply? Is the answer 12 + 8 = 20, or 3 x 12 = 36?

Well, mulitiplication goes before addition.

So, first we do "3 x 4" = 12 and next we add 8, so 3 x 4 + 8 = 12 + 8 = 20

Now, what if we had this instead: 3 x (4 + 8) = ?
Well, what's between parenthesis goes first, so we have 3 x (4 + 8) = 3 x 12 = 36

When you see any equation having multiplication, division, and adding or substracting, the order is:

(1) Between parenthesis first, (2) then Multiplication, (3) then Division, (4) then addition or subtraction.

(There is one missing actually, and that's "to the power of", which we will see later on).

So, for example:

2 x 7 + 8/2 + 4 x 3 + 5 = 14 + 4 + 12 + 5 = 35

3. Negative numbers.

If you would see these numbers, like 1, 7, 256 etc.. it all sounds really "normal".
Arithmetic and calculus uses "negative numbers" too, like -1, -20, -256, -1000 etc...

At first sight it may seem rather strange, but it's not really.
A few examples may illustrate that.

Example 1:

In physics we know that the electron has a (sort of) "negative" charge (electricity).
If we start with a neutral atom, and we radiate that atom, maybe one electron get's enough energy to leave the atom
and the result is that the atom now misses one "-" electric charge, and thus the atom is now "+1" or simply notated as "1".

On the other hand, if a free electron "clings" to an atom, it now has one extra negative charge, and as a whole
the charge is "-1".

(note: electric charges are normalized here).

So indeed, positive and negative results, captured in numbers, really exist.

Example 2:

Just think of a coordinate system in a flat plane, where we have 2 perpendicular "axes", and we try to pinpoint positions
in that plane. Take a look at the figure below.

Figure 1.



You see two axis, the horizontal one called "x axis", while the vertical one is called the "y axis".
The intersection of those axes, is called the "origin" (just a name for it).

Now, when you start from the origin, you can go 5 steps to the right. You are then at "x = 5".
However, from that point, you can take 7 steps to the left, which puts you at "x = - 2" (note the "-" sign).

I think you can find nummerous examples by yourself.
It's true, it's also a bit by "convention" (by agreement). For a coordinate system "to work", we simply need
"+" numbers and "-" numbers. Well, if you go to the right, you might say "I do +11 steps to the right",
but in such a case, "+" are usually not "written" down.

In any case, here are a few arithmetic examples:

+7 - 3 = +4 (but plus signs are left out, unless it's the "addtion operator", so we simply write it as: 7 - 3 = 4)
5 - 10 = -5
9 - 3 = 6
17 - 20 = -3

You can visualize such calculations, if you perform steps to the right or left, starting from the origin.


Example 3:

If you have 1000 euro's on your bankaccount, but you withdraw 2000 euro's, your saldo now is -1000 euro's.

Well, I hope I was able to illustrate the usability and neccessity of "negative numbers".

So, negative numbers are written as for example -1, -2, -3,..., -2000, -50000, etc...
Now, you might expect that "normal" (positive) whole numbers are written as +1, +2, +3,...., +2000 etc..., but that's almost
never done. When a number is positive, the "plus" is almost always simply left out.

Some more examples from arithmetic using negative numbers:

I think many will need some time to 'adopt' the way to add/subtract negative numbers.
That's normal. Just look at this:
If we want to subtract -3 from -9, the answer is -12.
If you take the analogy of the coordinate system, then at start you 'were' at position -9 at the x-axis.
Now, you go 3 more steps to the left (-3). So you end up at '-12'.
How do we write that down?

You may write:

-9 + -3

But usually that '+' is then fully left out. So it boils down to:

-9 - 3

Note that it's also equal to "-9 -3", but from an "operator view" (that is "+" and "-" operates on numbers),
people just write: -9 - 3

- Example 1: "-3 -7 + 4"

It might be read as: from the origin, we take 3 steps to the left, then 7 steps to the left. This places us at -10 (or x=-10).
From that position, me take 4 steps to the right (going in the positive direction), which places us at -6 (or x=-6).

So,

-3 -7 + 4 = -10 + 4 = -6

- Example 2: "3 x -7"

Using positive numbers, we know that 3 x 7 = 21.
You may also read it as: from the origin, take 3 times 7 steps to the right. This puts us at 21 (or x=21).

Now, 3 x -7 may also be read as: from the origin, take 3 times 7 steps to the left (the negative direction).
This will put us at -21 (or x=-21).

- Example 3:

3 x -7 + 3 x 7 = 0

This may be read as: take 3 times 7 steps to the left, then, take 3 time 7 steps to the right. The result is "0".

- Example 4:

4 - 3 + 8 - 9 = -1 + -1 = -1 - 1 = -2

Note:
You may wonder about how we write things here.
For example "-1 -1" means the same thing as "-1 + -1". You may interpret it as: starting from the origin,
take one step to the left, then, take another step to the left.

When you are only adding and/or subtracting numbers, you can choose your own ordering of subcalculations.
Here is an example:

- Example 5:

4 + 5 - 8 - 1 + 6 = (4 + 5) + (-8 - 1) + 6 = 9 - 9 + 6 = 0 + 6 = 6

Or, what end up to the same endresult:

4 + 5 - 8 - 1 + 6 = 9 - 8 - 1 + 6 = 1 - 1 + 6 = 0 + 6 = 0

At a certain moment, you get used to it, and you don't visualize "steps" anymore (if you ever did anyway...).

4. Types of numbers.

Up until now, we have only seen "whole" or "integer" numbers. They do not have a "dot" (USA), or "comma" (most of Europe),
to specify "fractions".

For example, the number 3.14 (e.g. used in the USA) or 3,14 (used in most of Europe), is not a "whole" number (or integer).
It has a certain amount "behind" the dot or comma (in this case: 0.14). As we know, numbers like 2, 3, 4, 5 etc...,
are "integers" and they do not have a fractional component.

Such a fraction behind the 'dot' (or 'comma'), is smaller than 1.
For example, the number 3.14 is exactly 0.14 larger than the whole number 3. And it is 0.86 smaller than 4.
You may also read 3.14 as:

3.14 = 3 + 0.14 = 3 + 14/100.

Note:

It's indeed rather unhandy (to say the least) that some countries uses "," (a comma) to denote fractions,
while in other countries, folks uses "." (a dot) to specify fractions.

We do not have another choice, then to take notion in which environment we are. For example, if you are in the US,
you know that people uses a "dot". But if you were in The Netherlands, folks use a "comma".

If a number is not a whole number, it has a certain fractional part which is smaller than "1".

For example:

3.14
17.9
1.6825534
2899.3356
-2.332
0.3333333333333333333333

Note that in the examples above, I always have used the "dot" to specify the fraction of the number,
which the part of the number which is smaller than 1.
In a countrly like The Netherlands, I would have used a comma.

Numbers which are not a whole number, might be called "real numbers", "or rational numbers", or "floats",
and other classifications go around too. That's not of interest to us, for now.

Now, let's calculate with "real numbers" !

Below I will use a dot to denote the fractional part in the number. In your country, it maybe a comma instead.
If so, then nothing fundamentally changes ofcourse.


First, it's very important to realize that a number like for example 6.45, can be witten as:

6.45 = 6 + 0.45

As another example, it's also true that a number as 11.15 can be witten as:

11.15 = 11 + 0.15

I am not sure if the above can help you in all circumstances, but anyway, it's essential that you
understand this fact.
Let's see some examples.

Additions and Subtractions:

Example 1:

6.45 + 11.15 = 17.60

One "trick" that you may use, is that you just add the whole numbers seperately, and the fractions behind the dots seperately.
If you like, you may write the calculation above as:

6.45 + 11.15 = 6 + 0.45 + 11 + 0.15 = 6 + 11 + 0,45 + 0.15 = 17 + 0.60 = 17.60 (or 17.6).

Example 2:

But what if the addition of the fractions are larger than "1"?
For example:

6.45 + 11.85 ?

6.45 + 11.85 = 6 + 0.45 + 11 + 0.85 = 6 + 11 + 0.45 + 0.85 = 17 + 1.30 = 18.30

The fractions 0.45 + 0.85 add up to 1.30 (1.30 = 1 + 0.3).
So, this "1" has to be added to the sum of the whole numbers.

Example 3:

6.45 - 3 = 3.45

Maybe you saw the outcome "at once". However, you may also write the calculation as:

6.45 - 3 = 3 + 3 + 0.45 - 3 = 3 + 3 -3 + 0.45 = 3 + 0.45 = 3.45

Multiplications:

Example 1:

10 x 1.65 = 16.5

10 x 3.13 = 31.3

3 X 0.33 = 0.99

For the last calculation, it may help to see that 3 x 0.33 = 0.33 + 0.33 + 0.33 = 0.99
Or, if you like:

3 X 0.33 = 3 x (0.3 + 0.03) = 3 x 0.3 + 3 x 0.03 = 0.9 + 0.09 = 0.99

Somewhat more complex examples are possible, and, (yes it's true) absolutely needed,
to give a more complete "picture".

However, I think it's best to go to "fractions" first, meaning the numbers like ¼ or ¾
But first, I like to illustrate two important theorems from calculus.

5. A few important theorems.

Theorem 1:

Suppose A, B, and C, are some random numbers. So, they litterally can be any number.
Ofcourse, A, B, and C are just "letters", but here they stand symbol for any number.

Then:

A x (B + C) = A x B + A x C

Indeed, even this is true:

A x (B + C + D) = A x B + A x C + A x D

So, for example:

5 x (2 + 8) = 5 x 10 = 50. So here we first calculated (2 + 8) and then multiplied 10 by 5.

But:

5 x (2 + 8) = 5 x 2 + 5 x 8 = 10 + 40 = 50. So here, the answer is 50 too.

Is this in conflict with section 1? There I told you that a calculation between "(" and ")"
should be performed first. That still holds, but multiplying the terms between "(" and ")
seperately, works too.

By the way: We will see plenty examples soon, where it is shown that calculating stuff between "(" and ")"
is really critical.


The following is also true:

A x (B - C) = A x B - A x C

So, for example:

5 x (8 - 2) = 5 x 6 = 30. So here we first calculated (8 - 2) and then multiplied 6 by 5.

But:

5 x (8 - 2) = 5 x 8 - 5 x 2 = 40 - 10 = 30. Here, the answer is 30 too.

This can be a handy theorem. However, sometimes the "otherway around" maybe helpful too !
For example, consider this situation. You are asked to calculate:

6 x 345

Then, maybe you find the following equivalent easier to calculate:

6 x 335 = 6 x (300 + 30 + 5) = 1800 + 180 + 30 = 2010

It's a bit "otherway around", since we started out with "6 x 335" and we rewrote it to "6 x (300 + 30 + 5)".

Theorem 2:

If you have any calculation where a division is part of, then the following may help enormously.

If you have a fraction like:

A / B

Where A and B again stand symbol for any number,
then you may multiply the top number (A, the nominator), and the number below (B, the denominator), by the same number (C).
That is:

A / B = C x A / C x B

By the way, there are different ways to "notate" fractions. Sometimes we see a notation like "A / B"
but in other occasions we see:

A
-
C

Let's see some examples:

10 / 5 = 2

Thus 3 x 10 / 3 x 5 = 30 / 15 = 2

We find the same results !

As another example:

32 / 8 = 4

Let's multiply the top- and lower number by 2:

2 x 32 / 2 x 8 = 64 / 16 = 4

In the next section, we will explore "fractions" a bit more, like how to add fractions etc..

6. Fractions.

It's essential to master arithmetic with fractions, so here we go...

First, a few obvious statements.

When you split "something" (like 1), in 4 equal pieces, each piece is 1/4 of the original amount.
When you split "something" (like 1), in 5 equal pieces, each piece is 1/5 of the original amount.
When you split "something" (like 1), in 3 equal pieces, each piece is 1/3 of the original amount.

If we take the number 1 as the "original amount", then we can take a look at the following calculations:

1/4 + 1/4 + 1/4 + 1/4 = 1
1/4 + 1/4 + 1/4 = 3/4
1/2 + 1/2 =1
1/3 + 1/3 = 2/3
1/3 + 1/3 + 1/3=1
1/12 + 1/12 + 1/12 = 3/12
1/12 + 1/12 + 1/12 + 1/12 + 1/12 = 5/12

But what's true for whole (integer numbers) works too for fractions:

1/3 - 2/3 = - 1/3
5/15 - 7/15 = - 2/15

Remember, there are different ways to "notate" fractions. Sometimes we see a notation like "A / B"
but in other occasions we see:

A
-
C

It get's a little more complicated if we want to calculate something like (1/3 + 3/5), since here
we want to add fractions with "different denominators" (denominator is the number below the "-" symbol,
or on the rightside of the "/" symbol if we use the "A / B" notation).
To accomplish that sort of calculations, we rely heavily on "Theorem 2" of the former section.

Take a look at figure 2 below:

Figure 2.



If needed, re-read Theorem 2 again from the former section.

In figure 2, we will take the 4th calculation as an example.
Here, we want to add

1/4 + 3/6

We know that it is easy to add fractions which have the same denominator.
So, let's rewrite 1/4 and 3/6 to be fractions in form "x/12", thus with "12" as the denominator.
In figure 2, you see that happen. According to theorem 2, we may multiply the nominator and denominator
of any fraction, with the same number.

So, we muliply the top- and bottom numbers of 1/4, with "3". This gives us 3/12.
And, we muliply the top- and bottom numbers of 3/6, with "2". This gives us 6/12.

Now we simply have the addition: 3/12 + 6/12 = 9/12

Ofcourse, 9/12 is the same as 3/4, and it's quite neat to promote that result as the outcome of our calculation.

Note:
By the way, did you noticed that "3/6" is the same as 1/2? And that is the same as "2/4".
So, it was actually quite easy to add 1/4 with 3/6, because it's the same as 1/4 + 2/4 = 3/4.


Now, please try to follow the 5th calculation of figure 2, all by yourself.

7. Square numbers, and "to the power of...".

To take the square of a number:

This is real easy. Namely, how to square numbers, and calculate "the power of" some number.

You know that 3 x 3 = 9. I can keep going on providing examples, like 4 x 4 (=16), or 9 x 9 (=81).

If you multiply a number, by itself, it's called taking the "square of".
We also have a simple notation for it. You simply put a "2" above it.
So:

2 x 2 = 22
5 x 5 = 52
10 x 10 = 102

etc...

You see? It's really easy.

So, if "𝒳" is a certain number (it can be any number), then:

𝒳 x 𝒳 = 𝒳2

To "the power of...":

We know about taking the square of a number, but .iIt can be generalized:

For example:

2 x 2 x 2 = 23
5 x 5 x 5 = 53
10 x 10 x 10 = 103

In these examples, we take the "power of 3" of such a number.

You will not be amazed that this is true also:

2 x 2 x 2 x 2 x 2 x 2 = 26
5 x 5 x 5 x 5 x 5 x 5 = 56
10 x 10 x 10 x 10 x 10 x 10= 106

In these examples, we take the "power of 6" of such a number.

So, if "𝒳" is a certain number (it can be any number), then:

𝒳 x 𝒳 x 𝒳 x 𝒳 x 𝒳 x 𝒳 = 𝒳6

So, for example, what's a million? It's 1000000 or 106 (=10x10x10x10x10x10)

How can we write 10 million? it's 10 x 1000000 = 10 x 106 = 107

How can we write 1000 million? it's 1000 x 1000000 = 10 x 10 x 10 x 106 = 109

8. Arithmetic using negative numbers revisited.

I like to be sure that you can handle negative numbers alright.

- Addition and Subtractions:

I think that you are comfortable with the following types of calculations.
We have seen enough examples by now. But, here are a few more:

5 - 3 = 2
9 - 5 = 4
7 - 9 = -2 (note the "-" sign).

(I) How to view or interpret it:

For the last example: It's like: In the Origin, on the x-axis, do 7 steps to the right (+ direction),
then do 9 steps to the left (- direction), which will put you in x = -2.

-2 - 3 = -5 (note the "-" sign).

For this example: It's like: In the Origin, on the x-axis, do 2 steps to the left (- direction),
then do 3 steps to the left again (- direction), which will put you in x = -5.

(II) Alternative view:

7 - 9 = -2

You can also see that this way: on the x-axis, you are at "7", then you do 9 steps to the left,
which places you at x = -2.

-2 - 3 = -5

You can also see that this way: on the x-axis, you are at "-2", then you do 3 steps to the left,
which places you at x = -5.

Both ways to view matters, will work at all additions or subtractions. In fact they are the same anyway.

- Multiplication:

If we just visualize positive numbers by "+" for a moment, and negative numbers by "-", then:

+ x + = +
+ x - = -
- x + = -
- x - = +

So, for example:

5 x 5 = 25 "=Take 5 times, 5 steps to the right (from the Origin). That's 25".

5 x -5 = -5 "=Take 5 times, 5 steps to the left (from the Origin). That's -25".

Now, "-5 x 5 = 5 x -5", since if we have any two numbers A and B, it holds at any time that A x B = B x A

As the last example we must explain that for example "-5 x -5 = 25", thus resulting in a positve number.
If we indeed use "-5 x -5", I can explain it as follows:

-5 x -5 = -( 5 x -5) = -(-25) meaning, take the opposite of 25 steps to the left, which is 25 steps to the right.

Bottom line: please remember that (a negative number) x (a negative number) = positive.
This can be important for my next notes.

9. The "square root" of a number.

In section 7, we talked about taking "the square of a number". That goes like for example:

22 = 2 x 2 = 4
52 = 5 x 5 = 25
102 = 10 x 10 = 100

Here, we are only concerned about "taking the power of 2".

Doing exactly the opposite, is called taking "the square root of a number".

So, suppose we have the number 25. What is "the square root of 25"?

It's √25 = 5

Note that we use the "√" symbol, to designate that we want to calculate the square root.

Suppose we have the number 16. What is the square root of 16?

It's √16 = 4. This is so because 4 x 4 = 16. So, √16 = 4.

Calculating the square root of for example 39.5, or for example 6.3, is not easy, for nobody.
That's why we have calculators!

But for "easy" numbers, it's not too hard.

Again an example:

We know that 8 x 8 = 82 = 64

Now, doing the "opposite":

√64 = 8

So, the square root of 64, written as √64, is 8.

10. Addition of "powers".

We know that for example:

5 x 5 = 52
5 x 5 x 5 = 53

Now watch this:

(5 x 5) x (5 x 5 x 5) "is the same as" 52 x 53 "is the same as" 5 x 5 x 5 x 5 x 5 "is the same" as 55

Yes, when multiplying numbers, of which each is raised to some power, then simply add those powers.

So, for example:

2n x 2m = 2n+m

In general:

Let "y" be any number.

Then:

yn x ym = yn+m

Example:

y3 x y4 = y7

Often, in literature, "x" plays the role of "any number" instead of "y". In such case, using the "x" as multiplication symbol too,
is then somewhat unhandy. So, often, "x" will be in another font, or characterset is used.
Or, the multiplication symbol "is simply left out", and the multiplication is implicitly assumed.

Example:

x3 x4 = x7


11. Introduction of some mathematical symbols.

Here I introduce some mathematical symbols. If you only want to learn some elementary arithmetic,
then this material is not so important for you.
If you like to go through my next notes, then it would be great if you would browse through the table below.

The examples below, use, for example, '0' and '5' as some "markerpoint", but it can ofcourse be any number.
The whole purpose is to introduce the ">" (larger than), and "<" (smaller than) symbols, and a few other symbols,
which can be of great help to "label" certain ranges.

Examples:

x > 0 means: x is larger than '0'
x ≥ 0 means: x is larger or equal to '0'
x < 0 means: x is smaller than '0'
x ≥ 0 means: x is larger or equal to '0'
x ≤ 0 means: x is smaller or equal to '0'
-5 < x < 5 means: x is smaller than '5', but larger than '-5'
a ∧ b means: a AND b
a ∨ b means: a OR b
∀ x means: for all x (in some set)
∃ x means: there exists some x (in some set)



That's it ! Hope you liked it.

The next note is more "mathematical" of nature,
and is a super quick intro in "linear equations".