Example 1:

Example 2:

+7 - 3 = +4 (but plus signs for numbers are often left out, so "+4" will be written as "4".
7 - 3 = 4
5 - 10 = -5
9 - 3 = 6
17 - 20 = -3

a - b: first do "a" steps to the "right", then "b" steps to the left.
a + b: first do "a" steps to the "right", then "b" steps to the right.
-a + b: first do "a" steps to the "left", then "b" steps to the right.
-a - b: first do "a" steps to the "left", then "b" steps to the left.

Example 3:

-9 + -3

-9 - 3

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

3 x -7 + 3 x 7 = 0

3 x -7 + 3 x 7 = -21 + 21 = 0

4 - 3 + 8 - 10 = 1 + -2 = 1 - 2 = -1

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

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 called "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".

So:

In the US and other countries you might see for example: 5 + 0.32 = 5.32
In most of Europe you might see for example: 5 + 0,32 = 5,32

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

Some examples:

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.

I will "stick" to the dot notation from now on. But there is absolutely no fundamental difference
if you would take the "," instead.

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 = 11.00 + 0.15 = 11.15

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 see 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 = 0 + 3 + 0.45 = 3.45

What do you think is the outcome of this: 7.03 + 8.006?
It is:

7.03 + 8.006 = 15.306

Next, let's see how this all works with multiplications.

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 = 0.33 + 0.33 + 0.33 = 0.99

or, do it 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 arbitrary 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 indeed really critical, to perform first.

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 are symbols 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 4^th 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 5^th 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
9 x 9 = 9²

etc...

You see? It's really easy. The notation A²

is just a shorthand for "A x A".

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

A x A = A²

To "the power of...":

We know how to take the square of a number, like for example:

11² = 11 x 11 = 121

This can be generalized. Suppose A is a number, then:

A x A = A²
A x A x A = A³
A x A x A x A = A⁴
etc..

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

a x a x a x a x a x a = a⁶

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 2^m = 2^n+m

In general:

Let "y" be any number.

Then:

yⁿ x y^m = y^n+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. Some further remarks on working with roots.

We know that, for example, √16=4, since 4²=16.

Here it was not too difficult, since we could easily "reckognize" 4 as the answer, since
we knew that 4x4=16. Thus √16=4.

However, √5, or √71, are not immediately solvable, unless you have a calculator.
Ofcourse, √5, is a little more than "2", but less than "3", since 2²=4 and 3²=9.
But, it is fully accepted in math to leave √5 "as it is", since that is the most accurate
representation of the solution.

So, in tasks, or homework, in math, you most often do not have to calculate √5 any further.
But, in engineering, which requires final answers, you must use calculators or computers
to reach final answers.

Now, we must further learn how to deal with larger expressions involving roots.

-Additions and subtractions:

For example, how can we simplify an expression like "2√3 + 3√5 + 7√3"?

In the upper example, there a two terms with √3. So, we can simply add them together, thus
resulting in:

2√3 + 3√5 + 7√3 = 9√3 + 3√5

So, in expresssions, where there are similar roots, then simply add (or subtract) them.

This is most often sufficient, unless the task explicitly states that you must find
the most compact expression, or that that you must end up with only one term with a root.

Here is another example:

3√3 - √3 + 7√100 = 2√3 + 7√100

However, √100 = 10, so the answer can be simplified to 2√3 + 70

Sometimes, the task is to reduce an expression, to have one sort of root, only.
For example:

8√3 + 2√27

It looks like as if we cannot "lump" together similar roots.

But, in this case, we will use a rule, that we will study in the next subsection.
You need to know (see below), that actually √27 is equivalent to √(3x9).
So, we then have:

8√3 + 2√27 = 8√3 + 2√(3x9) = 8√3 + 2√3 √9 =
8√3 + 6√3 = 14√3

-Multiplications:

Let's use an example to demonstrate that √ab = √a x √b

-Let's consider √16=4.

But √4 x √4 = 2x2 =4. So, in this example, √ab = √a x √b is true

Usually, any multiplication symbole like "x" is left out, so √ab = √a √b

-Let's consider √9=3.

But √3 x √3 = (√3)² =3. So, in this example, √ab = √a √b is true too.

Anyway: it's always valid that √ab = √a √b

Also note that √a √a = a. Indeed, in (√a)², then the "square root"
and "taking the square", cancel out.

Example:

4√6 + √2√3 + 7 = 4√6 + √6 + 7 = 5√6 + 7

-Quotients with roots:

The basic rule is √(a:b) = √a : √b.

The figure below gives a nice example:

Figure 3.

12. Working with percentages (%).

12.1 Calculating a percentage of an amount.

This is really easy.

⇒ Example 1:

Say, you have 100 of "something", like 100 apples, or 100 euro's, or 100 liters of water ec...

-Then 20% of 100 is:

20 --- 100

x 100

=

0.2 x 100

=

20

-Then 55% of 100 is:

55 --- 100

x 100

=

0.55 x 100

=

55

⇒ Example 2:

Say, you have 176 liters of some liquid (water, gasoline etc..).
Suppose you need to calculate 36% of that amount. We follow the same procedure as in Example 1:

36 --- 100

x 176

=

0.36 x 176

=

63.36

12.2 Adding a percentage to an amount.

This situation often occurs, if some article is priced without taxes. Or, you have a certain
amount of money, but you need to add some percentage of commission to that amount.

⇒ Example 1:

In a bikestore, a bike is priced as 1750 euro's. However, this is the price without taxes.
The taxpercentage is 7%. What is the endprice of that bike?

We need to calculate 7% of 1750. Then add that result to 1750.
So:

NewPrice = OldPrice + 7% of OldPrice.

So, 7% of the OldPrice is:

7 --- 100

x 1750

=

0.07 x 1750

=

122.50

Thus:

NewPrice   =   OldPrice + 7% of OldPrice   =   1750.00 + 122.50 = 1872.50

You may always do it that way. However, There is a slight shortcut to this method, which will help us
in a later section. Watch this:

NewPrice

=

OldPrice

+

7 --- 100

x OldPrice

=

100 --- 100

x OldPrice

+

7 --- 100

x OldPrice

=

107 --- x OldPrice 100

=

1.07 x OldPrice

Indeed, as we see now: NewPrice = 1.07 x OldPrice

So, the advantage in general is, that we can write: NewPrice = Total% x OldPrice

You may not see the advantage right away, but I hope it get's clear in the next section.

12.3 Subtracting a percentage of an amount.

This is very much like the example of section 12.2.

This time, however, we must calculate a percentage of an amount, and subtract that from that amount.

This might happen, for example, with an article which does not sell.
To arise customer interest, you decide to lower the price.

⇒ Example 1:

-Suppose you have a product you wanted to sell for 1200 euro's.
-There was no interest from customers, so you decide to reduce the price with 9%.
-So what is the new price?

It goes like this:

NewPrice   =   OldPrice - (9% of OldPrice)   =   1200 - 0.09 x 1200 = 1200 - 108 = 1092

Just like in section 12.2, we can also write the calculation as:

NewPrice

=

OldPrice

-

9 --- 100

x OldPrice

=

100 --- 100

x OldPrice

-

9 --- 100

x OldPrice

=

91 --- x OldPrice 100

=

0.91 x OldPrice

=

0.91 x 1200

=

1092

12.4 So you have the new value, but what was the former one?.

⇒ Example 1:

This is sort of the "reverse" of our "bike" example (see section 12.2).

-The tax included price of the bike was 1872.50.
-the tax percentage is 7%

So what was the former amount, thus without taxes?

Above we found:

NewPrice = 1.07 x OldPrice

The "NewPrice" is known, and it is 1872.50. And we want to know the unknown value "OldPrice".

So we have:

1872.50 = 1.07 x OldPrice

Thus:

OldPrice = 1872.50/1.07 = 1750

And indeed. This was the original value (before taxes) from the former example.

13. Optional: 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", or "indicate", 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".