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We will in this video go through the concept of numbers.
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We will in this video only shortly cover a few of the mathematic rules,
since the fundamental purpose isn't to walk through everything in detail.
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We want you to get an overall picture of what different types of numbers there are.
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We will go through what a number is, what a number system is
and also different types of numbers.
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We will also take a look at numbers in different forms and do some examples
on how the rules for calculating numbers can vary, depending on what type of number it is.
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What in fact is a number? What is it used for?
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A number is simply a way to describe the size of something.
The number can for example describe the weight, length and mass of something.
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This is then a way to describe how large, small or far something is.
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During a long period of time in human history, we have used numbers to describe quantities
and we have also used many different systems of numbers.
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For example the roman numerals where Romans
used letters to describe quantities.
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There is also the binary number system used by computers, and the number system
called the decimal number system, which is what we usually use today.
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So we use the decimal number system built
on the base of 10, but how does this number system really work??
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We can do an example with the number 123
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We can write this number as 1·10² + 2·10¹ + 3·10⁰
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1·10² is equal to 100, 2·10¹ is equal to 20 and 3·10⁰
is equal to 3 since anything raised to the 0-power is equal to one.
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We then get the number 123 by adding these terms together, i.e. 100, 20, and 3 = 123
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OK, now we have talked about what numbers are and also a little bit about different number systems.
We will now look at a few different number sets.
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That means that numbers can be divided into different types of groups.
Some numbers are included in several of these groups.
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This may sound very strange at first, but I
will walk you through it and show what I mean.
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The first number set, we can mention is the
natural numbers which are usually noted with a bold N.
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These are all positive integers, for example 0, 1, 2, 3, etc.
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If we first mark the numbers in the blue circle in the center of this picture,
I will shortly explain the purpose of placing the N right there.
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The second number set, is the one called integers and these numbers
are noted with a bold Z.
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These are all integers, both positive and negative, such as
-2, -1, 0, 1, 2 and 3
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The integers Z, we mark in the circle outside the natural numbers.
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This is to show that the integers also include the natural numbers.
The natural numbers (N) are thereby a part of the integers (Z).
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The reason why we place the integers Z outside the natural numbers N, is thereby to visualize how this set of numbers,
also includes the number set in the previous circle.
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In this way, we will continue to visualize how sets of numbers are connected.
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We continue with the next number set, which are the rational numbers,
or fractional numbers, which are noted with a bold Q.
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Rational numbers are numbers that can be written as a ratio of two integers.
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It can for example be three quarters (3/4) or one-fifth (1/5).
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The rational numbers include the integers Z and natural numbers N,
and thereby we place the Q in the circle outside of the two circles, which include these number sets.
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The next number set are the real numbers R.
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These are all the numbers on a number line, including those which can not be written as a fraction.
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Examples of the real numbers can be pi and the square root of 2, which both have
an infinite number of decimals.
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Numbers with an infinite number of decimals are called irrational numbers and is
thereby a part of the real numbers.
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That means all the numbers that can be placed on the number line.
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Let us mark the R in the figure and move on to the last number set
that I am going to mention in this video.
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You have now seen some different types of numbers or number sets,
but you can also write the numbers in different ways.
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Think of it as that the type of number is the same, but the form it is written in is different.
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I was going to mention three such forms in which you can write numbers. These are called
decimal form, percentage form and exponential form.
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There are other ways to write numbers, but these three will explain the idea of writing numbers in different forms just fine.
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One such form is a decimal, such as 0.6 or pi, written with
decimals, i.e. 3.14159 and so on.
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In Sweden, we use a comma when we write in decimal form, but computers, for example, use a regular dot, when writing in decimal form.
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Another way of expressing numbers is to write them in percentage form. Here you express the size or the amount in hundreds, and the symbol % is used here.
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Example, 60% = 60/100 = 0.6
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This is an excellent example of where you write the same exact number, but in
different forms.
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The different forms used to describe the same number are percentage form, fraction form and decimal form.
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Let`s look at one other way of writing numbers - the notation form.
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Here, numbers are written with a base and an exponent.
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For example, 3 raised to the 4th power. Here, 3 is the base and the 4 is the exponent.
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The exponent describes how many times we have to multiply the base by itself.
3 raised to the 4th power is therefore the same as 3·3·3·3 = 81
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So this is another way of writing numbers.
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We will mention one last thing about numbers in this video.
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Different types of numbers can have different rules when calculating them. These rules can be important to know.
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Again, we will not be able to go through all the rules
and the ones that we do go through, we will only mention briefly.
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What is important here is that you understand the importance of knowing which numbers that
exists and that there are different rules related to them.
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We will look at a few examples on calculating integers,
negative numbers and fractions.
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One of the first things you learn in school is how to count the integers.
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You learn how to add two numbers, such as 2 + 5 = 7,
where the result is called a sum.
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You also know how to subtract numbers, such as 7 - 2 = 5 where the result is called a differential.
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In multiplication, for example 3 · 6 = 18, the result is called a product and the numbers you multiply are the factors.
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When using division, you have a ratio, e.g. 25 / 5 = 5
where the result 5 then is a quota.
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These calculation rules you most likely recognize and they might even be printed in the back of your head.
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A little bit trickier is usually, when you are going to do
calculations with negative numbers.
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For example, subtraction of a negative number
becomes an addition.
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Thereby: 3 - (-7) = 3 + 7 = 10
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If two negative numbers are multiplied with each other, the result will be a positive number.
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Example: (-3) · (-7) = 21
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There are more rules for calculating with negative numbers,
and it can be very helpful to keep good track of them.
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We will mention a few more rules and examples for calculations, this time on
fractions or rational numbers
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If for example, we are going to add the fractions 3/4 to 1/6, we then first must write
both fractions with the common denominator 12
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We multiply the two fractions by 3 and 2
so that they both have the denominator 12
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We can now write the two fractions with the same denominator
9/12 + 2/12
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Now, since the denominator is the same, we are able to write the two fractions as one: (9 + 2) / 12 = 11/12
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If we instead multiply two fractions with each other, for example, 1 / 5 · 3 / 4,
we have to multiply the numerators with each other and the denominators with each other.
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In this example we get (1·3) / (5·4)
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This is then equal to 3/20
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These are some specific rules for calculating with fractions and now you are hopefully able to
practice this some more on your own.
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We have now taken a look at what a number is, what a number system is,
and different types of numbers.
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Finally, we have also taken a quick look at some examples of calculation rules for different types of numbers.
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The idea of this video has been to introduce you to, and give you an overall picture
of different numbers.