Sunday, September 30, 2018

Addition, Subtraction, and Whole Numbers

Background 

 Now that we have learned about Whole Numbers, Addition, and Subtraction, let's work a problem (there's more problems in the Practice Problem section)

 Question 
Find 5 + 2 - 3 - 1 + 6 - 0 + 0
Answer 
9
Analysis  

In prior entries, we've worked individual calculations, say like 5 + 2, both with and without the number line. We'll do the same here but we'll start on with the leftmost term and work our way through the different operations (the plusses and minuses).

We're evaluating 5 + 2 - 3 - 1 + 6 - 0 + 0

We can start with a number line and a red dot on the number 5:



So let's colour the 5 in our expression red: 5 + 2 - 3 - 1 + 6 - 0 + 0

Our first operation is plus 2 (+ 2). We move our point two numbers to the right, to 7 (I'll show that with blue):

5 + 2 - 3 - 1 + 6 - 0 + 0



Now starting at the blue dot at 7, we subtract 3 (- 3), or move 3 spots to the left. We'll land on the 4. I'll show that with green:

5 + 2 - 3 - 1 + 6 - 0 + 0



Starting at 4, we subtract 1 more to land on 3 (I'll show that in orange):

5 + 2 - 3 - 1 + 6 - 0 + 0



Now we add 6. Starting from the 3, we'll land on 9 (shown in purple):

5 + 2 - 3 - 1 + 6 - 0 + 0



And now we're asked to first subtract 0 and then add 0. Notice that when we add or subtract 0, the dot doesn't move! And so we leave our dot at 9.

This ability, to add and subtract 0 as many times as you want to any number and not change that number's value (like what we did with the 9 - we didn't change its value at all by adding or subtracting 0), gives 0 a special name - the additive identity.

Ok - so now let's work this same problem one more time, but this time without the number line. Here we go:

5 + 2 - 3 - 1 + 6 - 0 + 0

7 - 3 - 1 + 6 - 0 + 0

4 - 1 + 6 - 0 + 0

3 + 6 - 0 + 0

9 - 0 + 0

 9

 We start on the left and work our way to the right.

Vocabulary:
  • Whole Numbers - The set of numbers that starts with 0 and increases by 1 (0, 1, 2, 3,...)
For more information check out these links (comment to add your favourite link): 
Where might you have come from? 

 Fact-orials Index

Numbers:
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Graphing:
Where might we go?

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Saturday, September 29, 2018

Subtraction

Background 

 We've worked with the number line and math symbols to do addition, like 3 + 2 = 5. So what about taking away, also known as subtraction?

 Question 
What's 5 - 2? What are some words associated with subtraction?
Answer 
5 - 2 = 3. Subtraction, take away, deduct, subtract, eliminate, minus, less, difference
Analysis  

 I'll do the 5 - 2 first.

Let's start by taking a look at the number line. I'll go ahead and place a blue mark for 5:



When we subtract, we want to move to the left on the number line. In this case we're moving two to the left, so let's put a red dot on the spot 2 to the left of 5 (and show that movement with a red line):



This is the graph of 5 - 2 = 3

Now let's talk subtraction terminology.

 Words that tells us that we need to subtract include: subtraction, take away, deduct, subtract, eliminate, less, and minus. The numbers that we're using when we subtract are called terms. The answer we get from subtraction is called the difference.

 For more information check out these links (comment to add your favourite link): 

 Where might you have come from? 

 Fact-orials Index

Numbers:
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Where might we go?

Numbers:
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Greater/Lesser Than

Background 

 Now that we have a number line where we have numbers all in a line, we can start talking about numbers that are greater (i.e. bigger) and those that are lesser (i.e. smaller).

 Question 
Is 30 bigger than 10? How do I express that using math symbols?
Answer 
Yes it is. 30 > 10
Analysis  

 Let's look at a piece of a number line:



Let's just look at the numbers 10, 20, 30, and 40. Notice that as we move to the right along the number line, the numbers get bigger. This is true all along the number line, and so any number that is to the right of another number on the number line is greater. We can also say that any number that is to the left of another number on the number line is lesser.

Let's mark the 10 (in red) and the 30 (in green) on the number line:



The green 30 dot is to the right of the red 10 dot, so 30 is bigger than 10. We can also say it the other way, that 10 is less than 30.

So let's talk about how to write it using math symbols.

 I'm going to write the two numbers and leave a place for a symbol for "greater than":

30 ___ 10

Teachers I've had would talk about a hungry fish that lurks between those two numbers. It can only eat one of them and it wants to eat the bigger of the two. The mouth can either be turned to the left > or to the right <. In our case, we have:

30 > 10

and we say "30 is greater than 10".

We can also write it the other way:

10 < 30

and we say "10 is less than 30".

 For more information check out these links (comment to add your favourite link): 

 https://www.themeasuredmom.com/less-than-greater-than-math-activity-using-toys/

 Where might you have come from? 

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Whole Numbers

Background 

From the development of the Counting/Natural Numbers and the realization that 0 could and should be a number as well, we get a set of numbers that represents a small but significant shift in how we can express numbers.

 Question 
What are the Whole Numbers?
Answer 
Whole Numbers are the Counting/Natural Numbers and include 0, and so they are 0, 1, 2, 3,... on to infinity.
Analysis  

 There's not much to add here in terms of what the Whole Numbers are over and above what's been said above. The Number Line now includes the number 0:




















Vocabulary:
For more information check out these links (comment to add your favourite link): 

 Where might you have come from? 

 Fact-orials Index

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Graphing:
Where might we go?

Numbers:
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Thursday, September 27, 2018

The Number 0

Background 

We've covered Counting/Natural Numbers, which is the number 1 and then the next number is plus 1, so 1, 2, 3, 4, ...

 Question 
What about 0? 
Answer 
Zero took a long time to come into its own as a number. See below for a very brief history.
Analysis 

 While it's pretty straightforward to recognize the existence of a thing and then count it (like with the number 1), the concept of 0 took a lot longer to come into existence. And it makes sense - how do you count absence? Non-existence? Nothing?

Consider a situation where you have 2 eggs. You then make an omelette with those 2 eggs. You had 2 eggs and now you have... well... no eggs!

The history of the Counting/Natural Numbers is lost in the mists of time across a whole host of cultures, the same can't be said for 0. For many civilizations, there was no need for 0. The Romans, for instance, had no symbol for 0 at all - the way they wrote numbers didn't require placeholders. For instance:

I = 1
II = 2
III = 3
IV = 4
V = 5
VI = 6
VII = 7
VIII = 8
IX = 9 
X = 10

and the symbols continue with ones for 50, 100, 500, and so on. No need for zeros. (See the post on Roman numerals for more).

However, if you have a number system such as our current 10 decimal system (known as base 10) where we can make all of our numbers from the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and putting them into different places (1's, 10's, 100's, and so on), where the need for a 0 to hold a place would show up. Before 0, that particular place would simply be blank. And so the numbers 11 and 101 would look like this:

11
1 1

Is that second number really 101? Or is it 11? Perhaps a 1 and another 1?

Even with that, it was only the mathematically advanced civilizations of Sumeria, India, and the Maya who used the number 0. For the Sumerians and the Maya, 0 acted only as a placeholder in larger numbers - it was never a number in and of itself.

But it was 5th century India, where the concept of "emptiness" and "emptying the mind" were coming to the fore and becoming part of religious texts, that a symbol arose to help express that emptiness. If 1 is the number for the Self, then 0 is the number for the empty mind.

Once zero was introduced as a symbol for an actual value, it started being used in mathematics. Questions such as 2 + 0 = ? made sense - and it's this ability to ask that kind of question that leads to algebra (we'll discover that in later entries).

Vocabulary:
For more information check out these links (comment to add a resource to the list!): 

 Live Science article on the History of 0
http://www.newworldencyclopedia.org/entry/0_(number)

 Where might you have come from? 

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Wednesday, September 26, 2018

Addition

Background 

 Now that we have Counting/Natural Numbers and the Number Line, can we start to use them?

 Question 
How do I add? What are some words associated with addition?
Answer 
See below for how to add and also for terminology
Analysis 

Let's start with the Number Line to see how we add:



Let's put a blue dot on the number 1:



That dot means we have 1 of something. Perhaps we have one paperclip or one book or one channel we watch on Youtube. It's one of something.

What happens when we get another of that same thing? We now have 2 of them. To express this as addition, what we do is start with 1, add 1 (and we can use the + sign to indicate that), and then say that it equals 2 (and we show equals with the = sign). Ok - let's write this out step by step:

1 plus 1 equals 2

 1 + 1 = 2

 On the Number Line, we can graph this. We start at the blue 1. Since we are adding 1 (+ 1) to the blue 1, we move one spot to the right of the blue 1 and end up at the orange 2. I'll show the movement from the blue 1 to the orange 2 with a line connecting the two:



See how the first number tells us where to start? And how the amount that we add by tells us the number of numbers to move to the right?

Let's try another. Let's start with 3 and add 2 more. We put a dot (it can be any colour) on the number 3 and then we move up (to the right) by 2 more spots:



So that's 3 + 2 = 5

Let's talk about addition terminology:

The numbers that we are adding together (so both the starting point and the number(s) we're adding) are called terms. (They can also be called addends). The number that we get to as a result of adding (like the 5 above) is called the total or sum).

In word problems, look for words like sum, total, add, increase, and plus to indicate that addition is needed.

 For more information check out these links (comment to add a resource to the list!): 

 Where might you have come from? 

 Fact-orials Index

Numbers:
Where might we go?

Numbers:
Operations:
Operations with different kinds of numbers:
Properties:
Relations:
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Number Line

Background

 Now we have words and symbols that refer to amounts of things, but those words and symbols, unless arranged into something useful, is just a jumble of stuff.

 Question 
In what way can we arrange the Counting Numbers in a way that's useful? Something visual, perhaps.
Answer 
We certainly can - it's called the Number Line
Analysis 

 Take a moment and look at the keyboard of the computer (or texting app on your phone) and notice that the letters, numbers, and symbols are all arranged in a set manner. This is what is called standardization. (It's why, no matter where in the world you go, the burger you get in a McDonald's is the same. The burger in New York City is the same as in Tokyo is the same as in Abu Dhabi - it's thanks to hamburger standardization).

For mathematicians, having a standard way of arranging the Counting Numbers is important. Ideally, it'd be great to lay them out in a row, the next number being 1 higher than the number below it.

Enter the Number Line!

How do we make it? First we draw a horizontal (which means left to right) line. Then we make marks on that line to indicate where on the line the numbers go, then we write the numbers above those little marks. Overall, it looks like this:



I know there's a whole bunch of boxes on this view but I'd like to focus on the red line going across - that's the actual number line. I've made little blue marks to help place where the Counting Numbers are along the line.

Keep in mind that the Number Line continues to the right into infinity - if we wanted to, we could find the number 1,000,000 - we'd have to scroll a long long way but it'd be there.

We're going to use the concept of the Number Line a lot as we move forward with more and more entries.

Vocabulary:

  • Counting Numbers - The set of numbers that starts with 1 and increases by 1 (1, 2, 3, 4,...)

For more information check out these links (comment to add a resource to the list!): 

 https://apps.mathlearningcenter.org/number-line/

 Where might you have come from? 

 Fact-orials Index

Numbers:
Where might we go?

Numbers:
Operations:
Operations with different kinds of numbers:
Related Operations:
Relations:
Graphing:
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Counting/Natural Numbers

The Background

With the advent of the 1 and the accumulation of many of those 1's, we start to create larger and larger numbers of 1's - which get pretty clumsy to work with.

Questions
What are Counting Numbers? What are Natural Numbers? Where did they come from?
Answer
The Counting Numbers are the numbers we grew up counting: 1, 2, 3,... Natural Numbers are that same group of numbers and sometimes includes the number 0 as well. They developed from people's needs to be able to express larger numbers of things.
Analysis

Once the number 1 was established as a number, and people started looking at a certain number of "1s"or tick marks, the need to have more numbers results (although some cultures found it sufficient to use "1, 2, many").

At some point, the ability to express a certain number of "1s" would become important - and so words would be created to express larger and larger groups of "1s".

That then is how we get to the Counting Numbers - these are the numbers that arise from having 1, then having 1 more than that, then having one more than that, etc. The Counting Numbers are the numbers we grew up counting: 1, 2, 3, 4, ... all the way to however high up you want to go (true story - when I was quite young, I could count to 120 but not beyond. The older neighbourhood kids thought that was quite funny...).

The term Natural Numbers can also be used to express this same set of numbers (although sometimes people include the number 0 in with the Natural Numbers - yes folks, mathematicians will debate and deliberate on if 0 should be considered a Natural Number or not...). Because of this, when someone uses the term "Natural Number", make sure to know if they are including 0 or not.

The set of Counting/Natural Numbers is symbolized by .

While I used this image in The Number 1, I'll use it again here to show the first few numbers (0 through 9) and again this image is from quora.com:



 For more information check out the following (comment to add a resource to the list!):

The Hitchhiker's Guide to the Galaxy: Earth Edition's post on the history of numbers
Images of numerals in different countries

 Where might you have come from?

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Tuesday, September 25, 2018

The Number 1

The Background

 In many ways, this entry is The First Entry, at least for mathematics. Consider that there was a time when the concept of numbers, any number at all, just didn't exist. Time passes and human civilization develops and with that, the need to start using numbers is created.

Question
Where does the history of mathematics, and indeed, the history of numbers, begin?
Answer
Frankly, any answer is going to be speculation, lost in the mists of history. But we can speculate...
Analysis

When I was an accounting student, my Accounting 101 prof said that counting started when people began trading - perhaps a farmer sending goods, via a merchant, to town for sale. How do you keep track of the goods that change hands without the use of numbers?

Regardless of where numbers come from, and regardless of how proud that particular professor was at being a part of the origin of mathematics, I think we can state, without doubt, that the first number in existence was the number 1.

1 is a symbol for existence, for self, for a thing. It's easy to create a mark to indicate a single thing and then we can look at the number of ticks to count something up. The gathering up of ticks in groups of 5 gives the classic "4 vertical lines with a crosshatch fifth line" that we see in some parts of the world:



Check out the number 1 from different parts of the world in this image (courtesy of quora.com):



For more information check out the following (comment to add a resource to the list!):

Wikipedia article on the Documentary The Story of 1
Images of numerals in different countries

Where might you have come from?

Fact-orials index

The Start of Everything - Fact-orials Table of Contents

Where might we go now?

Numbers:
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Geometry:
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Fact-orials Index Page

Fact-orials Index

(Hyperlinked entries are active. All others show the direction the resource is going.)
New Content added daily!!!

The Welcome Page!
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The Start of Everything - the Fact-orials Table of Contents
  •  Accounting Principles
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  • Numbers - Defining what different types are...
  • Operations with Numbers - What can we do?
  • Operations with different kinds of numbers:
  • Exponentials and Rational Numbers
  • Exponentials and Rational Numbers - Practice Problems 
  • Addition, Subtraction, and Real Numbers
  • Addition, Subtraction, and Real Numbers - Practice Problems 
  •  Multiplication, Division, and Real Numbers
  • Multiplication, Division, and Real Numbers - Practice Problems
  • Exponentials and Real Numbers
  •  Exponentials and Real Numbers - Practice Problems
  •  Multiplication, Division, and Variables
  • Multiplication, Division, and Variables - Practice Problems
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  •  Exponentials and Variables - Practice Problems
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Monday, September 24, 2018

The Welcome Page

Welcome to Fact-orials!

So what is this site/blog/online thing-y?

You know how there are those sites that delve deep into topics, so much so that finding out what 1 + 1 is can be a real challenge? This isn't that kind of site.

Or how there are those sites that offer tutoring? Doesn't happen here.

This site/blog/online thing-y is here to be a resource that starts with First Principles and allows for navigation along sets of topics. Why is it that the next value after 9 (with one digit) is 10 (with two digits)? How is it that math and accounting are related? Where did the idea come from to develop negative numbers? What is square root of 4? And occasionally even looking at Who - like Who was it that found that equation that allows for finding the missing side of a right triangle?

This site is your site - feel free to comment, like, and share (or to jealously guard the secret of its existence from your friends...).

Seriously though, the site is here to be helpful for you in understanding the topics it addresses. And those topics will be math-y at first but can and will develop into other areas.

I hope you find your time here worthwhile!

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