The Start of Everything - the Fact-orials Table of Contents

Background

Not to sound trite or corny, but there was a time where our ancestors were mostly concerned with surviving - the time and effort (much less the brain power) needed to begin wondering about Life, Death, the Universe, and Everything In It, was just not important. Life was a struggle - wondering and thinking about things other than survival were a waste.

But as man evolved, as society developed, and as survival was no longer a 24/7 challenge, there was time to sit and wonder, to think, to speculate. Stories emerged to help explain some of the mysteries of life, such as the origin of the earth, the existence of different species and how they came about, the seasons, and more.

From those questions and the resulting answers came more questions and yet more answers. And so on. This is still the status quo of humanity - continually developing new social structures, new ways to understand self, the environment, the Universe, and everything within it.

It's from these first thoughts that Fact-orials tries to start - to follow the paths where human thought and endeavour have sprung and where they've gone.

Let's follow some paths.

One set of paths we can follow is the use of numbers to record, transmit, discover, and predict information:

Math Fact-orials - the story here starts with The Number 1, mathematical tools are developed to analyze many different aspects of the Universe.

• Accounting Fact-orials - starting with The Start of Accounting, numbers and math are used to describe financial information about an entity

Vocabulary used:

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

The Story of Mathematics

Where might you have come from?

Fact-orials Index

Where might we go?

• Accounting Fact-orials
• Math Fact-orials

Factorials in a Circle

Background

We know how to calculate the number of ways we can arrange distinct items (such as books each with different title) in a row (like on a shelf). Does anything change if we arrange those same items around a table?

Question

Adam, Barb, Carol, and Derek are going to play cards at a round table. The players don't care about which chair they sit in, but they do care about what order the people are around them. In how many different ways can the players sit?

Answer

(n-1)! = (4-1)! = 3! = 6

Analysis

Let's start with looking at the players, ABCD, can stand in a row (perhaps they are waiting outside the venue in order to play cards). From the Factorials entry, we know that there are 4! = 24 ways for them to stand in a row.

Now they approach the table they will play at. A will sit down first. Does it matter which chair he sits in? No - they are all equal. Where A sits has no bearing on anything.

B will now sit and gets to decide she'll sit on A's right, left, or straight across. So B has three choices as to where to sit.

C will then sit and can pick from the remaining two spots.

And then D sits in the last remaining spot.

And so the number of possible ways for the four people to sit is 3 x 2 x 1 = 6.

Remember that when we multiply natural numbers up to a given number, that's a factorial calculation that we can symbolize using an exclamation symbol. So in this case we have:

3! = 3 x 2 x 1 = 6

And in general, when we seat people at a round table, the number of ways we can seat them, looking only at the relationships of the people to each other, we can say that the number of ways they can be arranged is:

(n-1)!

Vocabulary used:

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

Where might you have come from?

Fact-orials Index

Combinatorics:

• Factorials

Where might we go?

Roman Numeral Multiplication Table

Background

Just for fun, what would a multiplication table look like using Roman numerals?

Question

Do a multiplication table using Roman numerals

Answer

See below...

Analysis

To make this table fit, I had to shrink the image a bit. The resolution isn't the greatest, but hopefully this will get the point across that using the Roman numeral system to multiply will lead to difficult to interpret results:



Vocabulary used:

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

Where might you have come from?

Fact-orials Index

Numbers:

• Roman Numerals

Operations with different kinds of numbers:

• Multiplication Table

Where might we go?

Multiplication Table

Background

Now that we've talked about how to multiply, let's make a resource that makes it easier to multiply.

Question

Make a 10 x 10 multiplication table

Answer

See below:

Analysis

Let's just go ahead and make that table:



Notice that we don't care if we put the red number x blue number, or if we do it the other way. This is because of the Commutative Property.

Before closing out this entry, let's point out what happens when we multiply by 1: we get the same number we started with. This makes 1 the multiplicative identity - or in other words, we can multiply a number however many times we want with the number 1 and we'll always get that same number. Like this:

5 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 = 5

Vocabulary used:

• Commutative Property - the property that says a x b = b x a

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

https://www.helpingwithmath.com/by_subject/multiplication/mul_tables_charts.htm

Where might you have come from?

Fact-orials Index

Numbers:

• The Number 1
• Prime and Composite Numbers

Operations:

• Multiplication

Properties:

• Commutative Property

Where might we go?

Numbers:

• Roman Numeral Multiplication Table