Roots

Background

We've covered what an exponential is and how it works. We can also go the other direction...

Question

Evaluate:

1.  
2.  

Answer

1. 
2.  

Analysis

Before we dive into what a root is and how it works, let's start with an exponential. Remember that we can take a number, say 2, and multiply it with itself a certain number of times, say 2. That looks like this:



We can go the other way, starting with 4 and working our way back to 2 (we'll use exponential rules to do it):



Let's look at the blue terms above, and in particular the . What is this?

It's called a root and is the inverse relation to an exponential.

There are various ways to express the operation of taking a root. One is to use the exponential form I used above - where we use an integer to express taking something to an exponential power (so the power 2 says we take a number and multiply by itself twice), we use a fractional exponent the root we want to take over 1 (and so the fraction 1/3 in the exponential position means we want the number that when multiplied by itself 3 times gives us the given number).

Another way to express a root is to use the symbol that looks sort of like a dividing sign (below is the square root of 2):



and the cube root of 27:



Notice that if there isn't a small number sitting in the "crook" of the root sign, assume you are taking the square root, or the exponential fraction 1/2.

Now let's talk about some things about the numbers that result from taking a root.

Let's start with even roots.

Notice that if we look at the square root of 4:



I'm looking for the number or numbers that, when multiplied by itself twice, gets me to 4. What numbers might they be?

One is 2: 2 x 2 = 4. Great! There is another number I can use, though, to get to 4: -2 x -2 = 4. And so there are two answers when I take the even root of a number - one will be positive and the other negative. Because of this, sometimes people will express the root as an absolute value:



When taking an even root, this will always happen!

This also means that if I take the even root of a negative number, I won't end up with a real number. For instance, if I take this number:



there are no numbers I can multiply by itself to get to -4. (To get around this, mathematicians use Imaginary Numbers - that'll be in further entries).

So what happens if I take an odd root?

Let's look at the cube root of -27:



If I multiply 3 x 3 x 3, I get 27. Which means that the cube root of 27 is 3.

So what happens if I take the cube root of -27?



Notice that I can say -3 x -3 x -3 = -27, and so therefore the cube root of -27 is -3.

Whenever we take an odd root, whatever sign the number under the root sign is the sign of that numbers root.

Vocabulary used:

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

Where might you have come from?

Fact-orials Index

Operations:

• Exponentials

Associated Operations:

• Absolute Value

Relations:

• Inverse Relations

Where might we go?

Numbers:

• Imaginary Numbers

Operations with Different Kinds of Numbers:

• Exponentials and Rational Numbers

Pi

Background

The history behind finding the value of pi is quite long...

Question

How was the value of pi discovered?

Answer

See below for the history of finding pi...

Analysis

Let's first talk about what exactly pi is. Pi is a Greek letter that represents the value we're talking about. It is an irrational number and so the closest we can come to expressing it as a decimal or a fraction is some sort of approximation.

It is defined as the circumference of a circle divided by its diameter:


Currently, school kids are taught to use either 3.14 or  as an approximation for pi. How was this number found?

Direct Measure

One way ancient cultures found an approximation for pi was to directly measure the diameter and circumference of larger and larger circles. The Babylonians, roughly 4000 years ago, found pi to be 3.125, or . The Egyptians, around 1650 BC, found the value for pi as 3.1605.

As a geometrical approximation

Another way to measure pi is to find it through calculating its area. Since we know how to calculate the area of a square, we can draw a square and inscribe a circle (put a circle in the centre such that it touches the square's perimeter) and within that inscribe another square, like this:



The circle is radius 1. The outside square is therefore equal to 4 square units and the inside square is equal to 2 square units. But we can get closer if we use more and more rounded shapes:



Here we've used an octagon. Clearly, the area of the octagon is closer to the area of the circle than the area of the square. And as we add more and more sides to the figure inscribed inside the circle, the closer it gets to being a circle.

It was known how to calculate the exact value of polygons (plane closed figures with equal straight lines, like a square and an octagon) and so as the number of faces, or sides, increased, the value of the polygon's area would more and more closely approximate the area of the circle.

Both the Greeks and the Chinese used this method. Archimedes narrowed the number down to . Zu Chongzi approximated pi as . To put these into decimal form:

Archimedes: 3.14084507042< pi < 3.142857
Zu Chongzi: 3.14159292035

More modern calculations have brought pi even closer to its actual value. There are a list of other methods used to refine pi in the wikipedia link.

Vocabulary used:

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

This link, to exploratorium.com, has more on the history.
This link, to mathopenref.com, has more on the use of polygons to find pi.

Where might you have come from?

Fact-orials Index

Numbers:

• Irrational Numbers

Associated Operations:

• Estimation and Rounding

Geometry:

• Defining a Circle and Right Angels
• Squares

Where might we go?

Update - 13 Jan 2019

Hi all,

This week I worked in a few different areas:

Geometry:

• Perimeter and Area
• Squares

Expressing values:

• Scientific Notation 
• Estimation and Rounding 

Operations:

• Addition, Subtraction, and Variables - Practice Problems 

Accounting:

• Balance Sheet 

An entry I've been building to is the discovery of the value of pi - everyone knows the value 3.14, but how did people find out the value? We'll work it out this week. Another entry will be on the mathematician Euclid - the Father of Geometry. I'm hoping to get that done this week as well. I'll expand a bit on the balance sheet in the accounting section, and I anticipate getting into roots and logs also this week.

Thanks again for all the support! If you'd like to help in any way, please follow this link: Would You Like to Help?

Until next week!

Parz

Squares

Background

Let's talk about squares...

Question

What is the definition of a square? What is its perimeter? What is its area?

Answer

A square is a plane figure that has four equal sides and four right angles.

Perimeter is the sum of its edges. Since each edge is equal, we can say the perimeter = 4s, where s is the measure of one side.

Area is the space enclosed within the perimeter. Area is the base times the height, but since they are equal, we can say that A = s^2  

Analysis

Squares are one of the first geometric shapes we learn in school. It looks like this:



The perimeter is the sum of the sides. Notice in this example that each side is a measure of 1 unit. Therefore, the perimeter is 4 units:

1 + 1 + 1 + 1 = 4 (1) = 4 units

Since each side is equal, we can say the Perimeter = 4s, where s is a side of the square.

The area is the space the perimeter encloses. We can either count squares with sides of 1 (there's 1 in our diagram), or we can multiply the "bottom" side by the "side". Since both of these are 1, we end up with:

Area = base x height = 1 x 1 = 1^2 = 1 square unit

Since the base and height are equal, we can say the Area = s^2

Vocabulary used:

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

Where might you have come from?

Fact-orials Index

Geometry:

• Perimeter and Area

Where might we go?

Numbers:

• Pi

Perimeter and Area

Background

When we deal with a closed shape we can draw on a piece of paper (known as a two-dimensional figure), we can talk about amount of distance the edge of that shape occupies, and the amount of area the shape encompasses...

Question

What is a perimeter? What is an area?

Answer

The perimeter is the length of the boundary of the shape. The area is the space enclosed by the shape.

Analysis

Let's start with a line segment that is 4 units long:



If we take this length and bend it at point 3 into a 90 angle in the up direction, we get this:



We can do this twice more (keeping the right angles made in the prior moves), we get this:



Let's talk about the distance around the figure - which is known as the perimeter.

As we saw when we started with the initial line, it's 4 units long. We can verify that with this current figure by adding up the individual distances. From (0,0) to (0,1) is 1 unit, (0,1) to (1,1) is 1 unit, (1,1) to (1,0) is 1 unit, and then back again from (1,0) to (0,0) is 1 unit. 1 + 1 + 1 + 1 = 4.

Now we can talk about the space within the 4 line segments. That space within the figure is called the area.

To calculate the area, we can count the number of squares with sides equal to 1. Just to be clear, we've made a square in the graph above (a plane figure with four equal sides and four right angles). There is one 1-unit square in the graph, and so we say the area is 1 square unit, or 1 unit square (can be said either way).

Let's look at another figure:



The perimeter, the distance around the edge of the figure, is 8 units (2 + 2 + 2 + 2).

The area is 4 units square. We can count the squares - we can see them in the figure. Another way we can find the area is to multiply one side of the figure by another side at a right angle to it. In this case, we have 2 units left to right and 2 units up and down, so that's 2 x 2 = 4 square units.

Let's talk about why this works.

If you take an ice cube tray, you can see that there are a certain number of spaces for ice going across and another number going up and down (mine are six long and two up and down). We can multiply the 2 and the 6 together (2 x 6 = 6 x 2) and get 12 spaces for ice. And so we can do the same for measuring the squares inside a square (and, in fact, some other shapes - we'll talk about those in further entries).

Vocabulary used:

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

Where might you have come from?

Fact-orials Index

Geometry:

• Points, Line Segment, Line
• Intersecting, Parallel, Perpendicular Lines
• Defining a Circle and Right Angles

Where might we go?

Geometry:

• Squares

Scientific Notation

Background

Sometimes, we want to get a sense of a number, an approximation, that tells just how big it is. Scientific Notation can be used to do that.

Question

Express these in scientific notation: 3000, 0.0005, 45245, 0.00345

Calculate 3000 x 0.0005 

Answer

3000 = 3 x 10^3, 0.0005 = 5 x 10^(-4), 45245 = 4.5245 x 10^4, 0.00345 = 3.45 x 10^(-3)

3000 x 0.0005 = 1.5 

Analysis

When we're looking at numbers, particularly very big and very small numbers, they can be hard to manage. For instance, what's the difference between 1,000,000,000,000 and 1,000,000,000,000,000 - notice that we have to count zeros and apply words to it - it's difficult to deal with. What would be great is if we could see in an easy way the size of a number. And we can.

We can take a number and express it so that there is an integer between 9 and -9, with the remaining digits behind the decimal point. The size of the number can be seen as a power of 10.

For instance, looking at the number 3000, we can express it as:

3 x 1000

and we can rewrite 1000 in terms of an exponential:

3 x 10^3

We can do the same with a very small number, say 0.0005. We can express it as:

5 x 0.0001

and we can rewrite 0.0001 in terms of an exponential:

5 x 10^(-4)

This is less useful if we have a lot more digits or is smaller. Look at 45245:

45245 = 4.5245 x 10,000 = 4.5245 x 10^4

Or this number: 0.00345

0.00345 = 3.45 x 0.001 = 3.45 x 10^(-3)

We can use multiplication rules to multiply scientific notation numbers in an easy manner. We can take, say 3000 x 0.0005, and through conversion to scientific notation we can easily multiply these numbers:

3000 x 0.0005 = 3 x 10^3 x 5 x 10^(-4) = 15 x 10^(-1) = 1.5 x 10 x 10^(-1) = 1.5

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:

• Real Numbers
• Expressing Values and Bases

Associated Operations:

• Estimation and Rounding

Where might we go?

Balance Sheet

Background

Let's dive in some more on the Balance Sheet, also known as the Statement of Financial Position...

Question

What is reported on a Balance Sheet? How is it organized?

Answer

A Balance Sheet shows what a company owns (assets), owes (liabilities), and what its "book value" (equity) is.  Accounts are organized by more liquid/shorter time frame to less liquid/longer time frame.

Analysis

The Balance Sheet organizes the financial position of a company. It generally will have assets on one side of the sheet and liabilities and equity on the other:



One of the basic equations of accounting is that Assets = Liabilities + Equity and the format of the report reflects this.

In further entries, we'll build the Balance Sheet with different kinds of accounts.

Vocabulary used:

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

An article from Investopedia.com on how to read a Balance Sheet.

Where might you have come from?

Fact-orials Index

Accounting Principles:

• Accounting Reports

Where might we go?

Estimation and Rounding

Background

Sometimes we want to use exact numbers in a calculation or use an exact number as a result. Oftentimes though, we want to use an estimate or get an approximation as a result.

Question

Find the following:

1. Round to the nearest whole number: 6.2, 6.5, 6.7
2. Round up to the nearest whole number: 4.2, 4.5, 4.7
3. Round down to the nearest whole number: 7.2, 7.5, 7.7
4. Estimate 4.32 x 2.1 

Answer

1. 6, 7, 7
2. 5, 5, 5
3. 7, 7, 7
4. 4 x 2 = 8 

Analysis

When we're asked to round a number, we look at the digit position we're asked to round to. In Question 1, we're rounding to the nearest whole number, and so we're rounding to the 1's position:



We're looking at the Ones position to see if changes or not. To see if it does change, we look at the next position to the right, in this case, the Tenths.

When we're rounding (sometimes called "rounding off"), we keep the Ones as is if the Tenths is either 0, 1, 2, 3, or 4. We'll increase the Ones by 1 if the Tenths is either 5, 6, 7, 8, or 9.

In 6.2, the tenths place is 2, so we round to 6.
In 6.5 and 6.7, the tenths place is 5 and 7, respectively, and so we round up to 7.

Question 2

Sometimes we need to round up, which means that if the number we're looking at is even a little bit over, we round to the next value up.

Looking at our numbers in question 2 and rounding up to the next whole number, we can look at the numbers in question:

4.2 is over 4 so we round up to 5.
4.5 is over 4 so we round up to 5.
4.7 is over 4 so we round up to 5.

Question 3

Sometimes we need to round down, which means that if the number we're looking at is even a little bit under, we drop down to the next value down.

Looking at our numbers in question 3 and rounding down to the next whole number, we can look at the numbers in question:

7.2 is under 8 so we round down to 7.
7.5 is under 8 so we round down to 7.
7.7 is under 8 so we round down to 7.

Question 4

When we estimate, we calculate using rounded or approximated numbers to end up with an answer that is close to what is the "true" answer. There isn't an absolute "correct" way to do this - but the better the estimate, the closer our answer will be to the "true" answer.

4.32 x 2.1

One way we can handle this is to round to the whole numbers: 4 x 2 = 8

We could round 4.32 to 4 but keep 2.1 as is to get 4 x 2.1 = 8.4

We could also round 4.32 to the nearest tenth and round 2.1 to 2: 4.3 x 2 = 8.6

In all of these cases, we can assume that the "true" answer is above 8 and approaching 9. (It turns out that the true answer is 9.072).

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:

• Real Numbers
• Scientific Notation

Where might we go?

Numbers:

• Pi

Addition, Subtraction, and Variables - Practice Problems

Background

Let's work some practice problems!

Question

Evaluate:

1. 5a + 3b + 6a - 2b 
2. 3 - 7c + 5x - 8 + 6y 

Answer

1. 5a + 3b + 6a - 2b = 7a + b
2. 3 - 7c + 5x - 8 + 6y = -5 - 7c + 5x + 6y

Analysis

When we're looking at adding and subtracting terms with variables, we combine "like terms". For instance, in Question 1, we combine the a terms and the b terms this way:

a + 3b + 6a - 2b

We can rewrite this as:

a + 6a + 3b - 2b

We can now add the terms more easily:

7a + b

Question 2

While there are a number of terms in this question, 3 - 7c + 5x - 8 + 6y, the only terms that can be worked with are the constants:

3 - 7c + 5x - 8 + 6y

3 - 8 - 7c + 5x + 6y

-5 - 7c + 5x + 6y

Vocabulary used:

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

Where might you have come from?

Fact-orials Index

Operations with Different Kinds of Numbers:

• Addition, Subtraction, and Variables

Where might we go?

Operations with Different Kinds of Numbers:

• Multiplication, Division, and Variables

Admin - 6 Jan 2019

Hey everyone,

Wow - 2019 already! Time flies when you're having fun!

This past week I worked in several areas:

Math (Inequality; Addition, Subtraction, and Variables)
Graphing (X Y Axes)
Geometry (Intersecting, Parallel, Perpendicular Lines; Defining a Circle and Right Triangles)
Accounting (Accounting Reports)

Going forward, I'll continue to develop ideas along these paths.

If you'd like to help in any way, please do let me know! Check out this post for details.

Thanks again for your support!

Parz

Inequality

Background

We talked about finding particular values on a number line, but how do we express a range of values on it?

Question

Graph:

1. x < 1
2. 
3. x > -2
4. 
5.  
6.  

Answer

See below for the graphs:

Analysis

When we're looking at a range of values as a solution to an expression, we indicate that a number is a solution by putting a solid dot over that particular value. For instance, if x = 0, we can put a dot over the number 0 on the number line:


If we have a series of numbers right next to each other, and the series of the numbers are real numbers (so there is no gap on the number line), we can indicate a range by having a series of dots one next to the other - and with them being right next to each other, we end up with a line.

For instance, for Question 1, we want to graph all the values on the number line that is less than 1 (we have the "less than" sign there). We'll therefore show a ray that starts at 1 and heads off towards smaller and smaller numbers. Since we don't want 1 to be a part of the solution, we don't put the full dot over it. What we do, however, is put a hollow dot over the 1 to show that the ray starts there but isn't part of the solution:



For Question 2, we do want the 1 to be part of the solution (the symbol means "less than or equal to", so we put a filled-in dot over the 1:



For Question 3, we do the same type of thing we did in question 1, but now we have the "greater than" sign. We put a hollow dot over -2 and run a ray towards bigger numbers:





And for Question 4, we put a filled dot over the -2:



Question 5 combines the two ranges we've been working with. It reads  and means we want all the values of x that are above or equal to -2 but less than 1. We can express it on the number line this way:



Question 6 reverses the direction of the signs. So let's look at it a little closer:




This says we want to graph all the points (x) that are either less than or equal to -2 or are greater than 1. That looks like this:



Oftentimes, because this notation is a bit confusing (with the two lines heading off in opposite directions), we'll write it as two separate statements. In this case we could say:



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:

• Variables

Relations:

• Greater/Lesser Than
• Equality

Graphing:

• Number Line

Where might we go?

X, Y axes

Background

With a number line, we can look at all the real numbers in a row. But what if I want to draw a shape or otherwise put things in a grid?

Question

Describe a method of laying out a shape on a grid that can be described mathematically.

Answer

Use a two-dimensional grid (in essence, use two number lines)

Analysis

We've started covering topics where having the ability to describe shapes in a manner such that a reader can repeat what a writer has described. And one way to describe points, lines, and other shapes, is to use a grid of perpendicular lines. We can build it this way:

First let's show the number line:



We can set up another number line perpendicular to the first one, running through the number 0:



These are called the axes. The horizontal one (the first one that runs left to right) we can call the "x axis" and the one that runs up and down is called the "y axis".

We can run parallel lines to each axis and in essence create little grids:



And now using these grids we can plot points.

The way we plot points is to use a format of the x value first (going left/right) and then the y value (going up down). We put the numbers into a bracket for easy reading. So for instance we can plot the number (0,0) (it's in blue):



we can plot (4,1):



And we can plot (-1,-2):



We can describe the format as (x,y).

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:

• Variables

Graphing:

• Number Line

Geometry:

• Intersecting, Parallel, Perpendicular Lines
• Defining a Circle and Right Triangles

Where might we go?

Defining a Circle and Right Angles

Background

We can define a line with two points. We can define a circle with one...

Question

Define a circle. Identify some parts of a circle.

Answer

A circle is defined as a figure defined by a single line (the circumference) such that any line segment from the centre of the circle to the circumference is equal to any other such line segment. See below for more terms.

Analysis

We can start with a point:



We can now draw a line around this point such that any line segment we draw from the centre point to the surrounding line is equal:



The purple line is the circle. It is also known as the circumference.

The blue point is the centre of the circle.

The red and black lines are called radii (radii is plural, radius is singular).

If we have a line that runs through the centre of the circle, that is known as a diameter (the black and red lines below):


Notice that the red line and the black line are perpendicular. This means that the circumference is divided into four even pieces. Where we have this situation, the angles between the red and black lines are equal - and are called right angles. We mark right angles with a small square mark:




Note that with the lines being perpendicular, with that one angle being shown as a right angle, they all are (it's understood that all of them are right angles and so all of them need not be marked).

Let's now talk about finding a point along the circumference. 

One way we can measure is by using something called degrees. If we say there are 360 degrees around the circle, then a right angle is 90 degrees (it's one fourth of the way around the circle).

Vocabulary used:

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

Where might you have come from?

Fact-orials Index

Geometry:

• Points, Line Segment, Line
• Intersecting, Parallel, Perpendicular Lines

Where might we go?

Numbers:

• Pi

Graphing:

• X, Y Axes 

Geometry:

• Perimeter and Area