Newton’s Clock, Part 3

So now I’m up to “4” on Newton’s clock:

So the expression

${\left({2\sin\frac{\mathit{\pi}}{2}}\right)}^{2}$

uses the sine function which has been talked about many posts before. Only this time, it is using radian measure of angles instead of degrees. If your calculator is in degree mode, you can substitute 90° in place of 𝜋/2 to get the same answer. The sine of 𝜋/2 radians or 90° is 1. So in the brackets we have 2 × 1 = 2. 2² = 4, hence its position on the clock.

Now let’s look at

$\sqrt[3]{125}$

This is the cube root of 125. This expression is asking the question: “What number multiplied 3 times equals 125?”. The answer to that is 5 because 5 × 5 × 5 = 125. So once again, the clock does not lie.

Now let’s look at 3! This is pronounced “3 factorial”. The factorial of a number is that number successively multiplied by a number which is 1 less. So 5! = 5 × 4 × 3 × 2 × 1 = 120. So 3! = 3 × 2 × 1 = 6. Factorials are used a lot in probability. I have touched on this before but perhaps there is another future post here.

Now let’s look at 01112. We are very familiar with decimal system way of counting. This system is a base 10 system because we use 10 distinct digits (symbols) to count: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When we run out of digits, like when we count up to 9, we add another place holder to the right of the number and put the starting digit 0 there: 10. And then successively increase it’s digits until we get to 9 again. then we increase the left digit by 1 and start over again: 20, 21, … . There are other number systems based on numbers other than ten.

Computers are composed of switches based on two states, on or off. We mathematically say that off is 0 and on is 1. Computers essentially count with just o’s and 1’s: a base 2 system. Counting in base 2 is done exactly as we do in base 10, we just have fewer digits to work with.

So we if we start counting we get 0, 1, but we’ve ran out of digits so we add a place holder to the right and start again: 0, 1, 10, 11. Ran out of digits again so add another place holder and start over: 0, 1, 10, 11, 100, 101, 110, 111. If you are keeping track, 111 in base 2 is equal to 7 in base 10. It is a convention to subscript a number with its base when dealing with other base systems, so 01112 means 7 in base 10. The leading 0 doesn’t add to the value but in computer maths, base 2 numbers are typically written 4 digit places at a time.

Percentages, Part 2

So how do you convert percentages to fractions and decimals and vice versa? This post will show examples of each.

1. Convert a percentage to a fraction:

This one is easy as if you remember, a percentage is already a fraction where the numerator is displayed and the denominator is 100. So you just create the fraction and simplify it (see my posts on fractions):

${40}{\%}\hspace{0.33em}{=}\hspace{0.33em}\frac{40}{100}\hspace{0.33em}{=}\hspace{0.33em}\frac{{2}{0}\hspace{0.33em}\times\hspace{0.33em}{2}}{{20}\hspace{0.33em}\times\hspace{0.33em}{5}\hspace{0.33em}}\hspace{0.33em}{=}\hspace{0.33em}\frac{2}{5}$

2. Convert a percentage to a decimal:

This one is just a matter of moving the decimal point, two places to the left. Keep in mind that the decimal point will not usually show at the end of integer percentage, but you can assume it to be at the end of the number:

37% = 37.% = 0.37

18.5% = 0.185

112% = 1.12

0.15% = 0.0015

Any 0’s at the end of the decimal, can be left off:

40% = 0.40 = 0.4

3. Convert a decimal to a percentage:

This is just the opposite of of the above: you just move the decimal point two places to the right, then add the % symbol:

0.25 = 25% (if an integer results, you can leave the decimal point off)

0.2786 = 27.86%

0.002 = 0.2%

2.345 = 234.5%

4. Convert a fraction to a percentage:

Here you multiply by 100/1, simplify, then multiply numerators together and denominator together. It is advisable to simplify before multiplying:

$\frac{3}{5}\hspace{0.33em}\times\hspace{0.33em}\frac{100}{1}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{\rlap{/}{5}\hspace{0.33em}\times\hspace{0.33em}{1}}\hspace{0.33em}\times\hspace{0.33em}\frac{\rlap{/}{5}\hspace{0.33em}\times\hspace{0.33em}{20}}{1}\hspace{0.33em}{=}\hspace{0.33em}{60}{\%}$

Sometimes though, not as much cancels and you will need to do some division in the end (long or short – see my post on long division):

$\frac{8}{9}\hspace{0.33em}\times\hspace{0.33em}\frac{100}{1}\hspace{0.33em}{=}\hspace{0.33em}\frac{800}{9}\hspace{0.33em}{=}\hspace{0.33em}{800}\hspace{0.33em}\div\hspace{0.33em}{9}\hspace{0.33em}{=}\hspace{0.33em}{88}{.}{89}{\%}$

In my next post, I will show how to do some of the more common problems using percentages.

Percentages, Part 1

I am confident that the chance of you liking this post is 100%. But what does 100% mean? I will talk about this in the next few posts.

First let’s review what a fraction is. The fraction

$\frac{3}{4}$

means that I have 3 pieces (the numerator) of some whole thing that was divided into 4 pieces (the denominator). So the denominator sets the size of the pieces (the bigger the denominator, the smaller the pieces) and the numerator sets how many pieces.

Being creatures that have 10 fingers, we naturally migrate to things that are powers of 10. Way back in the Roman empire days (I remember them fondly), the Romans frequently used fractions that had a denominator of 100. In fact, the word “percentage” comes from the latin (Roman) per centum which means “by the hundred”. That has continued to today, as a percentage is really a fraction where the denominator is fixed at 100. What you see in a percentage is the numerator:

${37}{\%}\hspace{0.33em}{=}\hspace{0.33em}\frac{37}{100}$

So a percentage like 37% means that if I divide something into 100 equal pieces, I have 37 of these pieces. Percentages are also used to indicate a probability. My first sentence in this post used a percentage in this way. If you remember, a probability of an event is a fraction. The probability of event A is expressed as P(A):

${P}{(}{A}{)}\hspace{0.33em}{=}\hspace{0.33em}\frac{{\mathrm{Number}}\hspace{0.33em}{\mathrm{of}}\hspace{0.33em}{\mathrm{times}}\hspace{0.33em}{\mathrm{event}}\hspace{0.33em}{\mathrm{A}}\hspace{0.33em}{\mathrm{occurs}}}{{100}\hspace{0.33em}{\mathrm{trials}}}$

So 100% of you liking this post means that if I take 100 random people who have read this post, 100 of them will like this post. This means all will like this post as that is what 100% means: the whole:

${100}{\%}\hspace{0.33em}{=}\hspace{0.33em}\frac{100}{100}\hspace{0.33em}{=}\hspace{0.33em}{1}$

Now percentages, fractions, and decimals are all ways to express a part of something. In my next post, I will show how to convert percentages to fractions and decimals and vice versa.

Algebra, Subscripts

In my tutoring travels, I notice that some students get confused when they see subscripts, for example ${x}_{1}$. As you know, there are only 26 letters in the alphabet. This is almost always enough to represent variables in algebra, but if a formula indicates a pattern, then this is difficult to do using just letters.A subscript is just a way of showing different unknowns using the same letter. ${x}_{1}$ is a different unknown than ${x}_{2}$ but the same letter x is used – only the subscript has changed. The subscript number just indicates an order and is not used in calculations.So for example, the method for finding the average of a set of numbers is to add up all the numbers, then divide by the number of numbers you just added. To show this in a maths formula:Average = $\frac{{x}_{1}\hspace{0.33em}{+}\hspace{0.33em}{x}_{2}\hspace{0.33em}{+}\hspace{0.33em}{x}_{3}\hspace{0.33em}{+}\hspace{0.33em}\cdots\hspace{0.33em}{+}\hspace{0.33em}{x}_{n}}{n}$Here, the pattern is easy to see. Each number in the set of numbers is given a different subscript. Since the subscript starts at 1 and ends in n, you can immediately see that there are n numbers, which is why the formula shows us dividing by n. The symbol “⋯” is called an ellipsis and indicates that you just follow the indicated pattern until you get to the last number, ${x}_{n}$. For any specific set of numbers, you know what n is, but since the formula is to apply for any set of numbers, we need to use the unknown n.Sometimes, the subscript is called an index. So in more complex formulas, you may see  ${x}_{i}$ to represent any of the unknowns. So we could say that the average is the sum of all  ${x}_{i}$‘s divided by n.

Algebra, Word Problems

Nothing strikes fear in a math student like the dreaded word problem. I have seen many students who are very good at solving equations but do poorly with word problems. The problem is that they lack the skill to convert english into an equivalent math language. In my last post, I started with converting english phrases into algebraic expressions. Let’s graduate to a full word problem and create the equivalent algebraic equation.

Karen is twice as old as Lori. Three years from now, the sum of their ages will be 42. How old is Karen and Lori?

As I suggested in my last post, let’s break this down. So here we have two unknowns: Karen’s and Lori’s ages. So a good first step is to assign letters to these unknowns.  Let’s let K be Karen’s age and L be Lori’s age. Now the first sentence in the problem has a word that means “=” in math. That word is “is”.  In other word problems, you may see words like “the same as”, “equals”, “was”, “will be”.

The first sentence in the word problem directly converts to an equation since we already assigned letters to the two ages:

Karen is twice as old as Lori: K = 2L

Now there are two unknowns here but that’s OK. We can’t solve anything yet, but there is more information in the word problem. As we read it, write down the equivalent math expressions.

Three years from now, what are their ages three years from now? Well three years from now, Karen will be K + 3 and Lori will be L + 3.

the sum of their ages will be 42. Another equation here because of the words “will be”. So we add the ages of Lori and Karen three years from now to get 42:

(K + 3) + (L + 3) = 42

The brackets are not really needed. I just put them there so you can see that I am adding Karen’s age 3 years from now to Lori’s age 3 years from now.

Now we have the equation but there are two unknowns. You usually cannot solve a single equation  with more than one unknown. But remember the first equation we wrote down: K = 2L? This equation means that algebraically, K is exactly the same as 2L. In the second equation, we can replace the K with 2L:

(K + 3) + (L + 3) = 42: (2+ 3) + (L + 3) = 42

Now we can solve this equation to find what is. I covered solving equations before, so I won’t do a lot of explaining here. I will start by removing the brackets and proceed:

2+ 3 + L + 3 = 42

3L + 6 = 42

3L = 42 – 6 = 36

L = 36/3 = 12

So Lori is 12. What about Karen? Again, look at the things we’ve written down so far. We have K = 2L, that is, Karen is twice as old as Lori. Since we already know Lori’s age, Karen must be 24.

So in most word problems, it will help if you first assign letters to the unknowns, then create expressions and/or equations from each part of the word problem. Have these all together and usually, the equation you need to solve will pop out.

Algebra, The Beginnings

So let’s leave statistics for a while and return to algebra. This post will be a bit more basic but it illustrates a skill needed when converting word problems to equations. First, a couple of definitions:

An Expression in algebra is basically anything you can write down in algebra without the “equal” symbol. You can think of an expression as either side of an equation. Examples are

${x}^{2}\hspace{0.33em}{+}\hspace{0.33em}{3}{x}\hspace{0.33em}{-}{1}{,}\hspace{0.33em}{2}{a}\hspace{0.33em}{+}\hspace{0.33em}{3}{b}{,}\hspace{0.33em}\frac{{x}\hspace{0.33em}{+}\hspace{0.33em}{1}}{{x}\hspace{0.33em}{-}\hspace{0.33em}{1}}{,}\hspace{0.33em}{5}$

They can be complex with more than one unknown or as simple as a number.

An Equation in algebra is two expressions with an “equal” symbol between them. Examples are

$\begin{array}{l}{{x}^{2}\hspace{0.33em}{+}\hspace{0.33em}{3}{x}\hspace{0.33em}{-}{1}\hspace{0.33em}{=}\hspace{0.33em}{2}{a}\hspace{0.33em}{+}\hspace{0.33em}{3}{b}}\\{\frac{{x}\hspace{0.33em}{+}\hspace{0.33em}{1}}{{x}\hspace{0.33em}{-}\hspace{0.33em}{1}}\hspace{0.33em}{=}\hspace{0.33em}{5}}\\{{y}\hspace{0.33em}{=}\hspace{0.33em}{7}}\end{array}$

So let’s look at creating expressions. In what follows, I am going to write an english phrase and follow that with the equivalent algebraic expression.

Double a number: 2x

One more than a number: x + 1

Half of a number: x/2

Seven less than triple a number: 3x -7

Take 5 more than a number then double it: 2(x + 5)

Other letters can be used to show an unknown number, but x is mostly used. Sometimes more that one unknown is needed:

Cost of 3 pears that cost each: 3p

Cost of 7 apples that cost a each: 7a

Total cost of the above fruit: 3p + 7a

The individual price 3 people pay at a restaurant if they split the bill: C/3

The total number of pencils in a classroom if each girl has 3 pencils and each boy has 2: 3g + 2b

It is usually a good practice to break down a word problem and write down the expressions first before generating an equation.

Let’s look at creating equations in my next post.

Exponents

Now let’s add another maths symbol: exponents. Now multiplication is really successive adding: 2 × 3 = 2 + 2 + 2 or 3 + 3. In other words, 2 × 3  can be thought of as adding 2 three times or adding 3 two times. Well exponents are indicating successive multiplication. Example:

$\begin{array}{l} {{3}^{2}\hspace{0.33em}{=}\hspace{0.33em}{3}\hspace{0.33em}\times\hspace{0.33em}{3}}\\ {{2}^{3}\hspace{0.33em}{=}\hspace{0.33em}{2}\hspace{0.33em}\times\hspace{0.33em}{2}\hspace{0.33em}\times\hspace{0.33em}{2}}\\ {{x}^{3}\hspace{0.33em}{=}\hspace{0.33em}{x}\hspace{0.33em}\times\hspace{0.33em}{x}\hspace{0.33em}\times\hspace{0.33em}{x}} \end{array}$

I will expand on this tomorrow.

My logo

Let me explain a bit about my logo. The foreground words are self-explanatory except for the 𝜋 symbol used in place of the “T”. 𝜋 is the Greek letter pi. Greek symbols are used extensively in maths and 𝜋 is the most common one used. You will see it used many times in subsequent posts.  The background equations not only have math equations, but symbols representing all sorts of math applications: astronomy, biology, chemistry, genetics, nuclear physics, electronics, … . I think this is a very appropriate logo for me. What do you think?

Why can we do maths?

Ever wonder why humans can do math? What was the evolutionary pressure that gave us the ability to do calculus? It turns out that our math abilities are a by-product of our language abilities. A good book that explains this is “The Maths Gene” by Keith Devlan. Check it out!

Introducing the DavidTheMathsTutor Blog

Hi all, Well after a bit of learning how to set up a blog site and addressing the security concerns and a lot of trial and error, I’ve finally set up the blog site. I’m sure some tweeking will occur in the early days, and I’m open to comments to improve the site. I will post my first official math related post soon. Hopefully, the facebook page will be automatically updated.