### Arithmetic Problem #6

Try this question!

$\frac{999999 \times 999999}{1+2+3+4+5+6=7+8+9+8+7+6+5+4+3+2+1} = ?$
Hint: Simplify it first! 🙂

### Arithmetic Problem #5

Saw this question on the internet and thought its rather interesting. A-levels students should be all buried other their workload, amidst the mock papers and ten year series. Hopefully this might help.

$\frac{100001 + 100003 + 100005 + \ldots + 199999}{1 + 3 + 5 + 7 + 9 + 11 + 99999} = ?$

Hint: try and group the number together.

### Trick to squaring numbers

Many students go wow when I evaluate workings, without a GC. I’m not showing off, but it is because I don’t really carry a calculator with me. haha. So some students do ask me how I evaluate the square of numbers so quickly. I thought, in light of A-level’s coming, I should share something interesting.

Firstly, we trace back to a formula that we saw in primary school.

$(a+b)^2 = a^x + 2ab + b^2$

This formula is really going to be the core of us solving any square of numbers.

Next, we just need to split the number that you want to square effectively. So consider,

$63 = 60 +3$
$63^3 = (60+3)^2 = 60^2 + 2(60)(3) + 3^2$

Some students will ask why not use $(a-b)^2$ instead. This is plausible, but we usually are better with addition than multiplication hah.

### Arithmetic Question #4

Here is an interesting question to test students if they know who to evaluate their square roots.

$\sqrt{(-8)^2}$

Ans: 8

First of all, students should know that this is positive square root so the resultant answer should be positive. 🙂 Arithmetically speaking, we need to value the parentheses first to get $\sqrt{64}$.

### The Modulus Sign #2

So now let us look at a simple relationship that many students memorise instead of understand. Below, we see a absolute function graph.

Now we all know that $|x| gives us $-a. So how did that happen? Consider drawing a horizontal line, $y=a$ in the graph above. $|x|$ is less than $a$ only when $-a. Similarly, $|x|>a$ is true only when $x>a$ or $x<-a$. I do hope that the graphical representation helps here.

### The Modulus Sign #4

I received this from a student of mine who was really confused with the transition from line 2 to line 3. Students should give it a thought before reading my explanation below.

Firstly, my student felt that $|(2x-1)+2yi| \le |(x+1)+yi|$ was being squared on both sides to remove the modulus and the next step should be $[(2x-1)+2yi]^{2} \le [(x+1)+yi]^{2}$. She is very mistaken, as $(2x-1)+2yi$ is a complex number, which is a vector, and not a mere scalar. So the transition from step 2 to step 3 was really an evaluation of the modulus. If we do not skip any steps, the following should clarify how we got from step 2 to step 3.

$|(2x-1)+2yi| \le |(x+1)+yi|$

$\sqrt{(2x-1)^{2}+(2yi)^{2}} \le \sqrt{(x+1)^{2}+(yi)^{2}}$

$\sqrt{(2x-1)^{2}+(2yi)^{2}} \le \sqrt{(x+1)^{2}+(yi)^{2}}$

$(2x-1)^{2}+(2yi)^{2} \le (x+1)^{2}+(yi)^{2}$

I do hope this raise awareness for students to treat modulus carefully and question themselves, when they are dealing with a scalar or vector.

### The Modulus Sign #1

A few weeks back, I conducted a weekend of lessons on the modulus sign (or absolute value sign) with all my JC2. Its to prepare them for the upcoming prelims and A-levels. Many students till today, are still having difficulties with the modulus sign. I thought I will start a little series, just like the integration to introduce some misconceptions.

So what does modulus do? We know that $|-2|=2$, so it pretty much makes everything negative, positive. Is that it? This is a very loose representation of modulus (absolute) function. I usually tell my students to think of modulus like magnitude. We are really interested in the magnitude to $-2$ here, and that will simply be $2$. Why do I choose such a representation? Simply because it is the actual meaning!! Applying a modulus, is seeking the magnitude.

It so happen that $-2$ is a scalar quantity. And when we talk about $|x+yi|=\sqrt{x^{2}+y^{2}}$, we are applying pythagoras’ theorem here to find the magnitude. This is slightly different because $x+yi$ is no longer a scalar, but a vector!

### Confusion on when to put ± sign

The $\pm$ sign definitely can cause a lot of confusion for students. For starters,

$|x|=\pm{x}$
$x^{2}={|x|}^2$

and the list goes on.

So my focus today is to explain when to put $\pm$ while dealing with $\sqrt{ }$. First, lets recognise a few things.

$x=\sqrt{9}$: This gives us $x=3$

$x^{2}=9$: This gives us $x=\pm{3}$

Yes, the square-root can be raised to a square for the x, but notice the difference in answer in both cases. The former is a $+3$ while the latter has $-3$ too. This is because the square-root is always a positive square-root and does not return $\pm{ }$. On the other hand, $x^{2}={|x|}^2$ and the $|x|=\pm{x}$ which causes the $\pm{ }$.

So next time, please be alert when you introduce the $\pm{ }$.

### How many chess games are possible?

The Shannon number, named after Claude Shannon, is an estimated lower bound on the game-tree complexity of chess of 10120, based on about 103 initial moves for White and Black and a typical game lasting about 40 pairs of moves.

Did you also know there are over 9 million different possible positions after three moves each. There are over 288 billion different possible positions after four moves each. The number of distinct 40-move games is far greater than the number of electrons in the observable universe.

### Mathemusic!

So we shared a bit on mathmagic by Dr. Benjamin a week ago. He showed off some amazing arithmetic skills. So today, we shall share a bit of mathemusic. How wonderful $\pi$ can get! This is something for all the music lovers and math lovers. It’s a song written to help memorize π.

Credits: David Macdonald

Can’t get enough of Math! 🙂