#### What is the second most irrational number?

The most irrational number out there is $\phi$. This is followed the second most irrational number, $\sqrt{2}$.

11/11/2019

The most irrational number out there is $\phi$. This is followed the second most irrational number, $\sqrt{2}$.

04/11/2019

$x=\frac{-b\pm\sqrt{b^{2}−4ac}}{2a}$,

for all values of $x$.

Just dervice the quadratic equation.

28/10/2019

$4y-2=2 \rightarrow 4y=4 \rightarrow y=1$.

21/10/2019

There are three picture cards and an ace per suit. That's four.

There are four suits.

That's four times four - sixteen.

There are fifty-two cards in a standard deck.

Sixteen out of fifty-two is the same as $\frac{4}{13}$.

14/10/2019

$12p-21=7p+9 \rightarrow 5p=30 \rightarrow p=6$..

07/10/2019

$2(x+3)+2(2x)=18 \rightarrow 4x+2x+6=18 \rightarrow 6x=12 \rightarrow x=2$.

30/09/2019

Sir Isaac Newton.

23/09/2019

6886

16/09/2019

$20\choose10=184756$

09/09/2019

3

02/09/2019

$x=5$ is the only solution.

26/08/2019

$x=3$ or $x=8$

19/08/2019

$x=-26$

12/08/2019

$8\choose3$ or 56.

05/08/2019

5 to 14. Work it out if you don't believe us. To see the workings, check the M4TH5 Forums.

29/07/2019

$3x = 16+5 =21$.

$x=\frac{21}{3}$.

$x=7$.

22/07/2019

They both satisfy the equation $x^{2} +x =72$. Can you explain why there are infinite relationships like this?

15/07/2019

The infamous unabomber.

08/07/2019

$x+2$.

$x^{2} \div x = x, \ x\left(x+3\right) = x^{2}+3x$.

$x^{2} +5x -\left(x^{2}+3x\right)=2x$.

$\frac{x^{2}+2x}{x}=x+2$.

01/07/2019

8.

$8+\frac{1}{2}8=12$.

24/06/2019

$\frac{11x+4}{20}$.

17/06/2019

56... we'll let you figure this one out.

10/06/2019

Ultimately, a prime number can be defined as a natural number with exactly four factors.

1) Positive itself

2) Negative itself

3) Positive one

4) Negative One

The number 1 only has two factors. Not four. It shares the pairs of positive factors and also the negative factors. This is why one is not a prime number.

03/06/2019

4-3x > -5, $x \in \mathbb{Z}$

$4-3x>-5$.

$\ -3x>-5-4$ ... take 4 from each side.

$\ -3x>-9$.

$\ \ \ 3x<9$ ... multiply both sides by -1 and reverse the inequality sign.

$\ \ \ x<3$ ... divide both sides by 3.

$\therefore x = 2, 1, 0, -1, -2, \ldots$.

27/05/2019

$7 \times 2.5 = 17.5$.

$17.5 \times 600 = 10,500$.

$10,500 \times 0.25 = 2,625$.

$\frac{2,625}{3} = 875$.

20/05/2019

Well, this can't really be answered. There is always a more complicated way.

We would approach this (while trying to save time) by splitting this down to the most simple units and adding them up individually.

16 + 16 = 8 + 8 + 8 + 8.

8 + 8 + 8 + 8 = 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4.

4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2.

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.

Now if you count the number of one's, you get the answer. 32.

So 16 + 16 = 32.

Obviously there are many many more ways of proving this that are more complicated. But this is our simple yet slightly (very slightly) complicated approach to the question.

13/05/2019

Yes. A simple addition question. Not all of these need to be difficult you know. Some are really simple. Like this one.

5 + 7 = 12.

06/05/2019

No.

The circle has an area of $\pi \ r^{2}$.

A square has a area of $L \times W$. This means that the length of one of the squares sides equals a multiple of $\pi$. This obviously is not possible as $\pi$ is an irrational number. So, no, you cannot square the circle.

29/04/2019

If, $\frac{x}{0} = x$ what does $\frac{0}{x}$ equal?

4.

Why? Well...

if $\frac{x}{0}=x$, then $x=0x$ (multiply each side by zero).

$0x=0$, so, $x=0$.

Therefore, $\frac{0}{x}= \frac{0}{0}$.

$\frac{0}{0}=4$ because when you multiply both sides by zero, you still get zero.

$0=0(4)$

Likewise, the answer is also 5, and 3, and 6, and any natural number you can think of.

Actually, $-1(0) = 0$, so it's not just natural numbers. It's intergers. Except, $2.5(0) = 0$, so it's any rational number.

Hold on, $\pi(0) = 0$, so it's any real number, and that's the answer.

But hold on, $\sqrt{-1}(0) = 0$ also. So it's any complex number.

In fact, it is undefined. There are too many different numbers it can be that we do not consider it a number. Some may consider it as one, but that doesn't make sense algebraically. We just don't do it. It brakes math as we know it.

22/04/2019

$\sqrt{.2^{-2}}^2$. That's one way we found. We know that looks confusing though.

Here it is in a slightly less ... confusing way.

15/04/2019

P + Q=6 | KP + Q = 18 | P + KQ = 30

KP + Q + P + KQ = 30 + 18 ... adding the final two systems.

K(P + Q) + P + Q = 48.

And since P + Q = 6, we get K(6) + 6 = 48.

K(6) = 42.

K = $\frac{42}{6}$.

K = 7.

08/04/2019

01/04/2019

25/03/2019

If it is divisible by 3 and also 5, then it must divisible by 3x5 = 15.

Since:

15 = 7 x 2 + 1

1 is the remainder. We want a remainder 5. So let's multiply the equation by 5.

15 x 5 = 5(7 x 2 + 1)

75 = 35 x 2 + 5

So the answer is 75.

18/03/2019

*a + b + c = x = abc*, so
*a + b + c = abc*.
* c = abc - a - b*.

Note:

Since

kc = 2kc - kc

The coefficent infront of

So

So

As above:

Divide by

1 = k

Put back into other equations.

ab = 2k

ab = 2(1)

So either

3 + c = 2c

3 = 2c-c

3 = c

1 + 2 + 3 =

So

11/03/2019

04/03/2019

Well, first you have to figure out how many seconds you have been alive for.

Well there's 60 seconds in a minute.

60 x 60 seconds in an hour.

3600 seconds.

3600 x 24 seconds in a day.

86400 seconds.

86400 x 365 seconds in a normal year.
**31536000 seconds.
**

31536000 + 86400 seconds in a leap year.
**31622400 seconds.
**

Next, how many leap years have you been alive for.

84$\div$4 = 21 years.

[If 85 years old, you'd divide 85 by 4 and round down. Same for 86 and 87. Unless the result is even, you round down, not up.]

Next, how many normal years were you alive for?

84 - 21 = 63 normal years.

So the number of seconds you were alive for is: [31536000 x 63] + [31622400 x 21]

2650838400 seconds you were alive for.

The 'n Choose r' function would tell you the answer, but if you put that into a calculator, it'll be too big. Luckily, thanks to the internet, there are High Precision Calculators.

We used this one to calculate the answer below: https://keisan.casio.com/calculator.

2.513402616591275829041783946019934930247267348224179879211788934269136263836500637385643571163009606E+104

25/02/19

16 x 10 x 5 x 261 = (208,800) number of seconds in a year.

The speed of light is 299,792,458 m/s

299,792,458x208,800 m/year

62,596,665,230,400 m/year

Divide by 1000 to make km/year

So a light year in this case is 62,596,665,230.4km

18/02/2019

To find out to what degree (power) this sequence deals with, we find the common difference.

4 15 40 85 156 259

\ / \ / \ / \ / \ /

11 25 45 71 103

\ / \ / \ / \ /

14 20 26 32

\ / \ / \ /

6 6 6

The 3rd difference is the same, so the sequence is in the 3rd Degree. This means that the sequence is written in the form An^{3} + Bn^{2} + Cn + D. So now we can use simultanious equations to find A, B, C, D.

T(1) = 4 = A(1)^{3} + B(1)^{2} + C(1) + D

= A(1) + B(1) + C(1) + D

= A + B + C + D

T(2) = 15 = A(2)^{3} + B(2)^{2} + C(2) + D

= A(8) + B(4) + C(2) + D

T(3) = 40 = A(3)^{3} + B(3)^{2} + C(3) + D

= A(27) + B(9) + C(3) + D

From here, we can calculate, through observation that

A, B, C and D = 1. So we can conclude that the formula is:

n^{3} + n^{2} + n +1

11/02/2019

04/02/2019

Out of the 4,531,985,219,092 different states of which a traditional 7x6 Connect Four game can have, there are 1,905,333,170,621 (just under two trillion) different ways of finnishing, from litterally dropping four down in a straight line, to filling the whole board and drawing. Out of this there is 1,904,587,136,600 ways to win.

It takes a long time to calculate, but computers can analyse the system to determine the figures you see above. If you have written a programme that can solve this question, Please feel free to share it with us at info@m4th5.ie.

28/01/2019

Well, logically it doesn't. That's upfront obvious. If you cut off at any point in this infinite series, the result will be exponential. However, using certain maths, this can be proven in a mathematical sense.

See here.

21/01/2019

Well, the most irrational number is the one with the most simple continuous fraction. This is:

1+(1/(1+1/(1+1/(1+1/(1+1/(1+1...))))))

To view as an image click here.

If we make this x, we can see that after the first fraction line, *x* is repeated.

Therefore we can write this as:
*x = 1+(1/x)*

Multiply accross by x and we get:
*x ^{2} = x + 1*

bringing accross [

Using the quadratic formula, we can simply determine that \(x = \)\(1+\sqrt{5}\over 2\) which is phi.

this is better shown here.

14/01/2019

Find the percentage error for the length of the rectangle. Give your answer to 2 decimal places.

The percentage error for length is the error devided by the true value, multiplied by one hundred.

If the error = 0.5cm

And the true value = 15.5 cm

Then:

(0.5/15.5)x100=3.23%

07/01/2019

a+b+c+d=x

3(a+1+b+1+c+1+d+1)=3(a+b+c+d+4)

=3(x+4)

=3x+12