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hi Chenfeng Liu

Welcome to the forum.

Also you may post questions here for help.

Bob

hi woodturner550

7/16 + 3/10 = 21/160

note. LCM of 16 and 10 is 80.

Bob

I tried googling that quote again and most hits attribute it to Galileo but a similar one "I cannot teach anybody anything, I can only make them think" is attributed to Socrates. I think you're right. Galileo would have read Socrates and the two quotes are very similar especially when you take account of translation variations.

A modern version is "You can lead a horse to water, but you can't make him drink."

You can also use the method to find the highest common factor.

eg. Find the HCF of 45 and 72

45 = 3 x 3 x 5

72 = 2 x 2 x 2 x 3 x 3

This time extract all the primes that occur in both. So 3 x 3

If you create a Venn diagram with all the primes and each circle enclosing those for one of the numbers then the lowest common multiple (LCM) is the union of the sets and the HCF is the intersection.

Bob

I don't think young children would understand what is going on. It's very abstract and learning starts with 'concrete' experiences. eg. Children learn to count by counting objects. They learn about subtraction by having some objects (eg. beads) and taking some away.

One problem with people with deductive logic understanding, they are harder to lead.

Isn't that a good thing rather than a problem.

Bob

1/3 2/9 1/4 3/16 3/10

Reduce each demoninator to its prime factors

3 = 3

9 = 3x3

4 = 2x2

16 = 2x2x2x2

10 = 2x5

Write the largest power of each prime and multiply them

(2x2x2x2) x (3x3) x 5

That's the only way to make sure that 16 and 9 go into it.

Bob

8÷2(2+2)=?

Not 8÷(2(2+2))=?

So I think there's only one interpretation.

Do the bracket first 8÷2(4)=?

Put in the missing multiplication sign

8÷2x4

Then evaluate in order

4 x 4 = 16

Wolfram Alpha agrees with this.

Bob

ps. rules such as PEMDAS, BODMAS etc are unclear that multiplication and division have equal status.

So 21 x 3 ÷ 7 and 21 ÷ 7 x 3 are the same.

hi paulb203

They mean the same. In number theory, division is defined as the inverse operation to muliplication.

So if a times b = c then c divided by b is a.

This is why division by zero has no meaning. [a times 0 = 0 for all a, so if we tried to calculate 0 / 0 we'd get a where a can be anything] and [ x / 0 would mean what number must we multiply 0 by, to get x as our answer. This cannot be answered.]

So an example would be what number when doubled gives 10 ? x times 2 gives 10 can then be re-written as x = 10 / 2

Bob

hi paulb203

That's what I was taught too. It was 'explained' as borrow ten for the top and pay it back at the bottom.

eg. 42 - 17 = = (40 + 10 + 2) - (17 + 10)= (40 - 10 - 10) + (12 - 7)

It was only when I came to teach it that I realised I had never questioned it and worked out for myself why it works.

I think it is technically known as subtraction by equal addition. (ie add ten top and bottom)

Primary schools at that time were encouraged to teach the decomposition method because it was argued it makes more sense and so pupils would be more likely to remember it.

eg. 42 - 17 = 30 + (10 + 2 ) - (10 + 7) = (30 - 10) + (12 - 7)

If, at secondary level, a pupil was struggling with subtraction, my first task was to find out which method they had been taught and then stick with it, explaining it properly so the pupil could work it out for themselves.

That principle of 'don't try to change their method' but 'make sure they can understand it' underpins most of my teaching philosophy.

Bob

hi woodturner550

I'd be interested to see some output from your program.

In Python, time.time() returns a UTC value ; in seconds and decimal fractions of a second. A computer's motherboard must save and increment this value using the computer's internal clock. As the motherboard keeps running even when the computer is switched off this will get out of step with the standard time and so the operating system must, from time to time, log in to a standard time somewhere to get the current UTC. This will take time (no pun intended) so the computer's version of UTC will never be exact. But that doesn't matter for your purposes, in fact, it helps because your version of UTC will be different from anyone else's.

Do you know how, for example, MS Excel makes up it's random numbers. Could it be that it already uses something similar to what you are proposing ? If it does then MS probably already has the copyright on this. It may also be that they don't publish this coding; precisely because they don't want it hacked.

If a computer virus were able to modify the system's record of UTC, could this be a weakness that would allow a hacker to predict your numbers ? If you can devise a way to be certain that the internal value ( of UTC) held by the computer's operating system is genuine; then you've got a way to be certain your method is reliable.

I think there are statistical tests for the reliability of randomness. You'd need a large set of random outputs.

you wrote:

I understand that this is not a programing forum

We do have a computer section called Coder's corner. I can move this thread across if you would like.

Bob

hi StephenDiaz

Welcome to the forum.

Sadly this forum is being effected increasingly by AI generated text. Your post acheived this result:

98% Probability AI generated

So we have two possibilities:

You're a robot and I'll ignore your post;

You're a human with a genuine request. If the latter I apologise for suspecting you of being a robot.

If you want specific help then post a question. What you have posted so far is way to vague I'm afraid.

Bob

hi woodturner550

Welcome to the forum.

I've used the 'random generator' on Excel quite a few times and the BBC Basic one, and I've never thought to question how it's done. Each time you 'call' the function it yields a different value so it looks random. But, you're right, it must be using a formula and hence results could be predicted by someone who knows how it works. Many years ago in the UK the Premium Bond organisation made a piece of equipment which they called ERNIE (= Electronic Random Number Indicator Equipment). Folk who buy premium bonds don't get a percentage interest, rather their numbers are entered in a draw, which pays out money to the lucky winners.

As far as I'm aware it has never been 'hacked'.

All digital computers run from an internal clock. If you can find out how to access this then you would have your stop watch moment. Since the clock cycles are very fast it would be very hard for anyone to predict the moment you did this.

Subsequent numbers could be generated from within your routine by the same means. The length of time that would elapse between numbers would probably be indeterminate too as the computer is doing thousands of other things whilst running your routine, which would be taking an unpredictable amount of time each, so the clock cycles would have progresed by an indeterminate amount each time.

Bob

hi luzbyrd

Welcome to the forum.

The one area I already knew about is the application of group theory in solving the Rubik's cube. If you're interested search for the name Singmaster.

Then I asked my pal google and came up with:

fields __ telecoms, medical imaging, navigation, power generation

complex geometry __ computer graphics, architecture, design

rings __ industrial control systems

linear algebra __ algorithms, graphics, data mining, machine learning.

At this point I decided to stop. You can complete your search by trying google yourself.

I should emphasize: I don't know the underlying theory for most of the above.

The other point that is worth making is this. Many areas of 'pure' mathematics seemed to be just there because mathematicians like to play around with ideas but who can predict when these theories might become useful in the real world? Who would have thought that a study of number bases and, in particular, base 2, would have become the basis for all computing or that prime numbers would be useful in encryption.

Bob

Meaning truncate:

WolframAlpha wrote:

1 | verb | replace a corner by a plane

2 | verb | approximate by ignoring all terms beyond a chosen one

3 | verb | make shorter as if by cutting off

4 | adjective | terminating abruptly by having or as if having an end or point cut off

That means that 6.25 truncated to 1dp is 6.2 not 6.3

Bob

Whoever wrote that answer needs to check the meaning of truncated. I'm with you on this one.

Doubt it'll come up in an exam but if it does I'm sure you'd get full marks.

Bob

We want /x^2 so factorise on the left and also take the 4 across to the RHS

Divide by x^2 and add 3

Bob

I've just thought again about my 'example' and I said it incorrectly. It's the gap that needs rounding down else a cupboard might appear to be ok but will not fit due to rounding. eg true gap is 50.6 cupboard is 50.8

I have a vague memory that a question like this did come up years ago and the mark sceme said award full marks for 14.5 or 14.4999999999...

Bob

Mostly .5 is rounded up but as its exactly halfway between the options you could make a case for rounding down.

Why is round up preferred?

(1) there only has to be one non zero digit after the 5 to tip the number over to round up.

(2) if you round down when the tenths digit is 0 1,2 3 or 4 and up if it's 6, 7 , 8, or 9 then that's 5:4 so choosing up for .5 gives even ups and downs. In a company such as an electric supplier this means bills are rounded up as often as down so the regulator is kept happy.

Example of rounding down. Various cupboards have to be pushed through a narrow gap. We know the gap is 50cm. Each cupboard is measured for width to see if it will fit through the gap. Round the widths down to the nearest cm

Why?

Bob

.

Let cuberoot(2) be c. Then c^3 = 2

Bob

If a=-5 then a^2 =25. Sqr(number) is taken to mean the positive square root which would be 5 not -5.

By writing |a| this difficulty is avoided.

Bob

For me, it's been a long day and I came home to discover you had been very rude to another member over an innocent remark and three members complaining about your behaviour.

In my haste I wrongly described your signature as offensive. I intended to edit this to inappropriate but you had already posted.

I have banned you for your failure to keep to our rules. One says that posts mustn't be promotional. Usually I delete those members from the forum

At the moment you are banned which is a lesser punishment since deleting means all posts get deleted too.

On Wednesday I will lift the ban. All future posts must be about your maths studies. The number should be moderate and you should avoid making unnecessary posts.

Bob

Factorial is only defined for whole numbers. There is a thing called the gamma function. See what wolfram says.

Bob

As both 5 and 8 go into both length and width you can put the tiles either way round and there's no half tiles needed.

Quickest is to use n times area of tile = area of floor.

Bob

Correct

Bob

A shopkeeper buys an item for C it sells it for R.

Their profit is the difference.

Bob

You can rewrite this as 2 times 2^(2x)

and use logs

Bob