Simple math you DON'T NEED to know.

I moved the original post down to my second post and made a quick index of everything I've done here so it's easier to see what's going on.

Memorizing Fractions: Halves up to Tenths
http://www.flopturnriver.com/phpBB2/...10.html#702810

Approximating Square Roots Mentally (second half of post)
http://www.flopturnriver.com/phpBB2/...10.html#702810

Approximating Base 10 Logarithms Mentally
http://www.flopturnriver.com/phpBB2/...24.html#703624

Approximating Sine Mentally
http://www.flopturnriver.com/phpBB2/...35.html#703635

Using Sine Approximations to Find Cosine, Tangent, Cotangent, Secant, and Cosecant
http://www.flopturnriver.com/phpBB2/...36.html#703636

Divisibility Tests for 2, 3, 5, 6, 9
http://www.flopturnriver.com/phpBB2/...42.html#703642

Divisibility Tests for 4, 8
http://www.flopturnriver.com/phpBB2/...46.html#703646

Divisibility Test for 7
http://www.flopturnriver.com/phpBB2/...12.html#703712

Multiplying Two Two-Digit Numbers Mentally
http://www.flopturnriver.com/phpBB2/...16.html#703716

Figuring a 15% Tip Mentally
http://www.flopturnriver.com/phpBB2/...56.html#704956

Squaring Numbers Quickly
http://www.flopturnriver.com/phpBB2/...08.html#705008

SOHCAHTOA imo.

It's pretty easy to memorize all of the fractions up from halves up to tenths.

These first ones are pretty easy and most people know them without having to think.

1/2 = 0.5
1/3 = 0.333..., 2/3 = 0.666...
1/4 = 0.25, 2/4=1/2=0.5, 3/4 = 0.75
1/5 = 0.2, 2/5 = 0.4, 3/5 = 0.6, 4/5 = 0.8
1/10 = 0.1, 2/10=1/5=0.2, 3/10=0.3, and so on.

The ones that people tend to have problems with are the rest of the lot.

With the sixths, you only need to learn 1/6 and 5/6 since 2/6=1/3 which you already know, 3/6=1/2 which you already know, and 4/6=2/3 which you already know.

1/6 = 0.1666... (easy to remember since the numbers are 1 and 6 ldo)
5/6 = 0.8333...

With the sevenths, (this is going to sound a lot harder than it is so keep reading) they do this thing with the repeating sequence 142857 which is cool since 7+7 = 14, the first part of the sequence, 14+14 = 28, the second part of the sequence, and 28+28 = 56 which is one less than the last part of the sequence. The way you do the sevenths is when you're wanting x/7 to start with the xth highest digit in the sequence. For example, with 3/7 I want to start with the 3rd digit number in the sequence which is the 4, so the sequence goes like this:

3/7 = 0.42857142857142857... and so on.

For 6/7 we would look for the 6th highest digit in the sequence (7) and start there:

6/7 = 0.7142857142857... and so on.

For the sake of completeness, here are the first six digits of all of the sevenths:

1/7 = 0.142857...
2/7 = 0.285714...
3/7 = 0.428571...
4/7 = 0.571428...
5/7 = 0.714285...
6/7 = 0.857142...

Now for the eighths, you only have to learn a few new ones (much like the sixths) because you already know that 2/8=1/4=0.25 and so on. If you know what half of 25 is, it helps:

1/8 = 0.125
3/8 = 0.375
5/8 = 0.625
7/8 = 0.875

The ninths are as easy as the tenths, you just repeat the number:

1/9 = 0.111...
2/9 = 0.222...
3/9=1/3=0.333...
4/9= 0.444...
5/9 = 0.555...
6/9 = 0.666...
7/9 = 0.777...
8/9 = 0.888...
9/9 = 0.999... = 1

I'll probably post more stuff later.




Okay dokey, time to learn how to approximate square roots in your noggin. This requires that you know your squares up to the number you want to approximate for (1, 4, 9, 16, 25, 36, and so on). I'll use examples and go through the process twice.

Example 1: Find the square root of 89.

Step 1: Identify the largest square under this number (81). This tells us that our answer is going to be like 9 and some change (9.xxx), and 9 is a number we will use in step 3.

Step 2: Take the difference between your number and that square (89-81=8). We will use 8 in step 3.

Step 3: Divide the number we found in step 2 by twice the number we found in step 1 (8/18). This is the fraction part of the square root.

Step 4: Put it all together. The square root of 89 is pretty close to 9 and 8/18, which we know is 9 and 4/9, which we know is 9.444... from above.

The actual square root of 89 is 9.434 so we're pretty damn close.

Example 2: Find the square root of 102.

Step 1: Identify the largest square root under 102 -- this is 100, so our answer will be 10-point-something.

Step 2: Find the difference between our number and that square 102 - 100 is 2.

Step 3: Divide step 2 by twice step 1: 2/20.

Step 4: This is about 10 and 2/20, which should be about 10.1.

As it turns out, the square root of 102 is 10.100.

For what it's worth, there are also algorithms that aren't too hard for calculating sine, cosine, tangent, logarithms of various bases, and so on, but I don't feel like typing that much.

Ni Han Sir

Originally Posted by spoonitnow

With the sevenths, (this is going to sound a lot harder than it is so keep reading) they do this thing with the repeating sequence 142857 which is cool since 7+7 = 14, the first part of the sequence, 14+14 = 28, the second part of the sequence, and 28+28 = 56 which is one less than the last part of the sequence. The way you do the sevenths is when you're wanting x/7 to start with the xth highest digit in the sequence. For example, with 3/7 I want to start with the 3rd digit number in the sequence which is the 4, so the sequence goes like this:

3/7 = 0.42857142857142857... and so on.

For 6/7 we would look for the 6th highest digit in the sequence (7) and start there:

6/7 = 0.7142857142857... and so on.

For the sake of completeness, here are the first six digits of all of the sevenths:

1/7 = 0.142857...
2/7 = 0.285714...
3/7 = 0.428571...
4/7 = 0.571428...
5/7 = 0.714285...
6/7 = 0.857142...

For sevenths and fourteenth's, notice: 7 x 14 = 98, which is almost 100. So...

1/7 is about 14%, and
1/14 is about 7%

Since these are touchdown numbers, American football fans shouldn't have much trouble with poker table estimates:

2/7 ~ 28%
3/7 ~ 42%, etc

Same for 14ths:

1/14 ~ 7%
3/14 ~ 21%
5/14 ~ 35%

No offense, spoon, but I would rather have my eyelids torn off than learn your way with 7ths

BTW, this method is easy enough to actually do it at the poker table which is NOT the point of the thread, I know. So, sorry spoon, for screwin' that part up.

Thanks for your contribution Robb

hahahaha, well played sir

*golf clap*

Originally Posted by Robb

No offense, spoon, but I would rather have my eyelids torn off than learn your way with 7ths

lol

That's just how I was taught to do it when I was in like the 6th grade or whenever my teacher made us memorize all that (also it's exact), but I know your way too =P

Originally Posted by spoonitnow

Originally Posted by Robb

No offense, spoon, but I would rather have my eyelids torn off than learn your way with 7ths

lol

That's just how I was taught to do it when I was in like the 6th grade or whenever my teacher made us memorize all that (also it's exact), but I know your way too =P

It's funny. I'm a math teacher, so I'm pretty good at mental arithmetic. But I hate doing it. If I can't think of the answer in like 2 seconds, I pull out a graphing calculator (which my office is littered with) or pop open the math software with floating point arithmetic that allows for 100+ significant digits, and let my fingers do the walking.

Originally Posted by Robb

Originally Posted by spoonitnow

Originally Posted by Robb

No offense, spoon, but I would rather have my eyelids torn off than learn your way with 7ths

lol

That's just how I was taught to do it when I was in like the 6th grade or whenever my teacher made us memorize all that (also it's exact), but I know your way too =P

It's funny. I'm a math teacher, so I'm pretty good at mental arithmetic. But I hate doing it. If I can't think of the answer in like 2 seconds, I pull out a graphing calculator (which my office is littered with) or pop open the math software with floating point arithmetic that allows for 100+ significant digits, and let my fingers do the walking.

Haha yeah I hear you.

I took Pre-Calc and Calc I in high school and our instructor was really big on teaching us how to use calculators to do all kinds of stuff but rather light on teaching us how to do stuff without one. Then when I got around to Calc II and beyond the instructors were very anti-calculator to put it lightly. Thankfully I was too poor to have an expensive calculator in high school so I was on par with everyone else

One of my modern algebra professors were really big on showing us how to do different stuff mentally, and knew algorithms for taking all kinds of roots and logs of different bases and other random stuff. I have it all in a notebook somewhere and if I can find it I might post it.

I was wondering this...
if 1/3 = .33333
2/3 = .66666666
3/3= .99999999?

actualyl... how is this not right?

Originally Posted by vaks

I was wondering this...
if 1/3 = .33333
2/3 = .66666666
3/3= .99999999?

actualyl... how is this not right?

Remember, it's .99999 repeating forever.

.9 is 1/10 from 1
.99 is 1/00 from 1
.999 is 1/000 from 1
.999999 is a millionth from 1
.999999999 us a billionth from1

Since this process repeats for ever and ever, out to infinity, 0.999 repeating is in fact = 1.

Okay I found one of the notebooks I had some of this stuff in. Here's a quick way to approximate logarithms (base 10) but I don't have how accurate this is in terms of %'s but it should be pretty damn close. The key is that you just need to memorize the approximations for the log of 1-9 if you haven't already. Here is a list and a 3x3 matrix of the results that I can still remember them by even though I haven't used this in over a year.

log 1 = 0
log 2 = 0.3
log 3 = 0.5
log 4 = 0.6
log 5 = 0.7
log 6 = 0.8
log 7 = 0.85
log 8 = 0.9
log 9 = 0.95

Code:

0     0.3  0.5
0.6   0.7  0.8
0.85  0.9  0.95

And for completeness, log 10 = 1.

For whatever reason I can remember this 3x3 by rows pretty easily so maybe you can too. Now that you know this, doing logs in your head is pretty easy as long as you know [remember] how to manipulate logs in the first place. Here are a few guidelines that you learn in algebra or pre-Calculus or where ever it's taught these days:

log(a*b) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(a^b) = b * log(a)

So here are a few quick examples.

Example 1: Find log(28).

Answer: log(28) = log(4*7) = log(4)+log(7) = 0.6 + 0.85 = 1.45.
(The real result is ~1.447).

Example 2: Find log(3000).

Answer: log(3000) = log(3 * 10^3) = log(3) + log(10^3) = log(3) + 3 log (10) = 0.5 + 3 = 3.5.
(The real result is ~3.477).

Example 3: log(36) = log(6^2) = 2 log(6) = 2 * 0.8 = 1.6.
(The real result is ~1.556).

So there you go. Maybe I'll do some trig stuff at another time.

Triptanes suggested I do some trig stuff, and since it's kind of early and I don't feel like playing or studying or getting my girlfriend up to go get some breakfast (she had a long night) I'll type this method out for approximating the sine of angles in degrees where the angle is between 0 and 90. Our answers will be a small fraction.

Warning: This is not nearly as hard as it looks, but it will take a few minutes of practice until you can remember the algorithm. It also helps if you can keep up with a number in your head for a moment or two if you're wanting to do this mentally. If you need a sheet of paper to do this that's fine because it's still going to be very useful.

Part 0: The sine of 0 degrees is 0 and the sine of 90 degrees is 1. So what we're going to be working with is a range of fractions with values between 0 and 1.

Part 1: For angles less than 35 degrees, you can just divide the angle by 60 and be pretty close. For example, sin(30) actually equals exactly 1/2. Also sin(10) ~= 10/60 = 1/6 which we know is 0.1666... and the actual value of sin(10) is about 0.174 so we're not too far off with this.

Part 2: For angles greater than 35 degrees we have a short algorithm that's going to appear to be much harder than it is to people who aren't used to seeing algorithms for this sort of thing. With about five minutes of practice, most people will be able to do these in just a couple of seconds. So here we go:

Step 1: 90 - our angle.
Step 2: Divide step 1 by 60.
Step 3: Square step 2.
Step 4: Take half of step 3.
Step 5: Subtract 1 - step 4.

Note: In step 4, doubling the bottom half of the fraction has the same result as taking half of the top half of the fraction and is generally easier.

Note: Don't try to simplify the fraction as you go along unless it's effortless, which will usually be right after step 2 when you divide by 60. The goal here is to keep up with what the top half of the fraction is and what the bottom half of the fraction is.

Note: A shortcut for step 5 is to replace the top of the fraction with the difference between the top and bottom of the fraction. In the first example below I do 1 - 9/32 as (32-9)/32 = 23/32 which is generally easier to compute mentally.

This probably looks kind of hard to a number of you, and it can be until you practice a little, so we'll do some quick examples.

Example 1: Find sin(45).

This is what my thought process is like when doing this approximation: 90 - 45 is 45 (step 1). Now our fraction is 45/60 which equals 3/4 (step 2). Now our fraction is 9/16 (step 3). Now our fraction is 9/32 (step 4). Now our fraction is 23/32 (step 5). And we're finished.

As it turns out, 23/32 ~= 0.72 and sin(45) ~= 0.71 so we're pretty close.

Example 2: Find sin(65).

Again, this is what my thought process is like when I do this approximation: 90 - 65 = 25 (step 1). Now it's 25/60 which is 5/12 (step 2). Now it's 25/144 (step 3). Now it's 25/288 (step 4). Now it's 263/288 (step 5). And we're done.

The fraction 263/288 ~= 0.91 and sin(65) ~= 0.91 as well, so we're pretty damn close.

Now you can approximate sine for angles between 0 and 90, and for my next trick, we'll learn how to turn that result into other fun stuff.

Approximating Cosine

If you know how to do the sine approximation for angles between 0 and 90, then you already know how to do the same for cosine because cos(x) = sin(90-x) meaning that if you want the cosine of 20 degrees it's equal to the sine of 70 degrees which you already know how to do from the above.

Approximating Tangent

We know that tan(x) = sin(x)/cos(x) and we know that cos(x) = sin(90-x) so we know that tan(x) = sin(x)/sin(90-x), which we know how to find from the sine approximation, though it's going to be a lot easier to keep things straight if you right down each of the results for sin(x) and sin(90-x) as you go.

Approximating Secant, Cosecant, Cotangent

We know that csc(x) = 1/sin(x), sec(x) = 1/cos(x), and cot(x) = 1/tan(x) so when we get that fraction at the end of our algorithm we can just "flip it" to get the other trig function values.

I'll see what else I can dig up.

I just thought of another thing that's pretty useful but I do it so much I take it for granted. I suspect more people know this one than the sine approximation algorithm.

Numbers that end in 0, 2, 4, 6, or 8 are divisible by 2, and numbers that end in 0 or 5 or divisible by 5. Pretty common right?

Numbers that add up to 3, 6, or 9 are divisible by 3. For example 123456 becomes 1 + 2 + 3 + 4 + 5 + 6 = 21, 2 + 1 = 3, so 123456 is divisible by 3 (and it's 41152).

Numbers that are divisible by 3 and are even are divisible by 6. From the above, 123456/6 = 20576.

Numbers that add up to 9 are divisible by 9. And yes, this works for all numbers, not just 2-digit ones. Take 63423 for example. 6 + 3 + 4 + 2 + 3 = 18, 1 + 8 = 9, and 63423/9 = 7047.

Forgot to add that 1000 is divisible by 8, so any number where the last 3 numbers are divisible by 8 will be divisible by 8. Example: 345235234534534523452345234523453064

Similarly 100 is divisible by 4, so any number where the last 2 digits and divisible by 4 will be divisible by 4. Example: 425654354982574892574589723489579823475923475934712

I know a way to test for divisibility by 7 but it's not very fast unless you're uber-familiar with it, and even then it's pretty fucked. However, I did a little research and found a decent test for 7 that I'll attempt to explain better than the source did (http://www.jimloy.com/number/divis.htm).

Step 1: "Remove" the ones digit from the number.
Step 2: Subtract twice that ones digit from the rest of the number.
Step 3: If the result is divisible by 7, the original number is as well. If you don't know if it's divisible by 7, then repeat the process until you find one that is.

We'll start with 133. As an example. 13 - 6 = 7, so 133 is divisible by 7.

Another example is 1467. We have 146 - 14 = 132, then 13 - 4 = 9 which is not divisible by 7 so 1467 is not divisible by 7.

That completes how to test divisibility for 2, 3, 4, 5, 6, 7, 8, and 9.

On the link above, there is a test for 11 as well but I didn't read it.

Quick way to multiply two two-digit numbers mentally (takes practice)

Say we were going to multiply 32 times 26. If you do this on paper, the algorithm basically everyone learns will make it look something like this:

Code:

     1
     32
   x 26
   ----
    192
    64
   ----
    832

But if you can keep up with a couple of numbers then you can find the solution mentally pretty quickly.

The ones place will be 2x6 = 2 with a 1 carrying over.
The tens place will be 2x2 + 3x6 + 1 = 4 + 18 + 1 = 3 with a 2 carrying over.
The hundreds place will be 3x2 + 2 = 8.

It's kind of hard to explain this one in text, but here's what's basically going on:

Code:

     ab
   x cd
   ----

So here our ones place will be bxd.
Our tens place will be axd + bxc + anything carried over from doing the ones place.
Our hundreds place will be axc + anything carried over from doing the tens place.

The same process can be expanded for multiplying any two numbers of the same (or even different) length but I think two-digit numbers are a start for anyone.

While I was eating lunch today I thought of something else I can add to this thread: How to quickly figure a tip. The general consensus is that a standard tip is 15%, and that's what I'm going to teach you to do here.

First thing you need to learn is how to find 10% of a number. It's easy: just move the decimal place one spot to the left. For example, 10% of $42.00 is $4.20, and the decimal place is just moved one spot to the left.

The second thing you need to learn is how to find 5% of a number. It's easy: just find 10% of the number as per above and take half of it. For example, 10% of $42.00 is $4.20, half of that is $2.10 which is 5% of $42.00.

So, find 10% and 5%, and add that together to get your 15% tip.

Example: Your bill is $23.57. Round that to $24, and find that 10% is $2.40. Half of that is $1.20. Now $2.40 + $1.20 = $3.60, which is close enough to 15% for our purposes. Note that 15% of $23.57 is actually $3.54 but nobody is going to leave some weird amount of change for a tip usually.

Example: Your bill is $140. Find 10%, which is $14, and half of that is $7. Now $14 + $7 = $21 which is 15%.

Squaring Numbers Quickly

Note: The following bit is an application of the fact that a^2 = (a+b)(a-b) + b^2 for any a, b in the real number system.

Note: This is a little weird at first so I'm giving plenty of examples. And yes, I'm doing these examples off of the top of my head.

Suppose we want to find the square of 14. The closest power of ten to 14 is 10, and since it's 4 more than 10, we add another 4 to 14 to give us 18 which will be the "left" part of our answer. Note our difference here was 4. The "right" side of our answer is the above difference squared, or 16.

Now we want to use addition to bring together the "left" and "right" parts of our answer. We know that 14 squared will be a 3-digit number, so moving 18 to the "left" and 16 to the "right" creates 180 + 16 = 196, which is 14^2.

Suppose we want to find the square of 7. The closest power of ten to 7 is 10, and since it's 3 less than 10, we subtract another 3 from 7 to give us 4 which will be the "left" part of our answer. The difference here was 3. The "right" side of our answer is the difference squared, or 9. So 7 squared is 49.

Suppose we want to find the square of 96. The closest power of ten to 96 is 100, and since it's 4 less than 100, we subtract another 4 from 96 to give us 92 which will be the "left" part of our answer. Our difference was 4, and 4 squared (16) will be the "right" part of our answer. We know that 96 squared will be a four-digit number, so we put our left and right parts together to give us the answer: 9216.

Suppose we want to find the square of 88. The closest power of ten to 88 is 100, and since it's 12 less than 100, we subtract another 12 from 88 to give us 76, which will be the "left" part of our answer. The difference was 12, and 12 squared is 144, the "right" part of our answer. We know that 88 squared will be a four-digit number, so we do 7600 + 144 = 7744 which is 88 squared.

you maths geeks make my head hurt.....

Originally Posted by spoonitnow

While I was eating lunch today I thought of something else I can add to this thread: How to quickly figure a tip. The general consensus is that a standard tip is 15%, and that's what I'm going to teach you to do here.

First thing you need to learn is how to find 10% of a number. It's easy: just move the decimal place one spot to the left. For example, 10% of $42.00 is $4.20, and the decimal place is just moved one spot to the left.

The second thing you need to learn is how to find 5% of a number. It's easy: just find 10% of the number as per above and take half of it. For example, 10% of $42.00 is $4.20, half of that is $2.10 which is 5% of $42.00.

So, find 10% and 5%, and add that together to get your 15% tip.

Example: Your bill is $23.57. Round that to $24, and find that 10% is $2.40. Half of that is $1.20. Now $2.40 + $1.20 = $3.60, which is close enough to 15% for our purposes. Note that 15% of $23.57 is actually $3.54 but nobody is going to leave some weird amount of change for a tip usually.

Example: Your bill is $140. Find 10%, which is $14, and half of that is $7. Now $14 + $7 = $21 which is 15%.

Standard tip amounts in some of areas are approaching 20% (or are there already). Either way, I just do 20% to not be a tip nit and it's easier to calculate imo: divide the total by 5 and there's your tip.

However, if the service is terrible I feel more inclined to go through the math and calculate the 15% :P

or move the decimal over one spot and double, which is a lot easier than /5 in your head

If the service is terrible I'll leave a very small tip in the range of 50 cents on a ~$40 bill. I feel this is more insulting than leaving no tip at all.