Re-Introduction

In this feat, someone gives you a date, and you quickly state the day of the week on which it fell. This new approach is updated for the 21st century, and employs new tips and tricks that help make this feat simpler to learn and quicker to perform.

Approach:

The Day of the Week For Any Date feat combines both memory and mental math. A relatively simple mastery of both, though, will create a response far out of proportion to the required work.

Before I describe the basics of the approach, I'd like to help you get a good idea of your goal, as well as what's possible, by seeing this feat performed by various people in the following videos:

• Dr. Arthur Benjamin performing at TED
• 8-year old Florian Osmani on German TV (in German)

Here's the basic principles, broken down into simple steps:

1) Day and Months Number Conversion: To work out the days of the week mentally, we need to convert them into numbers. We'll also need to convert the months into numbers, to adjust for their effects. These are taught in an easy-to-remember manner.

2) Addition of 3 Numbers: Without using a calculator, can you tell me what 6 + 6 + 31 is? That's about as difficult as the basic formula gets. If you're comfortable doing that, you won't have a problem working through the formula.

3) Subtracting Multiples of 7: Let's say you're asked about the 27th of a month. Regardless of the month or year, we can state with certainty that the 27th of a month will fall on the 6th (since it's 3 weeks, or 21 days, earlier). Since adding 6 is simpler than adding 27, and will give the same result, why not use 6? If you learn to subtract multiples of 7, this makes the arithmetic so easy that you won't have to worry about addition problems any tougher than 6 + 6 + 6!

4) Year Number Conversion: After becoming comfortable with all of the above when given dates in the years 2000 to 2003, you'll learn how to remember and adjust the leap years in the 21st century to key dates. After learning those, you'll learn a simple way to adjust for any year from 2000 to 2099, and even adjust for other centuries!

Running through all these principles, there will be an emphasis on recognizing and taking advantage of patterns. The quicker you can recognize a pattern, the quicker will be your calculation.

We'll start with the codes for the days of the week, since that is our goal. All the formulas and patterns you'll learn later will result in a number from 0 to 6. This number is turned into a weekday as follows:

Day of Week	Number	Mnemonic
Sunday	0	SUNday=NONEday
Monday	1	MONday=ONEday
Tuesday	2	TWOSday
Wednesday	3	Three fingers look like a W
Thursday	4	FOURSday
Friday	5	FIVEday
Saturday	6	SIXturday

To help get you comfortable with converting numbers into days, take the Weekday Codes quiz here. Once you can get a perfect score in a short time, then continue with this tutorial.

Next, you need to learn the codes for the months. Like the weekdays, they range from 0 to 6:

Month	Number	Mnemonic
January	6	WINTER has 6 letters
February	2	February is 2nd month
March	2	March 2 the beat.
April	5	APRIL has 5 letters (& FOOLS!)
May	0	MAY-0
June	3	June BUG (BUG has 3 letters)
July	5	FIVERworks
August	1	A-1 Steak Sauce at picnic
September	4	FALL has 4 letters
October	6	SIX or treat!
November	2	2 legs on 2rkey
December	4	LAST (or XMAS) has 4 letters

Take the Month Codes quiz here to help reinforce the mnemonics.

Note: It's important to note that, in leap years, January reduces by one to 5, and February reduces by one to 1. The other years don't change in leap years. Leap years will be discussed in more detail later.

Once you're comfortable with both the month and weekday codes, you're ready to start calculating your first dates. In the next tab, you'll learn how to use these codes to work out dates for the years 2000 to 2003.

Formula

Ready for the formula? Here it is: Month Code + Date + Year Code = Day of Week Code. It's a lot simpler than many people think, but there are some fine points to learn.

We haven't covered year codes yet, so I'm just going to teach you four simple ones with which to start out:

• 2000 = 0
• 2001 = 1
• 2002 = 2
• 2003 = 3

Those shouldn't be too hard to remember, should they? We'll learn more year codes later in this tutorial, but for now, we'll just focus on these years.

Let's start with a simple example. Let's figure out May 1, 2000. The code for May is what? The mnemonic is MAY-0, so May is a 0. The date itself is the first, so we use 1. The year code for 2000 is 0, so our problem is 0 + 1 + 0 = 1.

Which weekday has a code of 1? ONEday is MONday, so Monday is the day of the week on which May 1, 2000 fell. You can verify that at this site. Congratulations, you've just calculated your first date!

Let's try a date that's a little more challenging. Our next date is October 4, 2000. October is 6 (remember “SIX or treat”?), and 2000 is still 0, so that gives us 6 + 4 + 0 = 10. And the weekday that goes with 10 is...

Wait a minute, the weekday codes only range from 0 to 6! What do we do with a 10, or any result higher than 6 for that matter?

If you look on a calendar, the 10th of any month will always fall on the same day of the week as the 3rd, because the 3rd is 7 days earlier. So, with the result, or any number in the formula, we can subtract 7, or any multiples of 7 to reduce the answer. You may want to refresh yourself on the multiples of 7 with the help of Schoolhouse Rock here.

In our October 4, 2000 example, we got 10 as a result, so we can reduce that by 7. 10 - 7 = 3, so our result boils down to 3. Which day of the week is 3? Since 3 fingers looks like the letter W, that's Wednesday. Once again, you can check that here.

This reduction of the multiples of 7 can make the problem itself easier, as well. Let's try figuring out the day of the week for Halloween 2001, or October 31, 2001. October is 6, and 2001 is a 1, so the problem works out to be 6 + 31 + 1 = 38. The closest multiple of 7 to 38 is 35, so we do 38 - 35 = 3, which gives us another Wednesday.

That approach works, but it could be made simpler. When you hear that the date is the 31st, you can reduce that right away by working out that 31 - 28 (the closest multiple of 7 to 31) = 3, and doing 6 + 3 + 1 = 10. True, you would still need to reduce that 10 to 3 again to get Wednesday, but you'd need to subtract multiples of 7 either way.

Note that, by bringing the dates down by multiples of 7, you're making the problem you have to add much simpler. If 6 + 31 + 1 and 6 + 3 + 1 will both give you the same results, wouldn't you prefer to make it easier on yourself?

The scary technical term for subtracting multiples in this manner is modulo arithmetic, which is explained quite clearly at BetterExplained.com.

Let's try this with Valentine's Day in 2003. We start by making sure of the date, February 14, 2003. February is a 2 (remember the mnemonic?), and 2003 is a 3. The problem then becomes 2 + 14 + 3. However, if you spotted that 14 was already a multiple of 7, you should realize that you can drop it out completely! 14, or any multiple of 7, is the same as 0, so you can ignore them. For February 14, 2003, all you really need to add is 2 + 3 = 5. 5 is a FIVEday, or rather a Friday, so that's our answer!

Let's try one last problem before we go. What about February 2, 2000 (Groundhog Day)? February is 2, and 2000 is 0, so the problem we get is 2 + 2 + 0 = 4. 4 is a Thursday (remember FOURSday?), so February 2, 2000 should be a Thursday. Once again, we verify that information here and...OOPS! That site says February 2, 2000 is a Wednesday! What went wrong?

I briefly mentioned this at the end of the previous tab, but it needs to be repeated now. Whenever you're working in January or February dates in a leap year, you need to reduce the month code by 1 to compensate. Effectively the extra day, February 29, hasn't happened yet, so we're subtracting one to adjust for that fact. January becomes 5 (6 - 1) and February becomes 1 (2 - 1).

Since February 2, 2000 is a February date in a leap year, the code for February needs to be 1, not 2. Let's try the equation again, with that in mind. February in a leap year is 1, and 2000 is 0, so the equation is 1 + 2 + 0 = 3, which you should know by now is a Wednesday. As we saw when we originally made the error, Wednesday is indeed the correct day of the week.

Practice the Dates: 2000 to 2003 quiz here, making sure to keep an eye out for January and February dates in 2000, and adjusting your calculations accordingly. Practice these dates until you can calculate them with little trouble, and don't forget to subtract multiples of 7 to make your work easier!

Once you're comfortable with dates from 2000 to 2003, we're going to teach you how to better handle leap years in the next section.

Year Codes: Why?

You'll note that the year codes for 2000-2003 progressed in a nice, simple, 0-1-2-3 order. This is because a normal year consists of exactly 52 weeks (52 × 7 = 364) plus 1 day, to make 365 days. So, from one 365-day year to the next 365-day year, a given date in a given month will fall one day later.

However, a 366-day leap year means that everything must jump ahead 2 days. This also means that the year codes for leap years will jump ahead 2 instead of 1. You might expect 2004 to be a 4, but because it's a leap year, it jumps ahead to 5.

To get you comfortable with the strange nature of leap years, I'm going to start by teaching you the first 7 leap years.

First 7 Leap Years

Here are the year codes for the first 7 leap years, along with handy mnemonics by which to remember them:

Leap Year	Year Code	Mnemonic
2000	0	2000 is mostly 0s
2004	5	Count: 4...5...
2008	3	Right half of 8 looks like 3
2012	1	12 ÷ 12 = 1 (See below)
2016	6	16 ends in 6
2020	4	2 + 0 + 2 + 0 = 4
2024	2	24 ÷ 12 = 2 (See below)

With 2012 and 2024, you'll note that all you have to do is divide their last 2 digits by 12 to get their year code. This pattern keeps working all the way through 2096, which will give you a few extra leap year codes quite easily, assuming you know your 12 multiples:

• 2012 = 1
• 2024 = 2
• 2036 = 3
• 2048 = 4
• 2060 = 5
• 2072 = 6
• 2084 = 7 = 0 (Remember to drop any multiples of 7!)
• 2096 = 8 = 1 (Remember to drop any multiples of 7!)

First, practice just recalling the codes with the Leap Year Codes: 2000 to 2024 quiz here.

Once you're comfortable recalling all the codes, practice working out dates for those years with the Leap Year Dates: 2000 to 2024 quiz here. Don't forget that that the month codes for January and February are both reduced by 1 in leap years!

Once you've practiced those years, you're ready to learn how to handle any leap year in the 21st century!

All Leap Years

If you're given a leap year that ends in a multiple of 12, you can already handle those through 2096 quite easily, of course. What about the remaining leap years?

If there were no such thing as leap years, the pattern year codes would simply repeat every 7 years. Because of the effect of leap years every 4 years, however, the pattern of year codes usually repeats every 28 years.

I say “usually” because years ending in 00 are an exception. Years ending in 00 are only leap years if they're divisible by 400. So, 1600, 2000, and 2400 are leap years, while 1800, 1900, and 2100 are not.

Thanks to the 00 exception the calendar only repeats EXACTLY every 400 years. However, when you're dealing with a range of years in which EVERY (without exception) 4th year is a leap year, then you can still rely on the 28-year rule.

This means that you can depend on the 28-year rule for every leap year from 2000-2096! I'll break this down in a simpler manner, so you can see how this is useful.

For the leap years 2028 through 2052, all you have to do is subtract 28 years, and you'll get a year with the same year code! 2028 - 28 is 2000, which you already know has a year code of 0, so 2028's year code is 0. 2032 - 28 = 2004, whose year code is 5, and so on.

While doing 2028 - 28 = 2000 in your head is simple enough, some people find that working out problems like 2040 - 28 or 2052 - 28 during a performance to be a little challenging. There's a way to make it simpler.

If you're worried about subtracting 28 from a number, add 2 and then subtract 30 instead. For example, instead of doing 2040 - 28, work out 2040 + 2 - 30 = 2042 - 30 = 2012. 2012 is a 1 year, so 2040 is a 1 year as well! What year has the same year code as 2052? Add 2 and subtract 30, and you'll get your answer in no time.

Similarly, for the years 2056 to 2080, you subtract 56 to get the year code. The mathematical short cut here, if you feel you need it, is to add 4 and then subtract 60. What's the year code for 2064? 2064 + 4 - 60 = 2068 - 60 = 2008 = a year code of 3! How about 2068? You should get a year code of 1, just like 2012.

How about 2072? Did you start by adding 4? Stop! 72 is a multiple of 12, so we just work out that 2072 is 6 from the pattern of 12s above. Don't forget to take advantage of the 12 pattern when you can! You should ask yourself if a year is divisible by 12 first, before subtracting.

Finally, for the years 2084 to 2096, just subtract 84. For 2084 and 2096, of course, just use the 12 pattern we discussed earlier.

If you need a shortcut for 84 for the remaining years, simply subtract 4, then subtract 80. 2092 - 4 - 80 = 2088 - 80 = 2008 = year code of 3! I'm sure you have the idea by now.

You can practice this process with the Leap Year Codes: 2000 to 2096 quiz here. Again, don't forget to take advantage of the 12 pattern when you can.

Also, don't forget to practice actual dates in these leap years with this Leap Year Dates: 2000 to 2096 quiz.

Once you're comfortable working through both of these quizzes, it's time to learn how to determine the code for every year from 2000-2099 in the next section!

2000-2099

Are you ready to handle any date from 2000 to 2099? You're probably more ready thank you think!

Once you can handle leap years, the remaining years are simple.

When given a non-leap year, you need 2 pieces of information: The year code for the nearest leap year BEFORE the given year, and how far the given year is from that leap year. When you have these two pieces of information, simply add them together (remembering to drop any multiples of 7, as we've discussed before), and you have the year code.

For example, take 2009. The closest leap year is 2008, which has a year code of 3 (remember?), and 2009 is 1 year later. So, we work out 3 + 1 = 4, so 2009's year code is 4!

How about the year code for 2051? The nearest leap year BEFORE that is 2048, and we can use the 12 rule to determine that the year code is 4. Since 2051 is 3 years later, we do 4 + 3 = 7 = 0 (don't forget to drop out multiples of 7!), so 2051 has a year code of 0.

How about a tricky one like 2094? It's 2 years after 2092, which has a year code of 3 (remember how we know that?), so 2 + 3 = 5, so 2094 has a year code of 5.

Since you can't be more than 3 years after a leap year in any date from 2000-2099, this is a relatively simple adjustment.

To get practiced with this approach for determining year codes for any year, use this Year Codes: 2000 to 2099 quiz.

Once you get comfortable with that quiz, move on to the Dates: 2000 to 2099 quiz here.

Being able to determine the day of the week for any date in the 21st century is an impressive feat on its own. Once you're comfortable with doing that, you can move on learning how to handle dates in other centuries in the next section.

Other Centuries

Often you'll get asked about dates in the 20th century, especially if you're discussing someone's birthday. How do you handle those?

For dates from 1900 to 1999, simply work out the similar date for the 2000s, and then add 1. That's it!

For example, Let's say someone tells you they were born on January 20, 1985. Start as if you were working out January 20, 2085. 85 is a 1 year and January is a 6, so 1 + 6 = 7 = 0. Reduce 20 to 6 (cast out multiples of 7!) to get 6, and add 1 for the 20th century to get 7, which drops to 0. That 0 is the code for Sunday, and sure enough, January 20, 1985, was a Sunday.

Once you've worked out the day of the week for a given date in the 21st century, there's a simple pattern to alter the day for other centuries:

• 2300 to 2399 = add 1
• 2200 to 2299 = add 3
• 2100 to 2199 = add 5
• 2000 to 2099 = add 0
• 1900 to 1999 = add 1
• 1800 to 1899 = add 3
• 1700 to 1799 = add 5
• 1600 to 1699 = add 0

There is one VERY important note here for January and February dates in the years ending in 00: If a year ends in 00, it's only a leap year if it's divisible by 400. The years 1600, 2000, and 2400, and so on are leap years, while years like 1900, 1800, and 2100 are not. If you're given a January or February date in a year ending in 00, double check whether it's a leap year before you make the leap year adjustment.

Take a few minutes to study these adjustments, and then practice using them in the Dates: 1600 to 2399 quiz here.

Assuming you've put in the practice, you should be ready to give the day of the week for any date. In the next section, I'll provide a few tips and some background that can help improve your performance.

Calendar Background

The current calendar system we use is known as the Gregorian calendar, since it was introduced by Pope Gregory XIII. It was first put into use in 1582 by the Catholic countries, so the calculations you've learned aren't really effective for dates before 1600.

In addition, many non-Catholic countries didn't adopt the Gregorian calendar until much later. Britain and its colonies didn't adapt the calendar until 1752. The use of the Gregorian calendar as a worldwide standard, however, didn't happen until the 1920s!

Tips

• I can't emphasize enough the speed advantages of dropping multiples of 7, and becoming comfortable with that process. After you get use to doing this for dates from the 7th to the 31st several times, it almost becomes automatic.

• Carry a perpetual calendar! It's one thing to do this feat and know you're right. When you're doing it for an audience, they'll need some way to verify that you're correct. Originally, this meant carrying around a bulky book of calendars, but many mobile devices today make this much easier.

You'll generally want an app that mainly generates calendars for a wide variety of year, without appointment features, such as QuickCal for the iPhone and iPod Touch. iPad users can use YearViewer, and Android users can use Two Hundred Year Calendar or Day of Week.

• Want to practice on the go? Download these free mp3 files that give a date, then pause, then give the day of the week. The pauses range from 30 seconds down to 3 seconds, so you can challenge yourself as you get better. They're available in both DATE/MONTH/YEAR order (common in the UK, Australia, and Europe) and MONTH/DATE/YEAR (common in the US).

• As you've seen, working out year codes can take longer than just remembering the month, date, and week codes. When performing, the smart thing to do is ask for the year first, work out the year code as needed (including whether a leap adjustment will be needed), and only then ask for the specific date.

That way, not only do you get the year calculation out of the way, but you'll be able to determine the weekday more quickly and it will appear more impressive to your audience.

• When you're comfortable performing the feat this way, but you find you desire to be quicker, there is a more advanced step you can take. You can completely eliminate the calculations of the year code by memorizing the 100 codes needed for the years 2000-2099.

To do this, you'll need to be familiar with the Link System, the Shape Peg System and the Phonetic Peg System (AKA the Major System) (with images for 0 to 99).

Once you've practiced those systems and are comfortable with them, you use the Phonetic Peg System for images to represent the last two digits of the year (0 to 99), and the Shape Peg system for the year code (from 0 to 6). You then use the Link System to mentally link those two images together.

If you decide to go this method, here's a complete chart of the years from 2000 to 2099 with their corresponding year codes. Because the images people use with the above systems are so widely varied, I've avoided suggesting any mnemonics.

Year	Year Code
2000	0
2001	1
2002	2
2003	3
2004	5
2005	6
2006	0
2007	1
2008	3
2009	4
2010	5
2011	6
2012	1
2013	2
2014	3
2015	4
2016	6
2017	0
2018	1
2019	2
2020	4
2021	5
2022	6
2023	0
2024	2
2025	3
2026	4
2027	5
2028	0
2029	1
2030	2
2031	3
2032	5
2033	6
2034	0
2035	1
2036	3
2037	4
2038	5
2039	6
2040	1
2041	2
2042	3
2043	4
2044	6
2045	0
2046	1
2047	2
2048	4
2049	5
2050	6
2051	0
2052	2
2053	3
2054	4
2055	5
2056	0
2057	1
2058	2
2059	3
2060	5
2061	6
2062	0
2063	1
2064	3
2065	4
2066	5
2067	6
2068	1
2069	2
2070	3
2071	4
2072	6
2073	0
2074	1
2075	2
2076	4
2077	5
2078	6
2079	0
2080	2
2081	3
2082	4
2083	5
2084	0
2085	1
2086	2
2087	3
2088	5
2089	6
2090	0
2091	1
2092	3
2093	4
2094	5
2095	6
2096	1
2097	2
2098	3
2099	4

We'll start with the basics by introducting the concept of radians. What exactly are radians?

Radians are an alternative to degrees. Let's make sure that everyone is on the same page, with a quick and fun refresher course in degrees:



Most people are familiar with degrees. They're an absolute necessity when studying things like geography (where on Earth am I?) or astronomy (When will I be able to see a given star from where I am on Earth?). So why do we even need an alternative to degrees?

I'll highlight the problem with a smaller-scale example. Imagine a public park with circular running tracks of varying sizes, and two people, one who is trying to build his speed and stamina for a 100 meter race, and the other person who is timing the runner. They find a track, and the timer stands in the center of the circular track. The runner notes that the circular track is some weird amount, say, 142 meters in circumference, so he needs to know the location of the 100 meter mark.

The timer whips out his calculator, divides 100 by 142, getting 0.704225352. He multiplies this by 360 degrees, since the track is circular, and cheerfully exclaims that all the runner has to do is run 254 degrees (253.521127 degrees, to be more precise) to cover 100 meters on that track. At this point, the runner gives the timer a funny look.

See the problem now? 254 degrees is great when describing how far the timer, standing in the center of the circular track, has to turn to watch the runner. This doesn't help the runner much, as the runner isn't standing in the center. That's the whole thing with degrees; they're great for observers, such as the timer, but not so much for people or things moving around a circumference, such as our runner. This is why there's a need for an alternative to degrees.

Note that the absolute distance of 100 meters isn't of much help, either. On the different sized tracks in the park, that distance will be a different number of laps on each track. So, absolute distance and degrees aren't handy, thus we introduce radians. So what exactly are they?

Since the radius of a circle has a constant relationship with the circumference, that of Pi times twice the radius, how about counting off how far we've traveled around the circumference of a circle in terms of that circle's own “radius units”? It's a good idea, but “radius units” sounds weird (I think it's the repetition of the letter U), so we use the term radians instead!

It boils down to this: 1 radian is simply the length of a circle's radius traveled around that same circle's circumference.


That also gives us the formula to calculate radians: Radians = distance traveled / radius.

Math books like to scare you by writing this same formula as θ = s / r. θ is pronounced “theta”, and simply refers to the answer in radians. s means “arc length”, or, the distance traveled around the circumference. r, the only straightforward shorthand in this formula, means the length of the radius.

Going back briefly to the oberver's point of view, 1 radian translates into about 57.3 degrees (actually 57.29577951... and on and on). For the oberver, this certainly isn't a nice neat number, like 90 degrees, 180 degrees, or 360 degrees. As we've already learned though, the observer's viewpoint is not the point of radians.

However, in the next tab, we'll introduce the unit circle concept itself, and see how radians make things nice and easy for the person or thing that is moving around it.

Since we now have the scalable concept of radians to work with, we can now develop an entire scalable circle with which to work. We'll start by drawing on graph paper a circle centered at the (0,0) mark that has a 1 unit radius (remember cartesian coordinates?). It looks at it does below, with the coordinates marked where it crosses an axis:


That fact that it's a circle with a one-unit radius gives us the shorthand term for it: unit circle. Let's take a closer look at it what a unit circle can do.

Here's your first question: Since radians measure how far around a circle you've traveled, what's the distance around a full unit circle in radians? Let's work this out.

Our circle has a radius of 1 unit, so the circumference (our total distance traveled, in this case) is 2 times the radius times Pi, so we have 2 times 1 unit, giving 2 units, times Pi, resulting in 2 * Pi, or 2π for short (That π symbol is the Pi symbol, not a small letter n). That's just the total distance traveled, though.

To convert that to radians, we need to divide by the radius, of course. Our radius is 1 unit, so we work out 2π/1, which is simply 2π. In other words, the distance around a full circle is 2π radians in length!

Now you see why 1 radian is such a weird angle in degrees (1 radian = 57.29577951... degrees). When doubled and multiplied by Pi, it has to give a nice even 360 degrees. Again, we're going to stick to radians in this discussion, so just think of 2π radians as a full circle.

From here, it's not hard to see that π radians gives us half of a circle, and π/2 radians gives us a quarter circle. Three-quarters of a circle, then, would give us a distance of 3π/2 radians.


So, what distances are we traveling in radians when we traveling 45 degrees? 45 degrees is the centered observer's way of saying an eighth of a circle. The mover thinks of a circle as being 2π radians, so 2π times 1/8 would be 2π/8, or the same as π/4 radians.

Every 1/8 of a trip around the circle, then, would be in units of π/4 radians:

• 1/8 = π/4 radians
• 2/8 = 2π/4 = π/2 radians
• 3/8 = 3π/4 radians
• 4/8 = 4π/4 = &pi radians
• 5/8 = 5π/4 radians
• 6/8 = 6π/4 = 3π/2 radians
• 7/8 = 7π/4 radians
• 8/8 = 8π/4 = 2π radians

When you see the patterns and understand the process, it's actually not hard to understand.

Since angles of 30 degrees, or 1/12 of the distance around the circle, are also common, they're not hard to work out in radians. 1/12 of 2π radians comes down to π/6 radians. From there, we get:

• 1/12 = π/6 radians
• 2/12 = 2π/6 = π/3 radians
• 3/12 = 3π/6 = π/2 radians
• 4/12 = 4π/6 = 2π/3 radians
• 5/12 = 5π/6 radians
• 6/12 = 6π/6 = π radians
• 7/12 = 7π/6 radians
• 8/12 = 8π/6 = 4π/3 radians
• 9/12 = 9π/6 = 3π/2 radians
• 10/12 = 10π/6 = 5π/3 radians
• 11/12 = 11π/6 radians
• 12/12 = 12π/6 = 2π radians

Here's everything you've learned so far in one diagram, with the 90 degree units marked in black, the 45 degree units marked in red, and the 30 degree units marked in blue:


So, if you can remember that 2π radians takes you all the way around the circle, and that π radians, as well as 6π/6 and 4π/4 radians, takes you halfway around the circle, working out the rest of the numbers isn't difficult at all.

In the next tab, we'll discuss another amazing way the unit circle becomes useful when we focus on figuring out coordinates.

I have some quick refresher questions before we move on. When you've traveled π/2 radians, at what (x,y) coordinates are you? You're at (0,1).

Here's a tougher one: When you've traveled π/4 radians, at what (x,y) coordinates are you? That one's a little trickier. For this one, we're going to have to go back the viewpoint of the observer in the center, and thus back to degrees.

The radius of the unit circle is, of course, always 1 unit. If we construct a right triangle with the radius line as the hypotenuse, we could construct a right triangle for any angle in the circle:


Since π/4 radians from the central observer's point of view is 45 degrees, we'll create a 45 degree right triangle to help work out those coordinates:


Since we're looking for the (x,y) coordinate, the length of the side adjacent to our angle (the one running along the x axis) would give us the x coordinate. The height of the side opposite the angle (the one running straight up to meet the radius/hypotenuse) would give us the y coordinate.

If you remember all those lessons about right triangles, it's at this point where you begin to realize how many tools we can use here.

• Pythagorean Theorem: a² + b² = c²
• Sine of any angle = opposite/hypotenuse
• Cosine of any angle = adjacent/hypotenuse

(Don't forget: SOHCAHTOA)

Since the hypotenuse is 1, and both the sine and cosine divide by the hypotenuse, not only does this make the math nice and easy, but the answers will also be the exact coordinates we need!

The x coordinate is the length of the side adjacent to the angle, so we need to use the cosine formula. By cheating and using a calculator (make sure to set it in degrees, not radians!) we find that cos(45 degrees) = 0.707106781.... To get the y coordinate, we find the length of the side opposite the angle by using sin(45 degrees), which is also 0.707106781....

So, we have our coordinates for a 45 degree, or π/4 radian, angle: (0.707106781...,0.707106781...). Hmmm, that's another one of those numbers that goes on forever. Isn't there a better way to state that number?

Let's run through the Pythagorean Theorem approach, especially know that we know that sides a and b are equal, and see what we can come up with:



“OK,” you say, “so what? We just came up with the same numbers.” Let's back up a bit to to where a squared (and b squared, in the specific case of a 45 degree right triangle) was equal to 1/2. Instead of using .5, let's see where working with that 1/2 as a fraction can take us:



Hmmm...writing one over the square root of 2 is certainly much easier than that long string of numbers. Indeed, when you're tested on unit circles on many standardized tests, they'll usually ask you to write the π/4 radian or 45-degree coordinate in exactly that way.

Before moving on, though, I'd like to make that fraction a bit neater by making the bottom a whole number. We do that by multiplying both the top and bottom by the square root of two, and expressing the coordinates as fractions:



Not only is this a cleaner way to write the coordinates, it will make this and the other coordinates we work out much easier to remember, as you'll see in later sections.

The fractions may look funny, but all they're really saying is, “If you take the square root of 2, and divide it in half, that's a shortcut to working out this coordinate.” Or, put the opposite way, “Hey, if you take this coordinate, double it, and then square it, you get a nice, simple number, in this case - 2!”

We'll work out the coordinates and their corresponding fractions for the 30- and 60-degree angles in the next section, as well as their multiples.

So, now we not only know the coordinates for a 45 degree, or π/4 radian, point on a circle, we know that the cosine will give us the x coordinate and that the sine will give us the y coordinate on the unit circle. We also know that expressing these coordinates in fractions is much simpler than writing out the endless irrational decimals.

Let's work out the coordinates for the point at 30 degrees (π/6 radians):



Hey! That .5 is nice. That's easy to express as a fraction. But what about that other number? Well, everything else has been expressed over 2, so let's see what happens when we express that number over 2:



Ummm...yuck. Well, square roots seem to be popular in these fractions. Is 1.73205081... the square root of anything? Yep! It turns out that it's the square root of 3! So, the coordinates could be expressed this way:



Next, let's work out the coordinates for 60 degrees (π/3 radians):



Whoah! We get the same numbers as the 30 degree angle, only switched! That's not a coincidence. Remember that the internal angles of triangles always add up to 180 degrees. With a right triangle (a triangle with a 90 degree angle in it) that has a 30 degree angle in it, we find that 180 - 90 - 30 = 60 degrees, the remaining angle. Effectively, the 60 degree coordinates are just the 30 degree coordinates viewed from the other side of the triangle.

Let's review all the information, including the coordinates, that we know so far:


If you think of 1 as also being the square root of 1, you could write 1/2 this way:



So, at all the points where the circle crosses an axis, the coordinates involve 0, +1, or -1 - Simple! And at the oft-used 30° (π/6 radians), 45° (π/4 radians), and 60° (π/6 radians), all the coordinates involve these interesting fractions:



See that? Everything in the fractions is a square root of something over 2! Even better, there's a simple 1, 2, 3 progression! Even with all the complex math involved, it's all boiling down to 0, 1, 2, and 3.

That's easy enough to remember, but how do you remember which fractions are set at which coordinates, especially since it's not hard to make a mistake and switch around the coordinates for both 30° and 60°? Here's a video that will teach you quickly, using that 1, 2, 3 progression:



What about all the remaining major angles, such as 120°, 135°, and so on?

Well, you could go through and work out the coordinates laboriously as we did above, working out the sines and cosines, and so on. However, there's an easier way. We're measuring the 135° (3π/4 radians) angle from the 0° angle. When measured from the 180° (π radians) angle, it works out to be our old friend, the 45° (π/4 radians) angle.

One difference, though, is that we're now on the negative side of the x axis, so that the x coordinate will be negative. The means the coordinate for the 135° (3π/4 radians) angle will be:



Similarly, you can work out the 150° angle as a 30° angle, and so on. They work out to the coordinates we've already determined, but you need to make sure that you adjust the signs (+ or -) for the appropriate section.

Are you ready for this? Here's the entire unit circle worked out:


And here's how to draw it that so you remember all that data:



I like the approach used in this video of counting the π/3, π/4, and π/6 sections separately, so as to keep them mentally separate.

In the next tab, I'll review and break down the patterns for easier understanding. You'll also learn how to do some quick math on your fingers to get the coordinates.

It may seem like a lot that you've learned so far, but it boils down to a few basic things:

• Degrees deal with the observer's point of view. Radians deal with the mover's point of view.
• 1 radian = 1 radius length around the circle's circumference.
• Radians work very well with Pi, to the extent that π radians = 180 degrees, and 2π radians = 360 degrees.
• If you think of π radians as being 180 degrees, and you know your multiples of 30 and 45, the remaining degree angles aren't difficult to work out in radians.
• All the coordinate points where the circle cross an axis involve a 0, a -1, or a +1. Knowing coordinates makes it easy to work out which is which.
• All the coordinate points of π/6 (30°), π/4 (45°), and π/3 (60°) radians involve fractions easily remembered with the 1, 2, 3 progression taught in the video in the previous tab.
• The other major coordinate points can be worked out quite easily from those first 3 sets, as long as you adjust the signs properly.
• Cosine will always give the x coordinate on the unit circle, and sine will always give the y coordinate on the unit circle.

That last fact is especially interesting. Thanks to the simple progressions involved, you can actually work out the cosine and sine on your fingers with a simple trick! If you're familiar with tangents and cotangents, this video teaches how to handle those on your fingers, as well.



Betterexplained.com has a wonderful article you should read at this point, called Intuitive Guide to Angles, Degrees and Radians.

It really drives home the power of radians. Take this example from the article:

Let’s try a real example: you have a bus with wheels of radius 2 meters (it’s a monster truck bus). I’ll say how fast the wheels are turning and you say how fast the bus is moving. Ready?

“The wheels are turning 2000 degrees per second”. You’d think:

Ok, the wheels are going 2000 degrees per second. That means it’s turning 2000/360 or 5 and 5/9ths rotations per second. Circumference = 2 * pi * r, so it’s moving, um, 2 * 3.14 * 5 and 5/9ths… where’s my calculator…

“The wheels are turning 6 radians per second”. You’d think:

Radians are distance along a unit circle — we just scale by the real radius to see how far we’ve gone. 6 * 2 = 12 meters per second. Next question.

Wow! No crazy formulas, no pi floating around — just multiply to convert rotational speed to linear speed. All because radians speak in terms of the mover.

The reverse is easy too. Suppose you’re cruising 90 feet per second on the highway (60 miles per hour) on your 24″ inch rims (radius 1 foot). How fast are the wheels turning?

Well, 90 feet per second / 1 foot radius = 90 radians per second.

That was easy. I suspect rappers sing about 24″ rims for this very reason.

I'm not sure that's the reason they sing about 24″ rims, but I'll go along with it while it lasts.

Think of the power you have here. Sure, the unit circle seems complex at first, but the power and patterns it presents when remembered and understood make many things simple.

Scaling up to real-world circles and back down to the unit of circles, as well as determining their motion, becomes a simple matter of multiplication. You can work out sines, cosines, tangents, and cotangents on your fingers.

Converting from radians to degrees and back, with a little practice, isn't difficult.

Which brings to mind the question of practice. The simplest practice I can offer is to print out several copies of this PDF, and repeatedly fill it out until you're good enough to do so in under 5 minutes, just as it says.

Sporcle offer this ingenious quiz for practicing radians and degrees. With help from the videos from the previous tab, this shouldn't be too tough.

The most complete quiz on the unit circle that I've found, however, is Math Fanatic's Unit Circle and Trigonometry Quiz. The menu lets you select exactly what aspects you want to practice, and the quiz itself lets you practice each section for as long as you need or want. Since it covers most of trigonometry, and not just the unit circle, there will be some quizzes not covered in my unit circle tutorial.

I hope you find this useful, and I also hope it helped you better understand the unit circle.

Introduction

In this post, you'll learn how to square numbers from 1-100 in your head!

As a refresher, squaring a number simply means to multiply it by itself. For example, 4 squared is 16 because 4 times 4 is 16. You should know the squares of all the numbers from 1 through 10 by heart already.

Multiples of 10

If you already know your squares of the numbers 1 through 10, the multiples of 10 are easy. When a number from 1-100 ends in zero, simply drop the ending 0, square the remaining number, and then add 2 zeroes. For example, to work out 20 squared, drop the zero leaving the 2, square it to get 4, then tack on 2 zeroes to that 4, resulting in 400.

70 squared? 4900, because 7 squared is 49, and the two zeroes added make it 4900. 100 is trickier, but uses the same approach. 100 with the final zero dropped gives us 10. 10 squared is 100, and adding 2 zeroes gives us 10,000.

Multiples of 5

Multiples of 5 are almost as easy. You do need to make sure you know your multiplication tables up to at least 10 times 10. The method taught here is also taught in the root extraction tutorial, as well.

When given a number ending in 5, simply take the 10s digit, and multiply by a number one higher than itself. Take that answer, take a "25" on the end, and you've got the answer!

For example, let's say you're asked what 35 squared is. Take the 3 (the 10s digit), and multply it by 4 (which is one higher than 3), and you get 12. Tack a 25 on the end, giving you 1225. Simple, isn't it?

Let's try a higher number, like 75 squared. 7 times 8? 56. Tacking on the 25, gives us 5625!

Here's a slideshow to help explain this procedure in more detail:

Click here to view the Squaring Numbers Ending in 5 slideshow

To quiz yourself on squaring multiples of 10 and 5, click here. To learn the mental math approach to squaring the remaining numbers, click here. To learn the memorization approach to squaring the remaining numbers, click here.

Numbers from 1-25

For the approach using pure mental math, you'll need to know your squares from 1-25 by heart. From 1 to 10 you should already know, and from the techniques on the first page, 15, 20, and 25 will be easily handled, as well. That leaves these squares to learn by heart:

Number	Square of Number
11	121
12	144
13	169
14	196
16	256
17	289
18	324
19	361
21	441
22	484
23	529
24	576

The must be known by heart, because the method we're going to use to work out the remaining numbers requires that you can give the above numbers quickly.

Numbers from 26-50

To work out the numbers from 26 to 50, we're going to use an approach in which we multiply by 50.

Multiplying any number by 50 is easy – all you have to do is divide the number by 2, and add 2 zeroes (more accurately, you would move the decimal 2 places to the right). 48 times 50? Half of 48 is 24, and two zeroes added results in 2400, which is the correct answer. This method only involves multiplying even numbers by 50, so you won't have to worry about dealing with numbers like 24.5 (Half of 49).

When given any number from 26-50, you're first going to work out how far that number is from 50, then subtract that distance from the given number. For example, if you're given the number 47, it's easy to work out that it's only 3 away from 50. Subtracting that 3 from 47, we get 44.

Instead of solving 47 times 47, then, we're going to work out the much easier problem of 44 times 50, which is 2200. However, this isn't the same as the answer to 47 squared, so we need to make an adjustment.

From 47, we both moved up 3 to 50 and down 3 to 44. So, we square this 3 to get 9, and add that to the other answer we worked out, 2,200, to get a total of 2,209. This is the answer to 47 times 47!

So, when given a number, you work out how far the given is from 50, and find a number that's equally far below the given number (the “low” number), and also remember this difference. Multiply 50 times the “low” number, adjust it by squaring the difference you moved, and adding that amount, and the result will be the square!

Let's try this with 44, to help make this clearer. 44 is 6 away from 50, so we figure out that 44 - 6 = 38. 38 times 50 is easy, 1,900. We moved a difference of 6 in both directions, so we add 36 (6 squared) to 1900, to get 1,936!

How about 39 squared? That's 11 away from both 50 and 28. 28 times 50? 1,400. 11 squared is 121, and adding that to 1,400, we get 1,521!

How about 35? Trick question! That's made easier by the multiples of 5 technique from the first page. Don't forget to use the easier techniques in the easier cases.

Numbers from 51-75

The same technique is going to be used for numbers from 51 to 75, but with one minor change. You'll be moving down to 50, and up to another number (Previously, you moved up to 50, and down to another number). Other than that, the process is basically the same.

Let's try 56 squared. 56 is 6 away from 50 and 62. 50 times 62 is 3,100, plus 36 (6 squared) gives us 3,136!

How about 67? The distance makes this a little more challenging, but the process is still the same. 67 is 17 away from 50 and 84, so we multiply those two numbers to get 4,200. 17 squared is 289, so we work out 4,200 plus 289 to get our final answer of 4,489.

Numbers from 76-100

As easy as multiplying by 50 has been, multiplying by 100 is even easier – just add 2 zeroes!

For numbers from 76-100, we're going to adjust upward to 100, as opposed to using 50 as we have been. Wait until you see how easy this makes the process!

Let's try working out 98 squared. 98 is 2 away from 100 and 96, and multiplied together, that gives us 9,600. 2 squared is 4, and 9600 plus 4 is 9,604. That's 98 squared!

How about 91? We start out with 8,200 (do you see why?), and add 81 (again, do you see why?), to get 8,281.

The more you practice each of these stages, the more you'll get a feel for certain patterns. This will allow you to speed up your calculations.

Numbers from 100-125

By now, you've probably figured out that you can go up to 125 with just a minor adaptation, similar to that we used when going above 50.

What's 103 squared? It's between 106 and 100, so we multiply those to get 10,600. 3 squared is 9, so that added in gives us 10,609!

What about a toughie, like 124? That's between 100 and 148, so we start with 14,800. 24 squared is 576, so we add those together to get 15,376.

With a little practice, you should have this process down in a faster time than you may have ever thought possible.

To practice squaring numbers, click here. To learn an alternative approach using memorization for squaring the numbers from 1 to 100, click here.

Memorizing the Squares

It was Alabama math and science teacher Jim Wilder who first suggested the idea of memorizing the squares to me. The process is similar to the one I use for memorizing 400 digits of Pi.

First, you should learn the the multiples of 10 and 5 techniques from the first page, and you should still know the squares from 1-25 by heart, as those are still quicker than the memory approach. This also helps minimize the amount of links needed.

Prerequisites:

Link System
Major System

Links

With the exception of the multiples of 10 and 5, Jim Wilder put in some amazing work developing major system mnemonics for all the squares from 26 to 99 (Thanks again, Jim, for both your work and willingness to share it with us!):

Number	Square of Number	Number Mnemonic	Square Mnemonic
26	676	iNCH	SHaKiSH
27	729	kNocK	Key, NaP
28	784	kNiFe	CoVeR
29	841	kNoB	FoRT
31	961	MaiD	PuSHeD
32	1,024	MooN	DoSe NeaR
33	1,089	MuM	ToSS FiB
34	1,156	MaRRy	TighT LeaSH
36	1,296	MaTCH	DowN PuSH
37	1,369	MoCHa	DaMn, CHeaP
38	1,444	huMVee	TiRe RoaR
39	1,521	MoP	TaiL kNoT
41	1,681	RaT	TouCH, FiT!
42	1,764	RuN	TaKe SHaRe
43	1,849	RaM	TuFF RoPe
44	1,936	RoaR	TiP MatCH
46	2,116	ReaCH	NoT TouCH
47	2,209	RoCK	NoN SouP
48	2,304	ReeF	NaM, SiR!
49	2,401	RiB	uNRaiSeD
61	3,721	SHaDow	MaKe NighT
62	3,844	CHaiN	MoVe ReaR
63	3,969	CHuM	MoP SHiP
64	4,096	CHaiR	RiSe, PuSH
66	4,356	CHeeCH	RuM LuSH
67	4,489	CHeCK	RaRe FiB
68	4,624	CHeF	ReaCH NeaR
69	4,761	CHiP	RocK SHeeT
71	5,041	KiT	LooSe, RighT?
72	5,184	CaN	LeaD FeaR!
73	5,329	GuM	LiMe NuB
74	5,476	CaR	LuRe CaSH
76	5,776	CoaCH	LuCK, CoaCH
77	5,929	CoKe	LeaP, NaP
78	6,084	CaVe	CHooSe FiRe
79	6,241	CaP	SHiN, Right?
81	6,561	FaT	JeLLo JeT
82	6,724	FiN	CHiC NoiR
83	6,889	FoaM	CheF FiB
84	7,056	FouR	CaSe LatCH
86	7,396	FiSH	CoMb PuSH
87	7,569	FaKe	CoaL CHiP
88	7,744	FiFe	KicK ReaR
89	7,921	FiB	CaP NoD
91	8,281	PaT	FuN FighT
92	8,464	PaN	VeRy CHaR'd
93	8,649	PaM	FiSH RuB
94	8,836	PouR	ViVa MuCHo!
96	9,216	PiTCH	BaNNeD SHow
97	9,409	PuCK	PooR SOB
98	9,604	PuFF	PuSHeS aiR
99	9,801	PaPa	PuFFS iT

Memorizing the 50s

Like multiples of 10 and 5, the squares of numbers in the 50s have their own trick that is easy to remember.

Number	Square of Number
51	2,601
52	2,704
53	2,809
54	2,916
56	3,136
57	3,249
58	3,364
59	3,481

When given a number in the 50s, simply take the ones digit and add it to 25. Next, take the square of the digit in the ones place, and tack that on to the right of the previous answer.

For an example, let's use 53. The ones digit is a 3, so we add 25 to get 28. 3 squared is 9, so we add 09 to the end of the other digit to get 2,809.

57? 7 plus 25 is 32. 7 squared is 49. Therefore, 57 squared is 3,249. Once the pattern clicks, you'll find that these are quick and easy.

To practice squaring numbers, click here. To learn an approach using mathematics for squaring the numbers from 1 to 125, click here.