- Slides: 71
Magician or Math-a-magician?
Math Tricks and Tips
General Tips for Studying Mathematics 1. Go To Class regularly 2. Get to Class On Time. 3. LISTEN During Class. 4. Take Good Notes. 5. Ask Questions. 6. Listen When Others Ask Questions. 7. Review Notes After Class. 8. Make a Set of Index Cards. 9. Learn The (Proper) Notation. 10. Get Into A Study Group. 11. Note Due Dates. 12. Budget Adequate Time For Studying/Homework. 13. Do Homework After Each Class. 14. Do Homework Without Notes and Book. 15. Do More Homework. 16. Practice, Practice. 17. Persevere Keep Old Homework and Exam Papers. 18. Don’t Forget Your Textbook. 19. Seek Help If You Need It. 20. Seek Help If You Need It. You should always do the best that you can and strive for the best grade that you can possible get.
Study Tips for Math 1. Always read math problems completely before beginning any calculations. If you "glance" too quickly at a problem, you may misunderstand what really needs to be done to complete the problem. 2. Whenever possible, draw a diagram. Even though you may be able to visualize the situation mentally, a hand drawn diagram will allow you to label the picture, to add auxiliary lines, and to view the situation from different perspectives. 3. Do not feel that you must use every number in a problem when doing your calculations. Some mathematics problems have "extra" information. These questions are testing your ability to recognize the needed information, as well as your mathematical skills. 4. Remain confident! Do not get flustered! Focus on what you DO know, not on what you do not know. You know a LOT of math!! 5. If you are "stuck" on a particular problem, go on with the rest of the test. Oftentimes, while solving a new problem, you will get an idea as to how to attack that difficult problem. 6. In certain problems, you may be able to "guess" at an approximate (or reasonable) answer. After you perform your calculations, see if your final answer is close to your guess.
Fear of Maths is only mental I suggest: 1. Instead of saying DIVIDE BY 2, say HALF/HALVE IT. 2. Instead of saying MULTIPLY it by 2, say DOUBLE IT. 3. Never use more than two digit numbers to prove the working of a method. 4. Show the more interesting sides of maths, for example, show the beauty of the table of nine (which really looks cute, simple and well arranged). After these small things, leave the person to grow up inside herself, by herself. They’ll start with small victories, and keep gathering courage for bigger ones. Maths is easy and beautiful up to a certain level. Let’s all enjoy this beautiful, universal language.
Tough Multiplication If you have a large number to multiply and one of the numbers is even, you can easily subdivide to get to the answer: 32 x 125, is the same as: 16 x 250 is the same as: 8 x 500 is the same as: 4 x 1000 = 4, 000
Assorted Multiplication Rules v. Multiply by 5: Multiply by 10 and divide by 2. v. Multiply by 6: Sometimes multiplying by 3 and then 2 is easy. v. Multiply by 9: Multiply by 10 and subtract the original number. v. Multiply by 12: Multiply by 10 and add twice the original number. v. Multiply by 13: Multiply by 3 and add 10 times original number. v. Multiply by 14: Multiply by 7 and then multiply by 2 v. Multiply by 15: Multiply by 10 and add 5 times the original number, as above. v. Multiply by 16: You can double four times, if you want to. Or you can multiply by 8 and then by 2.
Assorted Multiplication Rules v. Multiply by 18: Multiply by 20 and subtract twice the original number v. Multiply by 19: Multiply by 20 and subtract the original number. v. Multiply by 24: Multiply by 8 and then multiply by 3. v. Multiply by 27: Multiply by 30 and subtract 3 times the original number v. Multiply by 17: Multiply by 7 and add 10 times original number. v. Multiply by 45: Multiply by 50 and subtract 5 times the original number v. Multiply by 90: Multiply by 9 (as above) and put a zero on the right. v. Multiply by 98: Multiply by 100 and subtract twice the original number. v. Multiply by 99: Multiply by 100 and subtract the original number
q. When solving absolute value inequalities: if the absolute value is greater than a number you must use the conjunction (OR), qwhen the absolute value is less than a number you q must use the conjunction AND. q. To remember this just remember two words "GOR""LAND, " which translate into "G(greater)OR" and "L(less than)AND. " q. When I introduce this topic I tell students that we are about to enter "GOR-LAND. " (no political implications intended)
Simple Multiplication Verification Method How do you verify your multiplication? Here is a simple method. Always reduce computations to a single digit. 43 x 92 3956 Add the digits of multiplicand i. e. 4 + 3 = 7 Add the digits of multiplier i. e. 9 + 2 = 11 then reduce to a single digit 1 +1 = 2 Multiply 2 x 7 = 14 then reduce to a single digit 1 + 4 = 5 Add 3+9+5+6 = 23 then reduce to a single digit 2 + 3 = 5 Both numbers (5) are equal, therefore multiplication is correct
Formulas For Easy Remberence Ø A=(pi)r^2 Apple pies r square Ø A=(pi)r*r Apple pies r round Ø C= (pi)d Cherry pie delight Ø I=prt I "am" p-r-t Ø pronounced I am pretty Ø rt = d rt are d Ø pronounced retard ØQuadratic formula : "X equals to negative b Plus or minus the square root, Of b squared minus four a c All over 2 a"
supplementary and complimentary angles v I teach middle school students. My students know that v Supplementary and complimentary angles are angles that equal 90 degrees and 180 degrees, but they get confused as to which is which. v They also know that 90 degree angles are right angles. v So I tell them that a compliment is the right thing to do, and right angles v equal 90 degrees, therefore complimentary angles are two angles that equal 90 degrees. A. Then they know that 180 degrees is the other one, supplementary. "Complementary" - early in the alphabet, so = 90 degrees. B. "Supplementary“ - later in the alphabet, so = 180 degrees.
Triangle Names o o o Equilateral triangles have 3 sides and 3 angles equal. Isosceles triangles have 2 sides and 2 angles equal. Scalene triangles have 0 sides and 0 angles equal. So, to remember them in that order, EIS, "Eat ice slowly"
Two given lines cut the coordinate axes in four concyclic points or not § just see whether the product of coefficients of x in both the equations is equal to that of coefficients of y. § If the given lines are ax+by+c=0 and bx+ay+d=0 § they cut the axes in concyclic points.
Metric Measurement my milli cat centi died deci unusually unit drinking deca hot hecta ketchup kilo
Example Convert 10 decameters to centimeters. Set up the columns as shown below so that the ones column comes under deca. Move the decimal point to the right of the column with centi. Add zeros until you are under centimeters. That is your answer. Kilo hecta deca unit deci centi milli 1 0 0 0 0 i. e. 10 dam = 10 000 cm
“King Henry Died Monday Drinking Chocolate Milk" Km Hm Dam M Dm Cm Mm To convert. . . 3. 75 Hm = ______ Cm It is 4 jumps to the right from Hm to Cm, Simply move the decimal 4 jumps to the right. 3. 75 Hm = 37, 500. Cm 0. 59 Dm = _______ Hm. I It's 3 jumps to the left from Dm to Hm, Simply move the decimal 3 jumps to the left. 0. 59 Dm = 0. 00059 Hm
TO FIND SQUARE OF A 3 DIGIT NUMBER LET THE NUMBER BE XYZ. SQ (XYZ) is calculated like this. STEP 1. Last digit = last digit of SQ(Z) STEP 2. Second Last Digit = 2*Y*Z + any carryover from step 1 STEP 3. Third Last Digit 2*X*Z+ Sq(Y) + any carryover from STEP 2 STEP 4. Fourth last digit is 2*X*Y + any carryover from STEP 3 STEP 5. In the beginning of result will be Sq(X) + any carryover from Step 4.
TO FIND SQUARE OF A 3 DIGIT NUMBER : EXAMPLE : SQ (431) STEP 1). Last digit = last digit of SQ(1) =1 STEP 2). Second Last Digit = 2*3*1 + any carryover from STEP 1 = 6 STEP 3). Third Last Digit 2*4*1+ Sq(3) + any carryover from STEP 2 = 2*4*1 +9= 17. so 7 and 1 carryover STEP 4). Fourth last digit is 2*4*3 + any carryover (which is 1) = 24+1=25. So 5 and carry over 2. STEP 5). In the beginning of result will be Sq(4) + any carryover from Step 4. So 16+2 =18. So the result will be 185761.
Special Numbers ü e to 15 decimal places e=2. 71828459045. . . Andrew Jackson was the 7 th president, elected in 1828 to two terms. Then tack on the 45 -90 -45 right triangle. • pi - first eight digits of pi by K. Mahadevan, PGT to get the first eight digits of pi, count the number of letters in each word of this phrase: • May(3) I(1) have(4) a(1) large(5)container(9) of(2) coffee(6)?
PROFIT AND LOSS : Suppose the price is first increase by X% and then decreased by Y% , the final change % in the price is given by the following formula. Final Difference % = X- Y – XY/100. EXAMPLE 1. : The price of T. V set is increased by 40 % of the cost price and then decreased by 25% of the new price. On selling, the profit for the dealer was Rs. 1, 000. At what price was the T. V sold. From the above mentioned formula you get : Final difference % = 40 -25 -(40*25/100)= 5 %. So if 5 % = 1, 000 then 100 % = 20, 000. C. P = 20, 000 S. P = 20, 000+ 1000= 21, 000.
EXAMPLE 2 : The price of T. V set is increased by 25 % of cost price and then decreased by 40% of the new price. On selling, the loss for the dealer was Rs. 5, 000. At what price was the T. V sold? From the above mentioned formula you get : Final difference % = 25 -40 -(25*45/100)= -25 %. So if 25 % = 5, 000 then 100 % = 20, 000. C. P = 20, 000 S. P = 20, 000 – 5, 000= 15, 000.
TRY THESE Now find out the difference in % of a product which was : 1) First increased by 20 % and then decreased by 10 %. 2) First Increased by 25 % and then decrease by 20 % 3) First Increased by 20 % and then decrease by 25 %. 4) First Increased by 10 % and then decrease by 10 %. 5) First Increased by 20 % and then decrease by 15 %.
TIME AND WORK: 1 If A can finish work in X time and B can finish work in Y time then both together can finish work in (X*Y)/ (X+Y) time. . 2. If A can finish work in X time and A and B together can finish work in S time then B can finish work in (XS)/(X-S) time. 3. If A can finish work in X time and B in Y time and C in Z time then they all working together will finish the work in (XYZ)/ (XY +YZ +XZ) time 4. If A can finish work in X time and B in Y time and A, B and C together in S time then : C can finish work alone in (XYS)/ (XY-SX-SY) B+C can finish in (SX)/(X-S) and A+ C can finish in (SY)/(Y-S)
PERCENTAGE TYPE 1 : Price of a commodity is increased by r%. By how much % should the consumption be reduced so that the expense remain the same. TYPE 2 : Price of a commodity is decreased by r %. By how much % can the consumption be increased so that the expense remain the same. Solution : TYPE 1 : (100* r ) / (100+r) TYPE 2 : (100* r ) / (100 -r)
Example TYPE 1 : Price of a commodity is increased by 60 %. By how much % should the consumption be reduced so that the expense remain the same. TYPE 2 : Price of a commodity is decreased by 60 %. By how much % can the consumption be increased so that the expense remain the same. Solution : TYPE 1 : (100* 60 ) / (100+60) = 37. 5 % TYPE 2 : (100* 60 ) / (100 -60) = 150 %
Geometry 1) Apollonius theorem could be applied to the 4 triangles formed in a parallelogram. 2) Area of a trapezium = 1/2 * (sum of parallel sides) * height = median * height where median is the line joining the midpoints of the oblique sides. 3)Let W be any point inside a rectangle ABCD. Then 4)Let ‘ a’ be the side of an equilateral triangle. Then if three circles be drawn inside this triangle touching each other then each’s radius =
Successive discounts v. Suppose in 1999 population increases by x% and then in 2000 by y% so the population in 2000 now is more that was in 1999. v. Suppose in 1999 population decreases by x% and then in 2000 by y% so the population in 2000 now is less that was in 1999. v. In 1999 population increases by 10% and then in 2000 by 5% so the population in 2000 now is 10+5+(50/100)=+15. 5% more that was in 1999. v If there is a decrease then it will be preceded by a negative sign and likewise.
Fibonacci Addition Trick Step 1: Choose two numbers Step 2: Form a Fibonacci sequence for ten numbers Example, I choose number 5 for my first number and 6 for my second number. Then I add the numbers to get a Fibonacci sequence. 5+6 gives my 3 rd number which is 11; 6+11 gives me my 4 th number which is 17. The entire sequence is as follows: 1 st – 5, 2 nd – 6, 3 rd – 11, 4 th – 17, 5 th – 28, 6 th - 45 7 th – 73, 8 th – 118, 9 th – 191, 10 th – 309. What is the sum of all these 10 numbers? (6 seconds) The answer will also be 803. Trick: Multiply 7 th number by 11 and the answer is 803. (It is true is any set of ten Fibonacci numbers)
Conversion of Fraction into Decimals and Percents Fractions 1/2 1/3 2/3 1/4 3/4 1/5 2/5 3/5 4/5 1/6 5/6 1/8 3/8 5/8 7/8 1/9 1/10 1/11 1/12 1/16 1/20 1/25 1/50 Decimals. 5. 3. 6. 25. 75. 2. 4. 6. 8. 16. 83. 125. 375. 675. 875. 1. 1. 09. 083. 0625. 04. 02 Percents 50% 33. 3% 66. 6% 25% 75% 20% 40% 60% 80% 16. 6% 83. 3% 12. 5% 37. 5% 62. 5% 87. 5% 11. 1% 10% 9. 09% 8. 3% 6. 25% 5% 4% 2%
Labeling Right Triangles Let’s put it all together. Given that angle B is the reference angle, here is how you must label the triangle: B (ref. angle) hypotenuse adjacent A C opposite
Labeling Right Triangles Let’s put it all together. n Given that angle C is the reference angle, here is how you must label the triangle: n B hypotenuse opposite A C (ref. angle) adjacent
TRIGONOMETRY is i Th te osi pp n ea . gle th so This is opposite the right-angle There are three ratios that you need to learn: This is next to the angle Where are the hypotenuse, adjacent and opposite lengths.
A PHARSE TO REMEMBER THE ABOVE DEFINITION OF TRIGONOMETRIC RATIOS SOME OLD HORSES CAN ALWAYS CAN HEAR TREIR OWNER’S APPORACH
Math Magic – Trick 1 Pick a number… any number! (keep it a secret though) say x Add 1 to that number i. e. x+1 Multiply by 3 i. e. 3 x + 3 Subtract your ‘secret’ number i. e. 2 x +3 Add 5 i. e. 2 x + 8 Divide by 2 i. e. x + 4 Subtract your secret number. i. e. 4 The answer is always 4
Everyone needs a little humor in their life Especially mathematicians Trick 1. Viral Math Challenge : Can you find the answer to this problem? Trick: 2. Write down a three digit number. The first and third digits must differ by more than one. Example: 264 Now reverse the digits to form a second number. we get 462. Subtract the smaller number from the larger one. 462 - 264 = 198 Now reverse the digits in the answer you got in step 3 and add it to that number. : 891 + 198 = 1089
Fast Multiplication ØSelect a four-digit number Example : 2345 ØAny 4 digit number is multiplied by 10001. what will be the answer ? Do you know the answer is amazing? ØThe answer will be the given four digit number is written twice in the same order. Ø 2345 10001 = 2345 ØSince 10001 = 73 x 137, 2345 73 137 = 2345 Øabcd 73 x 137 = abcd. ØEntering a four-digit number twice (2345) will be divisible by 73, 137, and the original four-digit number. Ø 1234 73 137 = 1234 ØCan you tell me 2685 is divisible by 73? ØCan you tell me 8194 is divisible by 137? ØCan you tell me 3829 is divisible by 3829?
Fast Multiplication/Division ØSelect a three-digit number Example : 234 ØAny 4 digit number is multiplied by 1001. what will be the answer ? ØDo you know the answer is amazing? ØThe answer will be the given three digit number is written twice in the same order. Ø 234 1001 = 234234 ØSince 1001 = 7 x 11 13, 234 7 x 11 13 = 234234 Øabc 73 x 137 = abcabc. ØEntering a three-digit number twice (234234) will be divisible by 73, 137, and the original three-digit number. Ø 123 7 x 11 13 = 1234 ØCan you tell me 685685 is divisible by 7? ØCan you tell me 194194 is divisible by 13? ØCan you tell me 829829 is divisible by 829? ØCan you tell me 529529 is divisible by 11?
Fast Multiplication/Division ØSelect a two-digit number Example : 23 ØAny 2 digit number is multiplied by 10101. what will be the answer ? ØDo you know the answer is amazing? ØThe answer will be the given two digit number is written thrice in the same order. Ø 23 10101 = 23 2323 ØSince 10101 = 21 x 37 13, 23 3 7 x 37 13 = 232323 Øab 37 x 13 21 = ababab. ØEntering a two-digit number thrice (232323) will be divisible by 37, 13, 7, 3 and the original two-digit number. Ø 12 3 7 x 37 13 = 121212 ØCan you tell me 686868 is divisible by 7? ØCan you tell me 191919 is divisible by 13? ØCan you tell me 828282 is divisible by 82? ØCan you tell me 525252 is divisible by 7? ØCan you tell me 757575 is divisible by 21?
MISSING DIGIT TRICK STEP 1: Ask participants to write down their mobile number. STEP 2 : Ask them to add the digits. STEP 3 : Ask them to subtract this number from the original one. STEP 4 : Ask them to select any digit from this new number and strike it out, without showing you. STEP 5 : Ask them to add the remaining digits and write down the answer they get. Example: 8+3+9+7+0 = 34 STEP 6 : Ask them to tell you the number they get (34) and you will tell them which number they struck out.
TRICK Hate 8 * §Ask your friend to choose a number between 1 and 9 except 8 Multiply the number by 9. 7 x 9 = 63 • Multiply the answer by 12345679 (no 8) 63 x 12345679 = magic! §Select another number say 6 6 x 9 = 54 54 x 12345679 = 66666 1 x 9 x 12345679 = 11111 2 x 9 x 12345679 = 22222 • Can you find the value of the following? 3 x 9 x 12345679 4 x 9 x 12345679 5 x 9 x 12345679
Dice Magic* §Find three dice and a friend. Turn your back. Ask your friend to roll the three dice so that you can't see the resulting numbers. §Multiply the number on the first die by 2. Add 5. Multiply by 5 * Add the number on the second die. Multiply by 10 *Add the number on the third die. Subtract 250. Now you are able to tell numbers on the top of three dice. Now for your magic prognostication. § Example §(5 x 2 + 5) x 5 + 3 x 10 + 4 -250 = 534 §Now you are able to tell numbers on the bottom of three dice. §i. e. 243 § Magic happens. . . all the time!
Calculating Dice* 1. Find three dice, a calculator and a friend. Roll the three dice and write down this number. Repeat this number. 2. Now roll one dice and multiply your number by this roll. 3. Divide this number by 11 Divide this number by 13 Divide this number by your single dice roll and divide by 7 4. Magic happens! 5. The magic is the three digit number appeared on three dice 6. Example: 123, 123123 7. 123123 x 5 = 615615 8. 615615/11= 55965, 55965/13 = 4305/5=861 , 861/7=123
Roll Them* Ask your friend to roll the dice without revealing to you the numbers Example : Your friend rolls a 4 and a 6. Ask your friend to multiply the number on the first die by 2 4 x 2 = 8. Add 5 8 + 5 = 13 Multiply by 5 13 x 5 = 65 Add the number on the second die. i. e. 65 + 6 = 71 You can now predict the numbers on the two dice. Here is what you must do: Subtract 25 71 - 25 = 46 4 = first die 6 = second die.
Mobile number Trick 1)Insert in the first five digits of your phone number (not the area code) Example : 94456 2)Multiply these three numbers by 80 94456 x 80 = 7556480 Add 1 7556480 +1 = 75564801 3)Multiply by 2500 75564801 x 2500 = 18891202500 4)Add to this the last 4 digits of your phone number 18891202500 + 06533 = 18891209033 5)Add again the last 4 digits of your phone number. 18891209033 + 06533= 18891215566 6)Subtract 2500 18891215566 - 2500 = 18891213066 7&. Divide number by 2 18891213066 / 2 = Magic phone number = 9445606533
Math mentalism to amaze everyone 1 - Ask a participant to choose a four digit number 2 - Write down your prediction. 3 - Ask participant to choose another four digit number 4 - Write down your own number under the participants number. 5 - Ask participant to choose another four digit number. 6 - Write down your own number under the participants number. 7 - Total the five numbers. 8 - Show your prediction and exchange high fives! You are now a Mathalism specialist. Example: i. 2345 ii. 22343 iii. 7123 iv. 2876 v. 5690 vi. 4309 vii. 2345 + 7123 + 2876 + 5690 + 4309 = Answer in step ii
Dice Math Trick i. 1)Use three dice. 2)Have a friend roll the dice and then stack them one on top of the other. 3)Tell your friend that you can not see five faces of the dice. 4) The 5 numbers you cannot see are : i)the bottom of the top dice; the top of the second dice ii) the bottom of the second dice; the top of the last dice iii) the bottom of the last dice You will now predict the total of the five hidden numbers. 5)In your head subtract the very top face of the three dice from 21. 6)The answer will be the total of the five hidden numbers. Reason is 7) The sum of the numbers of opposite faces on each die is 7
The Locker Problem
The Locker Problem As 500 students enter a school, they pass lockers that are numbered from 1 to 500. The first student opens every locker; the second student closes every second locker; the third student changes the position of every third locker (by opening the closed lockers and closing the open lockers); and the fourth student changes the position of every fourth locker. This pattern continues for all 500 students. Which lockers are open after all students enter the school?
The Hat Problem
The Hat Problem There are 100 people lined up on the steps of a stadium, each on a different step, all looking down toward the field so that they can see everyone in front of them, but no one behind them. Each person will be given either a red or black hat. We do not know the total number of red or black hats. Each person will not be able to see the color of his own hat (or the ones behind him), but will be able to see the colors of all the hats in front of him. Starting in the back, the last person will be asked what color hat he is wearing. If he guesses correctly, he will live; if he guesses incorrectly, he will be shot immediately. The second to last will be asked, and so on, until we reach the person on the bottom step. Each person will be able to hear what all the people behind him say, and will also be able to hear which people behind him were shot.
The Hat Problem Before we begin this process, the 100 people may meet to discuss a strategy. They can plan whatever they want, but once the line-up begins, they may no longer confer. At each person's turn, he may only say "black" or "red, " and no other words -- if he says anything else, all 100 people will be executed. He may also not use tone of voice, volume, etc. , to convey any meaning -- this will be detected and they will all be shot. What strategy will guarantee saving the maximum number of people? What is this number?
A Familial Math Equation
A Familial Math Equation A mother is 21 years older than her son In 6 years, she will be 5 times older than him Where is the father?
A Familial Math Equation Let x = the age of the mother (in years) Let y = the age of the son (in years) x = 21 + y x + 6 = 5(y + 6) Therefore, (21 + y) + 6 = 5(y+6)
A Familial Math Equation y + 27 = 5 y + 30 5 y – y = 27 – 30 4 y = -3/4 Remember that y is expressed in years…
Math Quiz Answers The obvious solution may be correct, but where’s the fun in always being right?
Math Quiz Answers 1) There are 8 apples on the table, you take away 3. How many do you have? 3 – The other 5 are still on the table
Math Quiz Answers 2) There are 10 birds in a field. If 2 are shot, how many are left? 2 – The others have flown away
Math Quiz Answers 3) Take away the first letter, take away the last letter, then take away all the other letters. What do you have left? The mailman http: //www. curiousmath. com
Math Quiz Answers 4) If you have 4 melons in one hand, and 7 apples in the other - What do you have? Big hands http: //www. curiousmath. com
Math Quiz Answers 5) A box holds nine ears of corn. A squirrel carries out three ears a day, but it takes him nine days to carry out all the corn. Why? He carries out one ear of corn in addition to his own two ears
Math Quiz Answers 6) Why do white sheep eat more than black sheep? There are more white sheep than black sheep.
Math Quiz Answers 7) It takes 7 men 2 hours to build a wall. How long does it take 3 men to build the same wall? Why bother? The 7 men have already built it.
Math Quiz Answers 8) I have 2 coins in my hand that add up to 60 cents. One of the coins is not a half dollar. What are the coins? A half dollar and a dime one (the dime) is not a half dollar
Math Quiz Answers 9) A man wanted to plant 4 trees, but all 4 had to be equal distances from each other. How did he do it?
Math Quiz Answers 10) A fisherman was asked the length of the fish he had caught. He said "it is 30 cms plus half its length. “ How long was the fish? 60 cms
Math Quiz Answers 11) What comes next in the following sequence ? 1, 4, 5, 6, 7, 9, 11, . . . 100 – the next number spelled without a t
Math Quiz Answers 12) In a scientific context, what could the following phrase mean? “How I want a drink, alcoholic of course, after the heavy chapters involving quantum mechanics…” π = 3. 14159265358979…