Barron's SAT Subject Test: Math Level 1, calculator

Barron's SAT Subject Test: Math Level 1, 7th Edition: With Bonus Online Tests 

– September 1, 2018, by Ira K. Wolf Ph.D. (Author)

The secret of mental math

Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks by Arthur T. Benjamin

Originally published: 2006

Summarize the method in three words: Left to right.

Since most human memory can hold only about seven or eight digits at a time, this is about as large a problem as you can handle without resorting to artificial memory devices, like fingers, calculators, or the mnemonics taught in Chapter 7.

Gauss’s desire to better understand Nature through the language of mathematics was summed up in his motto, taken from Shakespeare’s King Lear (substituting “laws” for “law”):“Thou, nature, art my goddess; to thy laws/My services are bound.”  (p20)

Chapter 0 Quick Tricks: Easy (and Impressive) Calculations

32 × 11

To solve this problem, simply add the digits, 3+2 = 5, put the 5 between the 3 and the 2, and there is your answer: 352  (p1)

314 × 11, 

the answer still begins with 3 and ends with 4. Since 3+1= 4, and 1+4=5, the answer is 3454. (p4)

• Square numbers: 

1. The answer begins by multiplying the first digit by the next higher digit.

2. The answer ends in 25.

square the number 35, we simply multiply the first digit (3) by the next higher digit (4), then attach 25. Since 3×4=12, the answer is 1225. Therefore, 35×35=1225. (p5)

We can use a similar trick when multiplying two-digit numbers with the same first digit, and second digits that sum to 10. The answer begins the same way that it did before (the first digit multiplied by the next higher digit), followed by the product of the second digits. For example, let’s try 83× 87. (Both numbers begin with 8, and the last digits sum to 3+7 =10.) Since 8 ×9 =72, and 3×7= 21, the answer is 7221. (p5)

Chapter 1 A Little Give and Take:.Mental Addition and Subtraction

there are many good reasons why it is better to work from left to right. After all, you read numbers from left to right, you pronounce numbers from left to right, and so it’s just more natural to think about (and calculate) numbers from left to right. When you compute the answer from right to left (as you probably do on paper), you generate the answer backward. That’s what makes it so hard to do math in your head. Also, if you want to estimate your answer, it’s more important to know that your answer is “a little over 1200” than to know that your answer “ends in 8.” Thus, by working from left to right, you begin with the most significant digits of your problem. If you are used to working from right to left on paper, it may seem unnatural to work with numbers from left to right. But with practice you will find that it is the most natural and efficient way to do mental calculations.

It also illustrates a fundamental principle of mental arithmetic—namely, to simplify your problem by breaking it into smaller, more manageable parts. This is the key to virtually every method you will learn in this book. To paraphrase an old saying, there are three components to success—simplify, simplify, simplify.

• Subtraction:

1. First subtract 20, then subtract 9:

86 - 29 = 66 - 9 = 57 (first subtract 20) (then subtract 9)

But for this problem, I would prefer the following strategy:

2. First subtract 30, then add back 1:

86 - 29 = 56  + 1 =  57 (first subtract 30) (then add 1)

Here is the rule for deciding which method to use: If a two-digit subtraction problem would require borrowing, then round the second number up (to a multiple of ten). Subtract the rounded number, then add back the difference.  (p22)

• Using Complements 

Quick, how far from 100 are each of these numbers?

57 68 49 21 79

Here are the answers:

Notice that for each pair of numbers that add to 100, the first digits (on the left) add to 9 and the last digits (on the right) add to 10. We say that 43 is the complement of 57, 32 is the complement of 68, and so on. 

the complements are determined from left to right. As we have seen, the first digits add to 9, and the second digits add to 10. (An exception occurs in numbers ending in 0—e.g., 30 + 70 =100)

Chapter 2 Products of a Misspent Youth: Basic Multiplication  (1:10:35)

GC- secret mental math, proof 11 * & 9*, adding left to right

The Secrets of Mental Math

DVD 510 SEC, 2010

Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks, 510 BEN, 2006

ARTHUR T. BENJAMIN, benjamin@hmc.edu

My comment: No mathematician bother explaining why. It is completely focusing on single-digit computation all the time. It is so ironic when I am watching this, youtube gives us this ad.

1: Math in Your Head! 

 adding from left to right, unlike what you do on paper and make using complement (the distance betwwen a number and a conveninent round number, typically, 100 or 1000. For example, the complemnt of 43 is 57, since 43 + 57 = 100.   (Using zero's property, B^n- structure)

When adding numbers, it is easier to add largest to smallest, rather than smallest to largest. Again, doing calculation from left to right is easier because that's the way we read numbers. Also it gives you better estimation.

Proof: 23 * 11= 23 * ( 10 + 1) = 230 + 23  = 253, thus  ( 3 + 2 = 5) 

2: Mental Addition and Subtraction 

3: Go Forth and Multiply 

Proof of distributive law

4: Divide and Conquer 

Turn now to the last fundamental operation of arithmetic: division. Explore a variety of shortcuts for dividing by one- and two-digit numbers; learn how to convert fractions such as 1/7 and 3/16 into decimals; and discover methods for determining when a large number is divisible by numbers such as 3, 7, and 11.

5: The Art of Guesstimation 

In most real-world situations-such as figuring out sales tax or how much to tip-you don't need an exact answer but just a reasonable approximation. Here, develop skills for effectively estimating addition, subtraction, multiplication, division, and square roots.

6: Mental Math and Paper 

Sometimes we encounter math problems on paper in our daily lives. Even so, there are some rarely taught techniques to help speed up your calculations and check your answers when you are adding tall columns of numbers, multiplying numbers of any length, and more.

7: Intermediate Multiplication 

Take mental multiplication to an even higher level. Professor Benjamin shows you five methods for accurately multiplying any two-digit numbers. Among these: the squaring method (when both numbers are equal), the "close together" method (when both numbers are near each other), and the subtraction method (when one number ends in 6, 7, 8, or 9).

8: The Speed of Vedic Division 

Vedic mathematics, which has been around for centuries, is extremely helpful for solving division problems-much more efficiently than the methods you learned in school. Learn how Vedic division works for dividing numbers of any length by any two-digit numbers.

9: Memorizing Numbers 

Think that memorizing long numbers sounds impossible? Think again. Investigate a fun-and effective-way to memorize numbers using a phonetic code in which every digit is given a consonant sound. Then practice your knowledge by trying to memorize the first 24 digits of pi, all of your credit card numbers, and more.

10: Calendar Calculating 

The fun continues in this lecture with determining the day of the week of any date in the past or in the future. What day of the week was July 4, 2000? How about February 12, 1809? You'd be surprised at how easy it is for you to grasp the simple mathematics behind this handy skill.

11: Advanced Multiplication 

Professor Benjamin shows you how to do enormous multiplication problems in your head, such as squaring three-digit and four-digit numbers; cubing two-digit numbers, and multiplying two-digit and three-digit numbers. While you may not frequently encounter these large problems, knowing how to mentally solve them cements your knowledge of basic mental math skills.

12: Masters of Mental Math 

Professor Benjamin concludes his exciting course by showing how you can use different methods to solve the same problem; how you can find the cube root of large perfect cubes; how you can use the techniques you've learned and mastered in the last 11 lectures; and more.

GC-Queen of the Sciences: A History of Mathematics lecture #13-24

13 Newton—Modeling the Universe

14 Leibniz and the Emergence of Calculus

15 Euler—Calculus Proves Its Promise

Raduis of 3438  *  2 * pi  = 360 * 60

16 Geometry—From Alhambra to Escher

17 Gauss—Invention of Differential Geometry

Negative curvative always takes pressure away. 

18 Algebra Becomes the Science of Symmetry

19 Modern Analysis—Fourier to Carleson

20 Riemann Sets New Directions for Analysis

21 Sylvester and Ramanujan—Different Worlds

22 Fermat's Last Theorem—The Final Triumph

23 Mathematics—The Ultimate Physical Reality

24 Problems and Prospects for the 21st Century

People from outside mathematics often focus on mathematics as problems, but for mathematicians, the subject is all about seeing how these patterns interrelate.  

GC-Queen of the Sciences: A History of Mathematics lecture #7-12

Very important Lesson learned

• 1. logarithm invented to facilitate multiplication calculation 
•  2. Inertia 

7 Chinese Mathematics—Advances in Computation

The Zhoubi Suanjing (Chinese: 周髀算經; Wade–Giles: Chou Pi Suan Ching) is one of the oldest Chinese mathematical texts. "Zhou" refers to the ancient Zhou dynasty (1046–256 BCE); "Bi" means thigh and according to the book, it refers to the gnomon of the sundial. The book is dedicated to astronomical observation and calculation. "Suan Jing" or "classic of arithmetics" were appended in later time to honor the achievement of the book in mathematics.

This book dates from the period of the Zhou dynasty, yet its compilation and addition of materials continued into the Han dynasty (202 BCE–220 CE). It is an anonymous collection of 246 problems encountered by the Duke of Zhou and his astronomer and mathematician, Shang Gao. Each question has stated their numerical answer and corresponding arithmetic algorithm.

The Nine Chapters on the Mathematical Art (九章算術) is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving mathematical texts from China, the first being Suan shu shu (202 BCE – 186 BCE) and Zhoubi Suanjing (compiled throughout the Han until the late 2nd century CE). Extend place value to 0.1, 0.01. Negative numbers did not recognize as numbers but as an intermediate step. 

Around A. D 1000, the Chinese would begin to use zero a placeholder

Liu Hui 劉徽 c. 225- c. 295

Haidao Suanjing (海島算經; The Sea Island Mathematical Manual) was written by the Chinese mathematician Liu Hui of the Three Kingdoms era (220–280) as an extension of chapter 9 of The Nine Chapters on the Mathematical Art. During the Tang Dynasty, this appendix was taken out from The Nine Chapters on the Mathematical Art as a separate book, titled Haidao suanjing (Sea Island Mathematical Manual), named after problem No 1 "Looking at a sea island." In the time of the early Tang dynasty, Haidao Suanjing was selected into one of The Ten Computational Canons as the official mathematical texts for imperial examinations in mathematics.

Zu's pi is accurate to a degree of less than 3 parts in 10 millions.

Sunzi Suanjing (孙子算经; lit. 'The Mathematical Classic of Master Sun/Master Sun's Mathematical Manual') was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still unknown but he lived much later than his namesake Sun Tzu, author of The Art of War. From the textual evidence in the book, some scholars concluded that the work was completed during the Southern and Northern Dynasties. Besides describing arithmetic methods and investigating Diophantine equations, the treatise touches upon astronomy and attempts to develop a calendar.

Li Chunfeng (李淳風) (602–670). Shibu Suanjing (十部算经) in 656. These were ten mathematical manuals submitted to the emperor.

Jia Xian (賈憲 ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song Jia was a palace eunuch of the Left Duty Group. 

Li Ye (李冶 1192–1279), born Li Zhi (李治), courtesy name Li Jingzhai published and improved the tian yuan shu method for solving polynomial equations of one variable. Along with the 4th-century Chinese astronomer Yu Xi, Li Ye proposed the idea of a spherical Earth instead of a flat one before the advances of European science in the 17th century. his work, Ceyuan haijing (測圓海鏡, Sea mirror of circle measurements, Mirror like the Ocean, Reflectign the Heaven of Calculation of Circles (written 1248)

Qin Jiushao ( 秦九韶;  ca. 1202–1261) courtesy name Daogu (道古)

Zhu Shijie (=朱世傑; pinyin: Zhū Shìjié; Wade–Giles: , 1249–1314), courtesy name Hanqing (漢卿), pseudonym Songting (松庭), . He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works have survived. Introduction to Computational Studies (算學啓蒙 Suan hsüeh Ch'i-mong), and Jade Mirror of the Four Unknowns. Siyuan yujian (四元玉鉴), also referred to as Jade Mirror of the Four Origins,is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie. Zhu advanced Chinese algebra with this Magnum opus.The book consists of an introduction and three books, with a total of 288 problems.

Chinese mathematicians disappears after 13th century. 

8 Islamic Mathematics—The Creation of Algebra

Al-Kwarizmi' name is also the origin of algorithm, which is used in mathematics to mean a procedure with clearly prescribed steps. The equal sign "=" was not invented until the 16th century in western Europe. 

9 Italian Algebraists Solve the Cubic

10 Napier and the Natural Logarithm

11 Galileo and the Mathematics of Motion

12 Fermat, Descartes, and Analytic Geometry

of Mersenne, 

2^82,589,933 − 1  (Dec 7, 2018)