MagicMath

The Magic of Math (Authur Benjamin)

  • Very interesting (some what basic level)
  • p13; Trick to squaring numbers in your head: N2 = (N-a)(N+a) + a2. Choose a such that the numbers (N-a) and (N+a) are easy to multiply, then add a2; Mental Models eg: 412 = 40*42 + 1 = 1680 + 1 = 1681
  • p14; The "right" way to multiply numbers is left-to-right. Focus earlier on the part the matters, can calculate then "forget" for the sub-leading digits. Mental Models ShouldTeachKids
  • p15: School spend much time on teaching calculation with pen and paper and almost no time on performing calculation in our heads. However in real life, doing them in your head is much more important. Techniques taught in school for pen-and-paper calculation are sub-optimal for calculations in your head. ShouldTeachKids
  • Mental Addition; "left to right"
  • Mental Subtraction: "Over subtract" and use compliments to turn hard subtraction problems into easy addition problems eg: 84 - 39 = 44 + 1 = 45
  • p19: Trick to multiply 2-digit numbers by 11. Add the digits and place the sum "between" the original digits. Mental Models eg: AB x 11 = A(A+B)B 44 x 11 = 484 if the sum is bigger than 10, then use the last digit and carry. eg: 48 x 11 = 528
  • p20: Order magnitude estimates are another important and simple tool that is not taught in school. Too much time is spent on calculating the answer and too little time on thinking about the important aspects of the problem. Its always more important to know the approximate size, rather than the last (or even the first digits). ShouldTeachKids
    • JA: easy if first convert to scientific notation
  • p21: When multiplying an N digit number by an M digit number the answer will either have N+M or N+M-1 digits. 10*10 = 100 / 99*99 < 102 x 102 = 1e4 (5 digits) -) If the product of the leading numbers is 10 or larger, the result will have M+N digits. -) If the product of the leading numbers is 4 or smaller, the result will have M+N-1 digits. -) In between requires more calculation
  • p22: An M digit number divided by a N digit number has M-N or M-N+1 digits. -) If the leading digit in the number being divided is smaller, then the result has M-N digits. -) If the leading digit is larger then its its larger choice. -) If they are the same, apply the same logic to the sub-leading digits.
  • p30: A negative number times a negative number is positive, because we want the distributive law to work for all numbers. [BROKEN LINK: Math] -5 x 0 = 0 = -5 * (-3 + 3) = X - 15 => X should be + 15
  • p52: A number is a multiple of 9 if and only if its digits sum to a multiple of 9
  • p54: "Cast-outing nines" the process of repeatedly summing digits of a number until you are left with a one digit number. eg: 34590 -> 31 -> 4 (4 is called the digital root of 34590) If an addition problem is done correctly the digital root of the numbers added add to the digital root of the sum.
  • p64: Earth takes about 365.25 days to go around the sun. Julian calendar (established by Julian Ceasar) was 365 days/year with a leap day once every four years. Works well. Problem year actually takes 365.243. Julian calendar slightly fast, off 3 days in 400 years. New calendar Gregorian adopted in 1582, in that year the Catholic nations removed to days from their calendar. (England didn't make the switch until 1752)
  • p140: powers of 2 are the "elementary particles" of addition. Prime numbers are the "elementary particles of multiplication.
  • p301: The cumulative law of addition can fail when adding an infinite number of numbers. Problem when negative and positive components each form diverging series.

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