Five is the Evenest Odd of Them All If I told you that dividing the number 74230 by 5 is about as easy as it gets, would you believe me? Clearly you can't rely on the old "five times WHAT is equal to 74,230," because nobody memorizes their times tables up to five digit numbers. So there must be a trick, right? Indeed there is. But first... Before writing this post I googled "what is dividing by five good for." Can you guess what I found? That's right. Nothing. Sure, it's handy if you need to figure out how to share a pizza amongst five friends and the like, but it doesn't have the gravitas of doubling or halving. Five is an odd number and odd numbers are appreciated for their symmetry-breaking aesthetics. Three is too few, and seven is too many. Five is just right! Well maybe it is or maybe it isn't, but it is darned easy to divide with. A number is divisible by five if it ends in zero or five. Depending on how you look at it, th

The Multiplication Workhorse Let's begin again with a problem. Can you easily multiply 484 ×  5 in your head? If so, then how about  3261  ×  5? By the end of this post, you will be able to easily multiply any number by five provided you have mastered doubling and halving discussed in my previous posts. Multiplication by five is the workhorse of martian mathematics. At least it is for the intermediate methods of multiplying times 3, 4, 5, 6, and 7. More on that in a later post. I think that everyone would agree that aside from multiplying by one, nothing is easier than multiplying by ten—just add a zero onto the end of the number. For example, 37  × 10 = 37 0 , 271  × 10 = 271 0 , and hell   × 10 = hell 0 . Well, maybe not that last one, but you get the idea: just tack a zero onto the back of the number when multiplying by ten.  Multiplying by five is almost as easy as multiplying by ten because five is just half of ten.  Using the commutative property of multiplic

Obscura: The Integral and Fractional Parts In my last post ( Foundations-Halving ), you learned that when halving an odd digit, you should treat it like the next lowest even number. For example, in 30 ÷ 2 we treat half of 3 the same as half of 2. Formally, this is accomplished using the mathematical function known as "the integral part." The integral part of a  is written with brackets, [ a ],  and identifies the unique integer a –1 <  [ a ] ≤ a . Similarly, the fractional part is denoted with curly brackets, { a }, such that { a } = a  – [ a ]. For example, the integral part of 3 ½ is 3  and the fractional part of 3 ½ is  ½ . When I say that half of three is one, mathematically I mean [3 ÷ 2] = [1 ½] = 1 . So, if it bothers you when I write  ½  ⋅  3 = 1, then just know that what I mean is [ ½  ⋅  3] = 1.

To Halve or not to Halve... That is the Question. As with the last post, let's begin with a problem. Try and mentally calculate the following division problem:  724898 / 2 .  As before, we will return to this problem at the end.  If you have an even number of items—let's say six jelly beans—and you halve them, then you divide them evenly into two groups—three jelly beans for you and three for a friend. The way we are using it here, you can't halve an odd number. Halving is dividing by two. Only even numbers are divisible by two. As martian mathematics is largely a single-digit technique, we really only need to know how to halve single-digit numbers. We will start with the easy digits: even numbers. Instead of dividing by two, I will denote halving by multiplying by a half. Halving a number with all even digits is as easy as doubling a number with all digits less than five. You can just read off the answer as quickly as you can read off the digits. Let's tr

Doubling is ×2 Before we begin, I would like you to try your hand at the following multiplication problem (in your head): 2  × 734829.  If you would like to be able to multiply this as easily as reading of the digits, then read on.  We are all familiar with doubling. If you have  N  items in a bag and you double it, then it is like adding a second identical bag so that now we have twice as many as we did before ( 2N ).  In martian mathematics you need to know the following doubles: What about the numbers 5-9? We will get to that in a moment, but first let's take what we have out for a ride.  If we want to double a number of any size then we just read it from left-to-right one digit at a time, doubling as we go. Easy, huh? Now let's tackle the other five doubles you need to know. For the most part, martian math is a single-digit technique and we can ignore anything not in the ones' place. For this reason, when we double a number greater than or equal to fi

Even and Odd Numbers Even numbers are the numbers of sharing jelly beans among two friends. You make sure that they are split evenly, grabbing two at a time and dropping one in each bowl. This is what it means for a number to be even. If you placed both bowls on the opposite ends of an old-fashioned scale, then they would balance. That is unless you accidentally drop an extra jelly bean in your own bowl, tipping the scales so to speak in your favor; then you have an odd number of jelly beans.  This is what it means for a number to be odd. The first five even numbers are 0, 2, 4, 6, and 8. The first five odd numbers are 1, 3, 5, 7, and 9. As martian mathematics is largely single digit in nature, these are the only ones that matter here. It is a happy coincidence that the evenness or oddness of a number—called parity—is betrayed by the ones' place value of the number in question  (I will show why this is in a later post) : 332 is even, while 241 is odd.  Knowing the e

Adding and Subtracting Three is as Easy as 147.  147? I told you it was weird. Hopefully you have done your homework since my last post and studied the telephone dialing pad. The first thing we need to do is rotate it 90° counter-clockwise and then move the zero to the top left next to the 3 key. Now, let's skip count in 3's starting at 0. Follow along the telephone sequence with your finger:  3, 6, 9, 1 2,  1 5,  1 8,  2 1,  2 4,  2 7,  3 0.  We are back at the beginning of the sequence, 0. Do you see what happened to the tens' place when we changed rows? It incremented. If you commit this sequence to memory, then you can easily add 3 to any number. For example, 34 + 3 = 37 because 7 comes after 4 in the telephone sequence and since it is on the same row, the tens' place value does not change. How about 28 + 3? Since the next number in the telephone sequence is 1 and it appears on the next row, we increment the tens' place value to get 31.  Let&

Adding and Subtracting Five is Easy I will frequently use the term "five's conjugate". The word conjugate means yoked together or coupled. The five's conjugate are pairs of numbers whose difference is five—that's the coupling. The five’s conjugate of 1 is 6, of 2 is 7, of 3 is 8, etc. The five's conjugate of 6 is 1, of 7 is 2, of 8 is 3, etc.  The easiest way to think of the five's conjugate is with your hands. Each hand has a relative finger count (1-5). Both hands combined have an absolute finger count (1-10). Martian mathematics is largely single-digit mathematics, so we are mostly interested in the single-digits (0-9); therefore, we think of ten as zero. Let's say you want the five's conjugate of 7. Hold up both hands with the left hand's fingers fully extended. That is five. On the right hand, extend the thumb and index finger. You now have seven fingers. Now, make your left hand into a fist. You are left with only two

Martian Mathematics  I call this Martian mathematics because if you saw someone doing math this way, you would probably think they were from Mars.  87 × 8, no problem. That's just 670 + 26.  Weird, huh? Not everything here is "out of this world," but a good bit of it will seem strange at first. With that said, I developed this system of mental mathematics for my children. Innumeracy is a growing problem worldwide and dependence on machines for calculation is the rate-limiting step in analytical thinking. These shortcomings are what I hope to alleviate with my system. In short, the methods you will find here are intended to unshackle mathematics from pen and paper. I am, of course, not the first person to do mental mathematics. There is much to be learned from Arthur Benjamin, Scott Flansburg, and Jakow Trachtenberg. I will visit them here as well. You should follow this blog if you are interested in fast mental math or homeschooling your children in mathematics.