# Napier's Rods

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John Napier invented his abacus rods in the early 17th century. Among other things, Napier's Rods can be used for multiplying two factors, and is an ingenious way to multiply multi-digit numbers.

## Step 1: Materials

You'll need about a dozen craft sticks, a ruler, a pencil, and some other markers if you want to gussy up your sticks.

## Step 2: Make Your Mark

Start by measuring and marking half-inch increments on the craft sticks. I keep extra space at the top. First, make a guide rod, which makes the first column, and is composed of digits one through nine.

## Step 3: More Lines

For all the digits zero through nine, make diagonal lines on the sticks from the top right to the bottom left.

## Step 4: Complete Your Set

You will then construct a times table for each digit. Multiply the stick's number -- the number at the top -- by the number on the guide stick to the left.

You can make as many sticks as you want, especially if you want to multiply pi to 12 places. If you do have duplicate digits in a factor, make sure you have enough rods. You can always use the back of a stick.

## Step 5: How to Use Them

Napier's rods were a refinement of the Arab lattice method imported to Europe by Fibonacci. Add the place values in each parallelogram, carrying when necessary, and you have a quick way of multiplying multi-digit by single digit numbers.

Example: 9 x 314 = 2 1000s + 8 100s + 2 tens + 6 ones.

## Step 6: Further Exploration

How is the lattice method different from standard multiplication? Is it just a different way to project the carried digit? Can you make binary base rods? Joshf made four-sided Napier's Bones. Can you make these with chicken bones? You could print the rods and paste them on to foam core. Here's a set of rods my wife made on a laser cutter -- for my birthday!

## 21 Discussions

5 years ago on Introduction

Both this and the linked Napier's Bones show an example of a multiple digit number multiplied by a single digit number but not a mutiple digit number multiplied by another multiple digit number. How would one multiply 314 by 97 for example?

Reply 5 years ago on Introduction

you can do it but you have to add the two numbers together

first 9*314=2826 insert a zero behind because you actually want to multiply with 90 then you have 28260 then you do the 7 7*314=2198 then you add your two results 28260+2198=30458

Reply 5 years ago on Introduction

Thanks for the reply. That's pretty much how I expected it would be done. I was hoping there was some "trick" inherent in the rods to accomplish this.

Unless for amusement's sake I guess I won't be making one of these anytime soon as I can already do the single digit multiplication easily in my head.

Reply 5 years ago on Introduction

I have no method other than taking out paper and pencil. Would be happy to know of others' methods.

5 years ago on Step 6

He must have been an interesting Geek.?

Reply 5 years ago on Introduction

Napier had his own castle, wrote apocalyptic literature, and worked with Henry Briggs to create the first slide rule. Yeah, I guess you could say he was a geek, or a nerd, depending on your definitions.

Reply 5 years ago on Introduction

Thank You. Good history to check out.

Reply 5 years ago on Introduction

He looks like an interesting guy

Wikipedia: "In addition to his mathematical and religious interests, Napier was often perceived as a magician, and is thought to have dabbled in alchemy and necromancy. It was said that he would travel about with a black spider in a small box, and that his black rooster was his familiar spirit."

http://en.wikipedia.org/wiki/John_Napier

http://en.wikipedia.org/wiki/Napier%27s_bones

5 years ago on Introduction

I'm trying to upload a pdf with a template for making Napier's rods. If I succeed you'll see the files below.

5 years ago

Fantastic! Many thanks for this great instructable.

5 years ago on Introduction

Very good teaching aids. If you don't want to buy a big pack of these sticks in a craft shop you can buy them singly in any pharmacy, just ask for "Tongue Depressors".

5 years ago on Introduction

In Step 4, I believe you forgot to tell how to make the other sticks. It is easy to see by looking at the picture, for each rectangle, multiply the number at the top of the stick by the rectangle's position in the list and write the result with one digit above the diagonal and one below. Thanks for sharing this, it is way cool!!!!!

Reply 5 years ago on Introduction

Thank you. I believe I'll edit that step based on your suggestions. And, yes, I do believe they would make great gifts.

Reply 5 years ago on Introduction

I forgot to mention that I think these would be a great gift for an elementary or middle school math teacher. They've probably never heard of these.

5 years ago on Step 6

This is a great invention that deserves to be better known! Thanks! Wonderful idea. Oh--it's easier to do the markings if you tack up a jig out of a base of spare plywood and a few extra sticks, to hold the bones -- we called these "Napier's bones" when I was a kid -- stable while you mark them.

Now show us how to do the cool lasered ones! Thanks again.

Reply 5 years ago on Introduction

I have tried, and failed, to make a jig out of craft sticks. If you can, please post examples or designs.

Drawing the diagonals freehand is easy, but I'm sure making the other lines could be speeded up. I could also see multiple sets being made by silk screening, printing, or stenciling.

Liz said that she'd be happy to make an Instructable, but not this week as she's preparing a TedX.

Reply 5 years ago on Introduction

These are indeed Napier's Rods. Napier's bones are three dimensional, square in cross section, with four different rods engraved on each one. These are related to the slide rule.

Regardless, what you call them, a nice instructable!

5 years ago on Introduction

I still use this method now for doing sums, i just find it easier... brilliant math based instructable :)

5 years ago

If I still taught, I'd use it in the class! This is fun.

5 years ago on Introduction

I love this!

I'd be nothing if it wasn´t for multiplying tricks: www.instructables.com/id/Tables-of-6-7-8-and-9-in-your-hands/