My Apologies

Wow!  My apologies, I have not been on here in a long long time.  I wiped my hard drive after the summer, and forgot what my password to this site was until now.  After I lost the password I forgot about the site completely, until the other day when I was googling vedic math and this site popped up!

I’m not sure what I’m going to do with this site now.  I may redo the lessons to make them better, or who knows.  Any suggestions?

Back to School

Just wanted to let everybody know that I won’t be posting for about a week (till next Thursday or so). I’ll be packing up my car all day tomorrow, and heading up to college again on Sunday. Four day camping trip, so no internet access for a little bit!

If you have any questions about anything send me an email and I’ll get back to you when I get internet connection again.

Hope you all enjoyed your summer!

Two-Digit Multiplication

Lesson 1.5, Part 2

In this post, we’ll be using the “Vertically and Crosswise” method of multiplying. We’ll be dealing with two 2-digit numbers.

How to use Vertically and Crosswise
Steps correspond to the picture below. In steps 3 & 4, if you end up with a number that has more than one-digit, you write the last digit down and carry the other digits to the left.
1) Write the two numbers out, one above the other.
2) Multiply the left-hand column vertically.
3) Cross multiply and add.
4) Multiply the right-hand column vertically.

(CLICK IMAGE TO ENLARGE)

Practice Problems:

1.    15               2.    17               3.    27               4.    32               5.    45
      12                     15                     22                     21                     13
    ------                 ------                 ------                 ------                 ------


6.    48               7.    61               8.    79               9.    92               10.   99
      24                     53                     68                     91                     97
    ------                 ------                 ------                 ------                 ------

Answers (Click here)

Other Parts of Lesson 1.5:
Part 1 – Intro. to “Vertically and Crosswise”
Part 3 – Three-Digit Multiplication
Part 4 – Extending to any number of Digits
Part 5 – Algebraic Multiplication
Part 6 – Extra Problems

Intro. to “Vertically and Crosswise”

Lesson 1.5, Part 1

I have previously posted about this topic (here and here), however I believe those posts may have been too long for most people’s liking. So what I’ve done now, is split this Lesson up into 6 smaller parts each with more practice problems so you can make sure you understand the material.

Urdhva-tiryagbhyam sutra
This sutra, as the title of this post suggests, translates to “Vertically and Crosswise”. This sutra is one of the best known of the Vedic Sutras, and has found many applications. In fact, as I mentioned in my first post on this topic, there is a whole book dedicated to this sutra. I highly recommend you check it out!

So what will we be using this sutra for?
We’ll be using this sutra to multiply two numbers together. I’ll begin by demonstrating two-digit multiplication, then move on to three-digit multiplication. By the time you’ve seen three-digit multiplication, you should be able to recognize the pattern and from there be able to extend it to multiplying any number of digits including uneven digits (for example….two-digit by three-digit, etc.). And finally, I’ll show you the algebraic proof so that you can understand WHY this works.

Other Parts of Lesson 1.5:
Part 2 – Two-Digit Multiplication
Part 3 – Three-Digit Multiplication
Part 4 – Extending to any number of Digits
Part 5 – Algebraic Multiplication
Part 6 – Extra Problems

Advantages of Left to Right

Lesson 1.2, Part 1

In Vedic Mathematics calculations can be performed from either right-to-left (conventional Western method), or from left-to-right. Using the method of calculating from right-to-left may be fine for calculating answers on paper, however, it is not very good for computing mathematics mentally.

Why calculate from left-to-right?
The main reason to calculate from left-to-right is because it makes it a lot easier to remember numbers this way. When we say numbers, and when we write numbers down we naturally start at the left and work our way right. This allows us to say the answer as we are computing it mentally.

Not only that…

Another advantage of calculating from left to right is that we may only want the first one, two or three figures of an answer, but working from the right we must do the whole sum and get the most significant figure last.
– Using Vedic Math

The majority of calculations that I do will be done from the left to the right, if however I do it the other way I will let you know ahead of time so it doesn’t confuse you.

Casting Out the 9’s

Lesson 1.1, Part 4

In Part 2, we discussed how to compute Digit Sums. If you did the practice problems (I hope you all did!), you should see how straightforward the process is. This is good and all, but what about when you want to find the Digit Sum of a numbers such as say…97637254. This could end up being quite tedious (in case you were wondering the Digit Sum is 7), but in today’s lesson I’ll show you how you can compute it quite easily!

One of the last things we learned in Part 3 (The Nine-Point Circle) was that adding ‘9’ to a digit sum was equivalent to adding ‘0’ to the digit sum…neither ended up changing the final Digit Sum. For example, 13 and 139 have the same Digit Sum.

13 —-> (1 + 3) —-> 4
139 —-> (1 + 3 + 9) —-> 13 —-> (1 + 3) —-> 4

Ultimately what this means is that when computing Digit Sums, you can throw-out/ignore any 9’s you see. This also includes any numbers that add up to ‘9’. For example, in the number 154, you can ignore the ‘5’ & ‘4’ since “5 + 4 = 9”.

So 154 becomes 154 (which means the Digit Sum is ‘1’). If you don’t believe me…

154 —-> (1 + 5 + 4) —-> 10 —-> (1 + 0) —-> 1

Now look at that number I gave you in the beginning, the one that looked like it would be tedious to compute the Digit Sum….97637254. After cancelling out the ‘9’s this number becomes 97637254, or just ‘7’ for the Digit Sum.

This works because each time you add ‘9’ to the Digit Sum you are doing one full-revolution on the Nine-Point Circle, bringing you back to the same starting point.

Practice Problems
Compute the following Digit Sums:
1. 237
2. 72892
3. 871362
4. 37226
5. 9438541
6. 1438273
7. 847382
8. 13956402
9. 57604
10. 848834

ANSWERS

(Side Note: If you end up with ‘0’ as a Digit Sum, meaning everything cancelled….you can count the Digit Sum as either ‘0’ or ‘9’ since they both lie in the same spot on the Circle)

Other Parts of Lesson 1.1:
Part One – What are Digit Sums?
Part Two – Computing Digit Sums
Part Three – The Nine-Point Ciricle

The Nine-Point Circle

Lesson 1.1, Part 3

This post is based on Rebecca Newburn’s video, she explains it really well in the video…probably better than I do here, but I will attempt anyways!

What is the Nine-Point Circle?
The Nine-Point Circle is a circle that has been split into 9 parts, as seen in Figure 1.a, and numbered from 1 to 9. This object has some special properties that are related to Digit Sums which you shall see.

Where would you place ‘0’ on this Circle?
Well if you think about it, in order to find out where zero goes you should count backwards until you get to the place where zero should be. In this diagram, we can see that the zero should be where the nine is, so we place it next to the number 9 as in Figure 1.b.

What does this mean? What is the importance?
Before I answer these questions, how about we continue filling in the Circle. So let’s continue counting upwards from 9 and placing the numbers in their appropriate spots, which can be seen in Figure 2 below.

In Figure 2, you can see that I continued counting all the way up to 28…really you can keep counting up to infinity, but 28 is enough for you to get the idea. Now look at the boxed section in Fig. 2, do you notice anything? The Digit Sum of each of those numbers is the same…3!

A big part of Math is noticing patterns like this and understanding what they mean. If you go around the circle and Compute the Digit Sums for each group, you’ll see that all the numbers in the group have the same Digit Sums (excluding 9/0 for now). What’s another thing you notice about the numbers in each of the groups, such as the boxed one above? Each successive number is 9 more than the one before it. For example, starting with 3…3+9 = 12….12 + 9 = 21, etc. This should be fairly obvious since we’re counting around a NINE-point circle, so naturally each number would be 9 above…but this has some important applications which I will talk about in Part Four.

So what about 9 & 0? Surely they aren’t the same number?
No of course they’re not the same number! But in a Digit Sum they act the same. As we saw above, adding 9 to a number didn’t change it’s digit sum…so in a way we could of said we added ‘0’. For example:

3 + 9 = 12, then compute this digit sum…..12 —-> (1 + 2) —-> 3. So we started and ended with ‘3’. This is the same as:

3 + 0 = 3.

Why this is important will become apparent in Part Four!

Other Parts of Lesson 1.1:
Part One – What are Digit Sums?
Part Two – Computing Digit Sums
Part Four – Casting Out the 9’s

Computing Digit Sums

Lesson 1.1, Part 2

In Part One I talked about what Digit Sums were, and why they are useful. Now I’ll show you some examples where I compute Digit Sums.

Example: 27
27 —-> (2 + 7) —-> 9
Therefore the Digit Sum of 27 is 9.

Example: 435
435 —-> (4 + 3 + 5) —-> 12

Since ’12’ contains two digits, and were trying to get down to one…we add them again.
12 —-> (1 + 2) —-> 3. Therefore the Digit Sum of 435 is 3.

Note: Notice that I didn’t use an ‘=’ sign because 435 does not equal 3, only the Digit Sum does.

Example: 102372
102372 —-> (1 + 0 + 2 + 3 + 7 + 2) —-> 15
15 —-> (1 + 5) —-> 6.
Therefore the Digit Sum of 102372 is 6.

Try to solve the problems below on your own first, whether mentally or by writing it down, before clicking on the link to the solutions.

PRACTICE PROBLEMS
1. 29
2. 32
3. 57
4. 354
5. 271
6. 10253
7. 27361
8. 56381
9. 1029301
10. 1425361

Answers to Practice Problems.

Other Parts of Lesson 1.1:
Part One – What are Digit Sums?
Part Three – The Nine-Point Circle
Part Four – Casting Out the 9’s

What are Digit Sums?

Lesson 1.1, Part 1

So what exactly are Digit Sums?
The Digit Sum of a number is the number you get when you add all the digits together, then add those digits together all the way until you have a single digit left between 1 and 9. You may be wondering about ‘0’, it has some special properties which we will talk about in a future lesson. As you will come to see, there are patterns involving digit sums and shortcuts to make computing Digit Sums easier.

Why are they useful?
Digit Sums can find use in many things, a couple examples being: checking your answers, checking divisibility, and for dividing by nine.

How do you get them?
In order to get a Digit Sum all you have to do is add the digits. If the number you get is a single digit then your done, if not then you add them again until your left with a single digit. Take the number 26 for example; the Digit Sum would be 8 because 2 + 6 = 8.

Other Parts of Lesson 1.1
Part Two – Computing Digit Sums
Part Three – The Nine-Point Ciricle
Part Four – Casting Out the 9’s

New Page Added

Just wanted to let you guys know that I added a Vedic Math page to the blog. You can access it either at the top of the blog, or from the side-bar on the right.

At the moment some of the pages are password-protected and/or hidden. This is because they are not finished yet, once they’re done I’ll open them up for everyone.

Hope you enjoy it! Let me know if you catch any errors, or have any suggestions.

UPDATE: Just opened up the “Table of Contents” & “Section 1” pages. Right now none of the links in Section 1 work because I haven’t written up the lessons yet. However, looking at it will give you a rough outline of whats to come. Yes the topics look basic, but you have to build a foundation first before you start doing anything else otherwise you will just get lost.