I well remember my wonder that I experienced when I learned about logarithms in Advanced Algebra. The thing that most caught my attention was that you could "multiply two numbers" through a not-so- simple addition process. To do this, you first find the logarithm of each of the two numbers, add them, then reconvert that back to a number that would be the product required. Sounds complicated, but it works!

*(For those readers who have not reached this level of math yet, do not turn away from this page until you've read a little more.)*

In algebra, we write the basic logarithmic property this way:

Of course, this is just a "taste" of a much larger and grander topic in mathematics, but it is meant to give a connection to what I want to show you this time, namely how you can **"MULTPLY by ADDING"**, using a much simpler, arithmetic basis.

**Triangular Numbers**

Before we can proceed, we must introduce what to some persons may be a new topic, that of **Triangular Numbers**. Square numbers have been discussed frequently in this website, but now we are going to
look at a family of numbers that form triangles. Observe:

10 is a triangular number, because 10 things can be arranged in a triangular array like this:

* * * * * * * * * *From this sort of picture it is easy to form and determine many other triangular numbers.

* * * * * * * * * *Here we see that 1, 3, and 6 are the first three triangular numbers. And OF COURSE with these few examples we can see a short cut for finding other triangular numbers.

10 = 1 + 2 + 3 + 4 6 = 1 + 2 + 3 3 = 1 + 2 1 = 1So going in the higher direction, we have 15, 21, 28, 36, 45, and 55, thus giving us the first ten triangular numbers.

To make discussing a particular triangular number a bit more convenient, we will use the following notation [using the 4th one as an example]:

The First 50 Triangular Numbers |
||||

T_{1} = 1 | T_{11} = 66 | T_{21} = 231 | T_{31} = 496 | T_{41} = 861 |

T_{2} = 3 | T_{12} = 78 | T_{22} = 253 | T_{32} = 528 | T_{42} = 903 |

T_{3} = 6 | T_{13} = 91 | T_{23} = 276 | T_{33} = 561 | T_{43} = 946 |

T_{4} = 10 | T_{14} = 105 | T_{24} = 300 | T_{34} = 595 | T_{44} = 990 |

T_{5} = 15 | T_{15} = 120 | T_{25} = 325 | T_{35} = 630 | T_{45} = 1035 |

T_{6} = 21 | T_{16} = 136 | T_{26} = 351 | T_{36} = 666 | T_{46} = 1081 |

T_{7} = 28 | T_{17} = 153 | T_{27} = 378 | T_{37} = 703 | T_{47} = 1128 |

T_{8} = 36 | T_{18} = 171 | T_{28} = 406 | T_{38} = 741 | T_{48} = 1176 |

T_{9} = 45 | T_{19} = 190 | T_{29} = 435 | T_{39} = 780 | T_{49} = 1225 |

T_{10} = 55 | T_{20} = 210 | T_{30} = 465 | T_{40} = 820 | T_{50} = 1275 |

**Magical Multiplying Method**

Finally we're ready to show you the magical way to multiply without multiplying anything. I call it the "Magical Multiplying Method" (or the "3M" way). First, I must confess we will do a little subtracting, too. Second, you will need a copy of the table above. I hope you don't mind too much.

**Let's take as an example 15 × 9.
**

- Take the larger factor 15 and find T
_{15}in the table above. It is 120.

- Subtract 1 from 9, the smaller factor, getting 8. Find T
_{8}in the table. It is 36.

- Subtract the two factors, 15 - 9; that's 6. Find T
_{6}. It is 21.

- Add the results of Steps #1 & #2, then subtract the result from Step #3. That's your product!

[*Well, I never said it was going to be easier, shorter, or anything like that. It's just an interesting idea by itself, don't you agree? And with practice, it's not too difficult.*]

We can summarize this by a little algebra. This is the formula for the steps given above.

Doesn't look so bad now, or does it?

**Rationale**

The reasons why I am presenting this activity are various:

- It's a different way to do a common thing.
- It's a good experience in "following steps" to achieve a certain goal.
- It shows a connection between advanced math (logarithms) and simple arithmetic, both of which are being used to multiply two numbers.
- It provides an experience in looking up information in tables to use for a certain purpose.
- It show how an algebraic formula can show the various steps in compact style.

**Extension Activity**

For students who are, or have been, in Algebra I, it would be a good challenge to prove that the formula indeed works for ALL numbers **a** and **b**. To do so, however, one needs to know the general formula for the triangular numbers. I will not show how it is derived, rather just state it here.

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