Thursday, April 28, 2011

Conversion exercises in binary and decimals. Part 1: Whole numbers.

The decimal numeral system, the numbers that start from 0 and end at 9, are of everyday use in our lives. Sometimes we tend to think that all other systems play a little roles because we do not use them or see them in action.

Well, the binary numeral system might appear to be of more importance than the decimal system; especially if you are talking to someone who is conversant with electronics, computer languages and logic. They can also be of importance to you. Unknown to me, until I read a wikipedia article on
binary numbers
, the Ifa priest makes use of this system in divination. I do not recommend though that you learn geomancy or divination because you want to understand the usefulness of maths.

I took the above picture because the positions of the birds is a good example I can use for the positions of the 0s and 1s in a binary system. If you represent a bird sitting on the power line with a 1 and an empty space a 0, the picture will produce the binary number: 10110000100110100110 base 2. Alternatively, if I represent each bird to be a ten in the decimal system, I will oblige your indulgence and make it number 90 base 10. So, for the exercises, we'll try converting the 90 base 10 to base 2 and then 10110000100110100110 base 2 to base 10.

From base 10 to base 2 (or from decimal to binary numeral systems):

A simple system of converting from decimals to binary is to divide the decimal number by 2 and if there is a remainder, place the remainder, which will be a 1, at the rightmost bit position but if there is no remainder, insert a zero. Then divide the resulting decimal number again and continue placing bit positions based on whether or not there are remainders until we arrive at decimal number 0. Then write down the 1 and 0s on the remainder column, starting from the bottom of the column such that the bottom is the leftmost bit position in binary numeral system. Your work is finished.

Let's show this with a table using the birds sitting on the powerline, or decimal number 90.

Converting 90 base 10 base 2.
Result : Decimal number % 2 Remainder (leftmost bit position)
90 / 2 = 45 0
45 / 2 = 22 1

Note our technique. Taking 90 and divide it by 2, we get 45 with a remainder of 0. So we insert 0 in the remainder position. Because the result is not zero, we continue the division. The second division, i.e 45 divided by 2 gives 22 remainder 1.

22 / 2 = 11 0
11 / 2 = 5 1
5 / 2 = 2 1
2 / 2 = 1 0
1 / 2 = 0 1
1011010 base 2

This dividing and inserting remainders continues to the seventh row when the result becomes 0. When you write down the converted number in binary, from the bottom, taking 1 as the left most bit, then the next is 0 and then 1 and until we get to the last remainder, 0, we get 90 base 10 converted to 1011010 base 2.

From base 2 to base 10 (or from binary to decimal numeral system):

This time, we'll convert the bird positions in binary,10110000100110100110 base 2 to base 10. How many positions are there? Let us count: the leftmost is a 1, then a 0 and counting from there we have 20 bit positions.

There is a system of binary to decimal conversions that uses bit positions and successive additions of their face values and place values to get to the result. The face value is the literal value for a bit position and the place value indicates the bit position, counting from the right starting from 0, to which we raise the number 2. 101 base 2, for simplicity has three (3) bit positions, with the rightmost bit 1, having a face value of 1 and a place value of 20. The next bit position has a face value of 0 and a place value of 21 and the last a face value of 1 with place value of 22. This simple system of conversions adds up the face and place values thus: 1 x 22 + 0 x 21 + 1 x 20 = 4 + 0 + 1 = 5 base 10.

If we apply this technique to a binary number with twenty (20) bit positions, how enormous the multiplications and additions that have to be done, and furthermore, mistakes will surely creep in.

There is a simpler technique, and it is very elegant. In fact, this technique, which I found while reading the wikipedia article on
binary numeral system
was what prompted this blog. I just was impelled to share it. It is called the Horner scheme.

These are the steps to apply the scheme:

  1. Create a prior value and give it a value of 0.
  2. Multiply the prior value by 2.
  3. Take the leftmost bit in the binary number and make it the Next Available Bit. Add the result of (2) to the Nest Available Bit.
  4. Let the result of (3) be the value of a box you will call, the Next Value.
  5. Now, assign the Next Value to the prior value, or make the Next Value be the value of the prior value.
  6. Provided there are still some bits that have not been manipulated in the binary number, or the next bit still have some candidates from the binary number, go back to (2).
  7. If the binary number bit positions is exhausted, then the result of the conversion is the value of the Next Value box.

We can illustrate the Horner scheme with a little table using the 20 bit positions of the birds sitting on the powerline representation.

Illustration: Horner scheme for 10110000100110100110 base 2.
Prior Value x 2 + Next Bit Next Value
0 0 x 2 = 0 0 + 1 (the leftmost bit) = 1 1
1 (the previous Next Value) 1 x 2 = 2 2 + 0 = 2 2
2 2 x 2 = 4 4 + 1 = 5 5

We started with a prior value of 0, multiplied it by 2, got 0, then took the leftmost bit as the Next Available Bit, which is a 1 and added it to the earlier result to get a 1 which was placed as the Next Value and then transferred to the prior value. Since there were still bit positions remaining, we continued by multiplying the prior value by 2, adding it to the next bit which is a 0 to get a 2 that was placed into the Next Value box. Continuing this algorithm for the third row resulted in a next value of 5.

Horner scheme for 10110000100110100110 base 2 continued
5 5 x 2 = 10 10 + 1 = 11 11
11 11 x 2 = 22 22 + 0 = 22 22
22 22 x 2 = 44 44 + 0 = 44 44
44 44 x 2 = 88 88 + 0 = 88 88
88 88 x 2 = 176 176 + 0 = 176 176
176 176 x 2 = 352 352 + 1 = 353 353
353 353 x 2 = 706 706 + 0 = 706 706
706 706 x 2 = 1412 1412 + 0 = 1412 1412
1412 1412 x 2 = 2824 2824 + 1 = 2825 2825
2825 2825 x 2 = 5650 5650 + 1 = 5651 5651
5651 5651 x 2 = 11302 11302 + 0 = 11302 11302

So far, we have calculated for fourteen (14) bit positions starting from the leftmost bit. I am sure you have realized how elegant Horner's scheme is compared to the earlier one that depends on face and place values? Especially if your bit positions are in the high numbers. Man, am I breaking out in sweat? Yet, the job has to be completed.

Horner scheme for 10110000100110100110 base 2 continued still.
11302 11302 x 2 = 22604 22604 + 1 = 22605 22605
22605 22605 x 2 = 45210 45210 + 0 = 45210 45210
45210 45210 x 2 = 90420 90420 + 0 = 90420 90420
90420 90420 x 2 = 180840 180840 + 1 = 180841 180841
180841 180841 x 2 = 361682 361682 + 1 = 361683 361683
361683 361683 x 2 = 723366 723366 + 0 = 723366 723366

So, finally, when there are no bit positions remaining, we arrive at the value of 723366 base 10.

Try out some exercises at home. You'll soon using this scheme yourself.

I will like to end this blog by noting that the Horner scheme is the conversion from base 10 to base 2 using division by 2 and noting remainders in reverse.

I leave this as an exercise for you to compare both methods. This underscores a trait in mathematicians that I admire very much: they have an open eye for patterns, even the queer variety. They have a knack for knowing what works most of the time if it has worked once in their life. They have an amazing memory recall system. I know it because I have experienced it myself.

Surprisingly, ancient India used binary numbers and applied binary to decimal conversions prior to 2nd century A.D, more than 1500 years before their discovery in the west.

My next post will cover fractional conversions, conversion exercise in decimal and binary systems, part 2 : fractions.

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Tuesday, April 5, 2011


If you are asked to mention one element in nature that is ambidextrous, I am sure that water will be one of them. The late Fela Anikulapo-Kuti sang about the varied uses that water can be made us of in one of his songs.

water is used in manufacturing various liquid beverages

Elements aside, one subject in life that is as versatile and can serve a trillion uses to human needs, that is as old as man itself, remember my post on counting?, is mathematics. One branch of mathematics that is prime to modern civilization, without which man would never have dreamt big dreams which have revolutionized science and technology, is calculus.

The count of the modern uses of calculus is endless, but I will attempt to mention some uses that will cover every faculty that man has endeavored to spend money and effort in his research endeavors. According to a professional resource on calculus,Calculus and Analytic Geometry by Thomas and Finney, Economists use calculus to forecast global trends, oceanographers use calculus to formulate theories about ocean currents, meteorologists use it to describe the flow of air in the upper atmosphere, biologists use calculus to forecast population size, medical researchers use calculus to design ultrasound and x-ray equipment for scanning the internal organs of the body, psychologists use calculus to understand optical illusions in visual perceptions and finally, drug companies use calculus to determine profitable inventory levels. The list can go on and on; I just have to stop here.

But whether calculus can help you to lose weight, or survive a zombie apocalypse, that is the subject of a book by Jennifer Ouellette, The Calculus Diaries: How math can help you lose weight, Win in Vegas, and survive a Zombie Apocalypse. You can get it from the bookstore online, or even Barnes and Noble.

The varied applications of calculus aside, the history of Calculus is laced with colorful characters lilke Isaac Newton, Baron Leibniz, René Descartes and Pierre de Fermat. The book also presents in vivid details how calculus can fit into your daily life; how you can use calculus to even calculate the stats on your blogger and Facebook Insights dashboard.

According to the reviews posted on the page for the book, I read that Jennifer Ouellette writes with a touch of humor, beauty and elegance. That her stint as a science writer and in physics gave her much material and experience to brace this daunting subject. She covers the basic principles very well. So if you are a student preparing for undergraduate degrees and you think an understanding of calculus is essential to your success, then you should lay hold of this book.

But the subject, calculus, has already defined the readership for the book, right? Don’t be in much of a hurry. Readers who are mathphobic, or afraid of mathematics, can even enjoy reading it. It doesn’t come with much mathematical formulas that could be daunting, one of the negative sides according to another reviewer but the writer did her majors in English and understands what mathphobic concerns her readers might and would surely face.

Why not get a hold of this book. I’d recommend it myself. I followed a link to her website and just subscribed to her blog feed.

So, for those who want to brave the world of calculus, I’d recommend the book:The Calculus Diaries: How math can help you lose weight, Win in Vegas, and survive a Zombie Apocalypse

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