awmathematics

[awmathematics]

Lesson 3. Fractions

The fractured number, used by Babylonian cultures as far back as 1800 BC, and appears in the Egyptian Rhind Papyrus, in hieroglyphic form from around 1650 BC, written as eyes with tears. The Egyptian system was very complicated and still isn't fully understood today, and they kept their system secret for the 'privileged' class. One thing we did learn, was to reduce fractions to the lowest common denominator. In classic fraction sense, the lower number (denominator) acts as the designated number of units or fractions that equal the whole. The upper number (numerator) represents how many units are actually present. 1/27, therefore, means if you broke a number into 27 parts, you'd have one of those parts. If you have more units than make a whole, like 14/5, it is called an improper fraction. Dividing 14 by 5 gets you 2 wholes with 4 units left over, expressed as 2 4/5.

The interesting thing about fractions is that they're easier to multiply than they are to add. To add 1/2 + 1/4, you have to convert them to the same denominator first. So you multiply the denominator by whatever variable converts it to the right quantity. In this case the denominators are 2 and 4, so you will convert 2 to 4. 2y = 4 y = 4/2 y = 2. Then you multiply the same variable to the numerator, so 1/2 becomes 1y/2y = 2/4. Now you can add the fractions 2/4 + 1/4 = 3/4. In cases where you cannot simply multiply one fraction by a number, you can multiply both. So if you had 1/2 + 1/5, you'd multiply them by whatever numbers give them the same denominator. Ie. 1/2 x 5 = 5/10. 1/5 x 2 = 2/10. Now you can add 5/10 + 2/10 = 7/10. This is the lowest common denominator. If you had chosen another common denominator (say 20), you'd have 1/2 x 10 = 10/20, 1/5 x 4 = 4/20. 10/20 + 4/20 = 14/20. Because both of these numbers can be reduced (divide both by 2), you get 7/10 as the lowest common denominator because 7 cannot be broke down evenly.

To multiply however, you simply multiply the numerators with each other and the denominators with each other. 1/2 x 1/4 = 1/8.

Converting fractions to decimals gets dicey enough when you start trying to divide numbers like 1/3, which will give you an infinite 0.3333..., so we write it 0.3 with a line over the 3 to represent a recurring decimal. What are some examples of you having used fractions without knowing it? What are some very practical applications to use fractions in? Pie charts are the first example, to indicate portions of data, or public opinion, etc. Can you think of some others?

[awmathematics]

Number Systems

Using Hindu-Arabic numerals, 1....9, and 0, and the decimal number system, we have 3444 + 394 = 3838. 3838 is 3000+800+30+8.

Using Roman numerals and a non-decimal system, we have MMMCDXLIIII+ CCCXCIIII = MMMDCCCXXXVIII. This MMM(3000) + DCCC(800) + XXX(30) + VIII(8).

That's basic arithmetic so imagine multiplication and division... Needless to say, the Hindu-Arabic system (it's called Arabic because the Arabs, who got it from the Hindus, passed it to the Europeans) of a cycling 1 thru 9, combined with the decimal system of keeping the places with 0's rather than creating a new numeral for each exponent of 10.

This system works wonderfully for all our practical purposes. But even it falls short when getting to extremely large or small numbers, in which case we shorthand the zeros in what we call scientific notation. 1,356,936,892 would be 1.35 x 109. (10 to the 9th power).

As we explore the universe and the quantum-verse the numbers we need to use become bigger and smaller respectively. How massive and how miniscule the numbers will continue to become is a fascinating mystery.

Try solving these problems without converting to Arabic numerals... XIII + VII = ? MMDXXI - DXX = ? VIII x XX = ?

[awmathematics]

Greeks and Romans, despite their advancements, started counting from 1 - and could not account for, or factor in, zero. Mayans and Babylonians, however, did. It wasn't until Brahmagupta in India finally put it into our mathematical system that the world began using it. 1 has no meaning without 0. The question has become, how do we use this concept in certain areas, like division? 0/7 = b. If it were 28/7 = 4, you can cross-multiply 4 (the whole units) by the denominator (units that make a whole), 7 and get the numerator (actual units present) 28. 28, thus, is shown here, 28 = 4 x 7. But using this for the first formula (0/7 = b), we get 0 = 7 x b. Therefore b = 0, and the solution of zero divided by anything is zero. This should be obvious if you think of the numerator as actual units. The real problem, however, is in reversing the numbers. If you take 7/0 = b, using cross multiplication, you get b x 0 = 7. That defies the rules of mathematics because we know anything times zero is zero. What do we do with it then? Currently they call 7/0 (or any numerator with the denominator zero), 'undefined' and disallow its use. This is not far off, as if you think about it, the smaller denominator (units that make a whole) the larger the answer. ie. 7/.1 = 70, while 7/.01 = 700. Getting smaller and smaller, the answer getting larger and larger, you approach nothingness on one end, and infinity at the other. How small it gets before zero is like asking how large it gets before it becomes infinity. Wrap your heads around that. Thoughts? Can you think of any real life applications where this would be a problem?