MASTERING FRACTIONS

MASTERING FRACTIONS

FRACTIONS are NUMBERS. Fractions are useful when dealing with numbers that are not WHOLE NUMBERS. Whole numbers are all the COUNTING NUMBERS, namely, 0, 1, 2, 3, etc., but excludes INFINITY.  Infinity is not a counting number because you cannot count Infinity.

Example of numbers that are not whole numbers are one half, a quarter, and a hundred and one tenth. You express these three numbers using ARABIC NUMBERS these ways: 1/2, 1/4, and 100 1/10, respectively.  These three numbers are called fractions because they are not whole numbers.

What is the meaning of 1/2?  ONE HALF means that instead of one whole number, there is only a half of that one whole number. In other words, you divided that one whole number, the number 1, by 2, that is, 1 divided by 2, or 1/2.  Remember that the SLASH SIGN, /, means "DIVIDED BY".

What is the meaning of 1/4?  A QUARTER means there is only a quarter of that one whole number. That means you are dividing that one whole number, 1, by 4, that is 1 divided by 4, or 1/4.

What is the meaning of 100 1/10?  A HUNDRED and a one TENTH means that you have 100 but you also have the fractional number 1/10.  One tenth is one whole number, 1, divided by 10, or 1/10. So writing down the whole number together with its fractional number, you write the whole number first followed by its fractional number, 100 1/10, in our example.

MATH EXERCISES:
Write the following alphabetically written numbers into Arabic numbers:
1. One third
2. One fifth
3. One hundredth
4. One and one tenth
5. Two hundred and three hundredth

Happy math hunting!

John

MASTERING MULTIPLICATION

MASTERING MULTIPLICATION TABLE

Any student or pupil of Arithmetic or Math should be able to remember the Multiplication Table by memory. You don't have to remember every product of the factors from 1 to 12, just as long as you know the product of the number or numbers before the factor that you need to multiply.

For example, say you need to multiply 8 x 7, and you don't know or have forgotten its product.  Here is what you can do to multiply them.

You should at least know all the products when a number is multiplied by itself, by memory or by heart, at least between 1 to 12.

For example, 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16, 5 x 5 = 25, 6 x 6 = 36, 7 x 7 = 49, 8 x 8 = 64, 9 x 9 = 81, 10 x 10 = 100, 11 x 11 = 121, 12 x 12 = 144.

So, since you know 7 x 7 = 49, all you have to do is add 49 + 7 = 56.  So 8 x 7 = 56.

All you have to do is to know the product of the least of the two numbers, and just add the same number once, or twice, if necessary.

For example, say you need to multiply 9 x 7.  You know 7 x 7 = 49, and you need two more 7's, so you just add 49 + 14 = 63 to get 9 x 7.

Remember, the key is to learn the products of the numbers from 1 to 12 when it is multiplied by itself, then just add the same number as many times to get the right result.

Learn math by applying the math you know!

John

MATH IS HARD?

MATH IS HARD?

Why do some students find math is hard?  Because knowing math means knowing a lot of stuff.

First of all, there are many numbers in math that you need to know by heart. There's 0, there's 1, there's 2, there's 3, and on and on and on till you learn that there's also the number 1,000,000,000.  So now you have a trillion things to know.  That is you have a trillion and one numbers to know: all the numbers from 0 to 1,000,000,000.  

Luckily, mathematicians have made it easier for students to know all these numbers without having to learn each one of them or even to say or write each one of them.  What is this thing that makes it easy to know all these 1,000,000,001 numbers?  The key to knowing all of them is that each number following another is only 1 more than the previous number.  For example:

1 = 0 + 1.

2 = 1 + 1.

3 = 2 + 1.

4 = 3 + 1.

5 = 4 + 1.

Do you see the pattern?  Each succeeding number is 1 more than the previous number.  So that means that 1,000,000,000 is this:

1,000,000,000 = 999,999,999 + 1.

So knowing only one thing makes you learn a trillion things.  And you can learn these trillion things in only 10 minutes, 1 hour, or 10 hours depending on how much fun you are having in learning a trillion things.

What about the rest of math?  Math is all about patterns.  Knowing one pattern can make you learn 10, 100, 1000, 10000 or a trillion things all at once, I mean, 10 minutes, 1 hour, or 10 hours.

Have fun discovering and exploring mathematics!

John

The Basic Operations On Numbers

THE BASIC OPERATIONS ON NUMBERS

There are two (2) basic operations on numbers.  They are ADDITION (+) and SUBTRACTION (-).  All other operations on numbers are different versions of addition or subtraction.

Also, if one is to do a deep analysis on numbers, à la NUMBER THEORY, there is only one operation on numbers -- addition.  Because, one can always add NEGATIVE NUMBERS and get the same results in addition as one would in subtraction.  But for the sake of clarity and brevity, we will say there are two basic operations on numbers.

The following exploration discusses the variations of HOW TO ADD NUMBERS.

HOW TO ADD WHOLE NUMBERS

Whole numbers starts with 0 followed by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, et cetera, excluding INFINITY.

Given: 1 + 2

The SYMBOL + means to ADD THE NUMBERS.

So we have 1 + 2 = 3.

The symbol = means IS EXACTLY EQUAL TO.

Summary:  When adding whole numbers, you just add the QUANTITIES the numbers represent and combine them into one number.

HOW TO ADD INTEGERS
Integers starts with the lowest integer next to the lowest infinity followed sequentially by the next higher integer, and so forth, with the middle integer being zero (0), and continuing on up to the highest integer next to the highest infinity.

Given: 1 + (-2)

The symbol “(“, OPEN PARENTHESIS, and “)”, CLOSED PARENTHESIS, means the enclosed number or EXPRESSION should be solved or evaluated first.

We will solve the GIVEN step by step.

Step 1: Write given.

1 + (-2) =

Step 2: Rewrite the given so that it is easily solvable or readable.  We use the COMMUTATIVE PROPERTY OR RULE OF ADDITION.

1 + (-2) = (-2) + 1

Step 3: Solve the number or expression inside the parentheses.

(-2) + 1 = -2 + 1

Step 4: Solve the easily solvable, readable, or simplified expression, or simplify it further.  We will simplify this expression further for those who wanted to simplify it further.  We will use the DISTRIBUTIVE PROPERTY OF MULTIPLICATION on -2, negative two.

-2 + 1 = (-1) (1 + 1) + 1

Step 5: Simplify the whole expression.  We continue using the distributive property of multiplication.

(-1) (1 + 1) + 1 = (-1) + (-1) + 1

Step 6: Simplify the whole expression.  Using the INVERSE PROPERTY OF ADDITION, we add up numbers that add up to zeroes (0).

(-1) + (-1) + 1 = (-1) + 0

Step 7: Simplify the whole expression. We will use the IDENTITY PROPERTY OF ADDITION.

(-1) + 0 = (-1)

Step 8: Simplify the expression and write the final answer.  We solve the expression inside the parentheses simply by releasing the parentheses because the expression is already in its most simplified form.

(-1) = -1

Given: 1 + (-2)

Answer: -1

Summary: When adding integers, knowledge and understanding of the commutative, ASSOCIATIVE, distributive, inverse, and identity properties of addition and multiplication will make you truly understand the dynamics and complexities involved in the simple process of adding integers.

Have fun learning!

John

BRANCHES OF MATHEMATICS

BRANCHES OF MATHEMATICS

Mathematics is the oldest surviving and living language in the world.  It is no wonder it has matured to many different branches of study.  Here are some of the major branches and their general topics of discussion.

MATHEMATICS = study of quantities using numbers.
   ALGEBRA = study of numbers and variable numbers.
      ARITHMETIC = study of numbers.
         INTEGERS = study of the whole numbers in a straight line.
         FRACTIONS = study of whole numbers and numbers that are not whole.
         DECIMALS = study of fractions using the decimal point.
         PERCENTAGE = study of fractions using the denominator 100.
         EXPONENTS = study of numbers multiplied by itself a number of times.
         ROOTS = study of numbers divided by itself a number of times.
      DETERMINANTS = study of matrix and matrices.
      GEOMETRY = study of numbers applied to any objects of any dimension.
         PLANE GEOMETRY = study of numbers applied to two-dimensional objects.
         EUCLIDEAN GEOMETRY = study of three-dimensional space without motion.
         TOPOLOGY = study of three-dimensional space after transformation.
      TRIGONOMETRY = study of triangles.
      PROBABILITY = study of likelihood of future events.
      STATISTICS = study of frequency of past events.
   CALCULUS = study of variables in a function.
      DIFFERENTIAL CALCULUS = study of the curve or line formed by a function.
      INTEGRAL CALCULUS = study of the area formed by a function.
   DIFFERENTIAL EQUATIONS = study of an equation's function and its derivatives.
   DISCRETE MATHEMATICS = study of discrete numbers.
   NUMERICAL ANALYSIS = study of the approximation of continuous numbers.
   REAL ANALYSIS = study of functions with real variables.
   COMPLEX ANALYSIS = study of functions with complex variables.
   VECTOR ANALYSIS = study of numbers in motion.
   TENSOR ANALYSIS = study of vectors applied over multiple coordinate systems.
   LINEAR ALGEBRA = study of systems of linear equations in motion.
   ABSTRACT ALGEBRA = study of theoretical numbers using algebra.
      NUMBER THEORY = study of the properties of integers.
      GROUP THEORY = study of numbers and their set of operations.
      GAME THEORY = study of games.
      RING THEORY = study of numbers and their behavior inside ring objects.
      FUZZY SET THEORY = study of the generalizations of uncertain sets.

The study of Mathematics is still evolving, much like the study of Physics is gravitating towards a Unified Field Theory.  There will be more branches to discover as long as humanity has the penchant toward abstraction.

Happy computing!

John

NUMBERS EXPLORATION

Given that you are a six year old child or at least you were a six year old child.  And you are intrigued by all these numbers all around your world, but all you know is the letters of the alphabet and how they make up words and sentences to talk to your mom or pop or sibling or friend.  How would you go about learning about these intriguing numbers and enter their universe?

The answer is the same way that you learned your alphabet and all the words that it makes.  Through familiarity, through hanging around with them, through making them a part of your everyday life.

So if you're a six year old child reading this and would like to explore the mystery of numbers, you should ask your mom or pop to buy you a book of numbers.  Then you can show your new book of numbers to all your playmates and relations.

Here's some numbers to get you started.

Arabic     English     German     Spanish     Roman     Stick     Binary
1              one            eins           uno             I              |             01
2              two            zwei          dos             II            ||            10
3              three          drei            tres            III          |||           11

Have fun with numbers!

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