Category Archives: Mathematics

Additional Math Concepts

November 24, 2011 – 11:50 AM

These basics are not presented in as rigorous a manner as the axioms, but they serve to clarify and build upon the axioms.

 (1) A unit is anything that can be grasped as an entity.

In mathematics, the most fundamental idea is that of a unit. A unit is commonly represented by the number “1” (one).

(2) A unit is counted one at a time.

Counting provides natural numbers that are commonly represented by the numbers, 1, 2, 3, 4, 5, and so on.

(3) All numbers are referenced from 0 (zero).

Any quantity present is relative to no quantity. Therefore, all numbers are referenced from the idea of “no units.” This concept is commonly represented by the numeral “0” (zero).

(4)  Therefore, the number n is fully defined by “0 + n”.

Therefore, the number 1 is fully defined by “0+1”; the number 2 is fully defined by “0+2”; the number 3 is fully defined by “0+3”, and so on.

(5) The number “0 + n” is abbreviated as n, or as the positive integer +n.

The number “0+1” is abbreviated as +1; the number “0+2” is abbreviated as +2; the number “0+3” is abbreviated as +3, and so on.  The numbers +1, +2, +3, +4, +5, etc. are called positive integers.

(6) If a number is n, then the next number is “n + 1”.

The next number is obtained by counting one more. This gives us the basic function of adding. Addition is represented by the sign “+”. Thus, the next number after 1 is “1+1” written as 2; the next number is “2+1” written as 3; the next number is “3+1” written as 4, and so on. One may keep on counting forward without limit.

 (7) If a number is n, then the previous number is “n – 1”. 

The previous number is obtained by counting one less. This function of taking away (subtracting) is the opposite of adding. Subtraction is represented by the sign “–”. Thus, the number previous to 3 is “3–1” or 2; the number previous to 2 is “2–1” or 1; the number previous to 1 is “1–1” or 0.

(8) The counts previous to 0 (zero) account for units that are missing.

As mentioned in (3) above, 0 (zero) represents the reference point of “no units”. The number previous to 0 is, 0–1; the number previous to 0–1 is 0–2; the number previous to 0–2 is 0–3, and so on. These counts define units that are missing. One may thus keep on counting backward without limit.

(9) A missing number is fully defined by “0 – n”.

“0–1”, “0–2”, “0–3”, etc., provide a count of units that are missing. An example would be a count of the money that one owes.

(10) The number “0 – n” is abbreviated as the negative integer –n.

The number “0–1” is abbreviated as –1; the number “0–2” is abbreviated as –2; the number “0–3” is abbreviated as –3, and so on. The numbers “–1, –2, –3, –4, –5, etc.” are called negative integers.

(11) The reference point zero (0) is neither positive nor negative.

Zero (0) is simply the reference point for quantities that are present, as well as for the quantities that are missing.

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By vinaire | Comments (12)

The Basics of Math

November 19, 2011 – 10:33 AM
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Mathematics presents “logical tools” for learning.

Arithmetic forms the first part of mathematics that presents the “number skill”. Arithmetic starts with counting.

Counting is a tool for learning how many things are there. Counting starts with one. The next count is one more.

A unit is the thing being counted one at a time. If one is counting houses, then each house is a unit. If one is counting inches of a length, then each inch is a unit.

The digits are the ten symbols – 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 – that are used to write numbers.

Numbers provide a system to represent the counts. A number is made up of one or more digits, just like words are made up of one or more letters.

Addition is counting together of numbers. Subtraction is opposite of addition.

Multiplication is repeated addition of a number. Division is opposite of multiplication.

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THE FIELD AXIOMS

Real numbers are undefined objects that satisfy certain properties.

AXIOM #1: CLOSURE PROPERTY OF ADDITION

If x and y are real numbers, then x+y is a unique real number.

Addition is an operation such that for every pair of real numbers x and y we can form the sum of x and y, which is another real number denoted by x+y. The sum x+y is uniquely determined by x and y.

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AXIOM #2: CLOSURE PROPERTY OF MULTIPLICATION

If x and y are real numbers, then xy is a unique real number.

Multiplication is an operation such that for every pair of real numbers x and y we can form the product of x and y, which is another real number denoted by xy or by x.y. The product xy is uniquely determined by x and y.

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AXIOM # 3: COMMUTATIVE PROPERTIES OF ADDITION & MULTIPLICATION

If x and y are real numbers, then x+y = y+x.

If x and y are real numbers, then xy = yx.

ADDITION: Two numbers may be added in any order. For example,

5 + 8       =       8 + 5

One may visualize the numbers as items of a one-dimensional array. For example,

$ $ $ $ $ $ $ $ $ $ $ $ $

One may count the items in this array as “5 first and 8 next”; or “8 first and 5 next”. The result is the same.

$ $ $ $ $      $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $      $ $ $ $ $

“Subtraction” is accounted by this law by treating the number being added as a negative integer. The sign moves with the following number. The first unsigned number is treated as having a positive sign. For example,

[8 – 5]     is           +8 –5    =     –5 +8

MULTIPLICATION: Two numbers may be multiplied in any order. For example,

5 x 8       =       8 x 5

One may visualize the numbers as items of a two-dimensional array. For example,

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

One may count the items in this array, as “5 rows of 8 each”, or “8 columns of 5 each”. The result is the same.

8+8+8+8+8     =     5+5+5+5+5+5+5+5

“Division” is accounted by this law by using the reciprocal (multiplicative inverse) of the divisor as the multiplicand.

[8 ÷ 2]     is           8 x ½    =     ½ x 8

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AXIOM # 4: ASSOCIATIVE PROPERTIES OF ADDITION & MULTIPLICATION

If x, y and z are real numbers, then (x+y)+z = x+(y+z).

If x, y and z are real numbers, then (xy)z = x(yz).

In addition, three things, arranged in the same order, may be associated in two different ways. The result is the same. For example,

3   +  (5 + 8)      =        (3 + 5)   +  8

Similarly, in multiplication, three things, arranged in the same order, may be associated in two different ways. The result is the same. For example,

3   x   (5 x 8)      =        (3 x 5)   x   8

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AXIOM 5: DISTRIBUTIVE PROPERTY

If x, y and z are real numbers, then x(y+z) = xy + xz.

Multiplication distributes over addition. For example, a factor may be multiplied by the other factor as a sum of two parts with the same outcome as follows.

5 x 17   =   5 x (10 + 7)   =   5 x 10 + 5 x 7   =   50 + 35   =   85

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AXIOM 6: IDENTITY ELEMENTS

The identity element for addition is 0, i.e., for any real number x, 

x+0 = x.

The identity element for multiplication is 1, i.e., for any real number x, 

x.1 or 1x = x.

There exist two real numbers, which we denote by 0 and 1, such that for every real x we have

0 + x  =  x + 0  =  x                    (the idea of adding nothing)

1 . x  =  x . 1  =  x                      (the idea of a single occurrence)

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AXIOM 7: INVERSES

A unique ADDITIVE INVERSE exists for every real number, i.e., for every x the additive inverse is -x such that

x     +     (-x)         =           0.

A unique MULTIPLICATIVE INVERSE exists for every real number, i.e., for every non-zero x the multiplicative inverse is 1/x such that

x      .       (1/x)      =           1.

For every real number x there is a real number y such that

x + y   =   y + x   =   0       (the idea of negating something into nothing)

For every real number x (except 0) there is a real number y such that

x . y   =   y . x   =   1         (the idea of reducing something to its unit)

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Note 1: 0 is an exception because it denotes nothing, whereas all other numbers denote something.

Note 2: 1 is unique because it denotes a unit, whereas all other non-zero numbers denote multiple units.

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By vinaire | Comments (0)

MILESTONE 4: Multiplication

September 22, 2011 – 6:06 AM

Multiplication is “repeated addition.”

To explore further logic that follows from the idea of multiplication as “repeated addition,” go to the link

MATH MILESTONES #A4, MULTIPLICATION

Many a time the problem with multiplication gets resolved by getting the student to count on the ten fingers “by two,” “by three,” “by four,” etc. When the student is counting on his fingers, say, “by three,” each finger has a value of three. As he counts “by three,” he adds three at each count. He may then write down the count in a column. This will then be the multiplication table for three.

By repeated addition, one may easily produce multiplication tables, and then use them to solve problems.

The proper approach is to create the multiplication table, as above, many times using “repeated addition.” Knowing the techniques of addition from MS 02: ADDITION can be of great help. You are building the skill in multiplication as an extension of the skill in addition. This approach is better than simple memorization.

The real shortcuts in math come from thinking with the basics, such as,

  1. A number multiplied  by zero  = the number “added repeatedly” zero times = zero

  2. A number multiplied by one =  the number “added repeatedly” once = the number

  3. A number multiplied by 10 = the digits shift one place value to the left = the number with a zero attached (e.g., 3 x 10 = 30;  12 x 10 = 120).  

Check out the above with repeated addition on abacus per MILESTONE 1: Numbers & Place Values]

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The following is the most amazing property of repeated addition.

4 x 17     = 4 added repeatedly 17 times

= 4 added repeatedly 10 times + 4 added repeatedly 7 times

This may be written as,

4 x 17  =  4 x (10 + 7)  = (4 x 10) + (4 x 7)  

One may now compute this mentally as 40 +28 = 68. Here, the key property is

4 x (10 + 7)  = (4 x 10) + (4 x 7)

This ia also written in its general form as

a x (b + c)  = (a x b) + (a x c)

And we recognize it as the Distributive Property.

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Since one operation of multiplication contains many opearation of addition, multiplication takes precedence over addition in mixed operations.  For example,

3  +  2  x  4  =  11  (not 20)

Here are some videos from the Khan Academy that explain multiplication:

Basic Multiplication

Multiplication 2: The Multiplication Tables

Multiplication 3: 10,11,12 times tables

Multiplication 4: 2-digit times 1-digit number

Multiplication 5: 2-digit times a 2-digit number

Multiplication 6: Multiple Digit Numbers

Multiplication 7: Old video giving more examples

Mulitplication 8: Multiplying decimals (Old video)

Lattice Multiplication

Why Lattice Multiplication Works

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MILESTONE 3: Subtraction

August 30, 2011 – 8:26 PM

The idea of subtraction as “reverse addition” is explored in greater detail in this document.

MATH MILESTONE #A3: SUBTRACTION

Subtraction is taking a quantity away from another quantity. For example, suppose you have 15 marbles; and you give 6 marbles to your friend. You can find the remaining marbles in two different ways:

  1. Take 6 away from 15:  start from 15 and count back 6 to see the number you end up with.
  2. Find 6 and “what” is 15: start from 6 and see how many counts it takes to arrive at 15.

The method in (a) requires a new skill of counting backwards. The method in (b) utilizes the already learned skill of addition. Both these methods work because

Subtraction is the opposite of addition.

The better one can do addition, the more skillful one gets at subtraction. In subtraction, one also learns the concept of regrouping the place values. The following videos from Khan Academy demonstrate subtraction.

Basic Subtraction

Subtraction 2

Subtraction 3: Introduction to Borrowing or Regrouping

Alternate mental subtraction method

Level 4 Subtraction

Why borrowing works

When we take away all the units, which are there, we are left with nothing. Thus, subtraction provides us with the idea of “nothing” as a “number.” We call it zero.

When we need to take away more than what is there, we are faced with a definite shortage. Thus, subtraction also provides us with a measure of “shortage.” We call it a negative number.

Calculators may be used to save time with subtraction; but still mental awareness is necessary to know if the calculated answer is correct. This awareness comes from the practice with mental math as outlined in the document above.

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MILESTONE 2: Addition

August 14, 2011 – 3:05 PM

One may easily learn to add numbers mentally up to one hundred using the procedures given here.

MATH MILESTONE #A2: ADDITION

Addition is an extension of counting. For example, by counting you know that you have 3 marbles. By counting again you know that you got 4 more marbles from your friend. You put all these marbles in a box and count them together. You find that you now have 7 marbles.

Addition is counting together.

In the initial stages one learns to add small numbers by counting on one’s fingers. For larger numbers, one may use abacus. The first concept that one learns here is “carry-over” across the place values. After gaining sufficient familiarity with objects, one may add on paper using digits.

It is easy to visualize addition with objects. We may also visualize addition using a number line. Basic addition is demonstrated in the following video from Khan Academy.

Basic Addition

When we have larger numbers we may add them more easily by column. See the demonstration in the following videos from Khan Academy

Addition 2

Level 2 Addition

Addition 3

Addition 4

Here are some comments on addition.

(1) Addition depends on the concept of the same units. For example, we may add 2 cats and 3 cats to come up with the sum of 5 cats. However, we cannot express the sum of 2 cats and 3 dogs in terms of a single unit, unless we change the unit to “animal”, which is inclusive of both cats and dogs.

(2) Mathematical units may be the same, but units in the real world are never exactly the same. For example, any two oranges would never be exactly be the same in all aspects including the number of atoms they contain.

Thus, mathematics may come very close to describing the physical universe, but it is never exact.

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