Vinaire's Blog

ARITHMETIC = Arithmos (number) + Techne (Skill) = Number skill

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(1)  In counting, the previous number is one less.

1 less than 3       =     2

1 less than 2       =     1

1 less than 1       =     0 (zero, nothing)

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(2)  Zero provides the reference point for all counting numbers.

1      =     0 + 1

5      =     0 + 5

9      =     0 + 9

If N is any number then,

N      =     0 + N

For measuring heights on earth, zero is the sea level.

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(3)  Then numbers may be counted both forward and backward from zero.

Numbers counted forward from 0 are positive numbers:  +N   =   0 + N

Numbers counted backward from 0 are negative numbers:  –N   =   0 – N

The numbers go on forever in both forward and backward directions.

This we call the set of INTEGERS (untouched, hence, undivided)

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(4)  The set of integers may be visualized as a number line.

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(5)    An integer is made up of a sign and an absolute value.

Absolute value of +3   =    |+3|    =    3

Absolute value of –3   =    |–3|    =    3

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(6)  POSITIVE affirms the existing characteristic.

Positive of a positive integer is that positive integer:    + (+1)   =   +1

Positive of a negative integer is the negative integer:  + (–1)   =   –1

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(7)  NEGATIVE indicates the opposite characteristic.

Negative of a positive integer is the negative integer:   – (+1)   =   –1

Negative of a negative integer is the positive integer:   – (–1)   =   +1

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(8)  A Rational number can be expressed as a ratio of two counting numbers.

Therefore, there is a definite unit on which a rational number is based.

This unit is the common factor of the two counting numbers.

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(9) An Irrational number cannot be expressed as a ratio of two counting numbers.

Therefore, there is no unit, however small, on which an irrational number may be based.

Please see Going Beyond Counting.

Thus, the basic nature of number being discrete comes under question.

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(10)  The following are considered as Real numbers:

Natural, whole, positive, negative, rational and irrational numbers.

Real numbers may be represented on a Number Line with a common reference point of zero.

The reference point of zero may be selected arbitrarily on the number line.

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(11)  The square of an integer of either sign is always a positive integer.

A positive integer has two square roots: one positive and the other negative.

A negative integer has no square root in the real number system.

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(12)  The square root of –1 is denoted as i (an imaginary unit)

Thus,      i²   =   −1

Therefore, the square root of –4 is written as 2i.

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(13)  The Imaginary numbers may be represented on a different number line.

The imaginary and real numbers lines are independent of each other.

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(14)  A complex number a + bi is made up of real and imaginary components.

The real component is a, and the imaginary component is bi

Here a and b are real numbers, and i is the imaginary unit satisfying i² = −1

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(16)  The Complex numbers may be represented on a complex plane.

A complex plane is determined by a real number line as the horizontal axis,

and an imaginary number line as the vertical axis.

The reference point of zero is shared by both number lines.

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