**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,** i ^{2} = −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

*are real numbers, and*

**b***is the imaginary unit satisfying*

**i**

**i**^{2}= −1.

(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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## Comments

Vin: (13) Thus, the basic nature of number being discrete comes under question.

Chris: How so? I am reading on to try and pick it up.

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Look at the reference just before it.

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

Chris: How so?

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They cannot be put on the same number line. That is why we have complex numbers represented by a plane.

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This is a travel advertisement?

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Looks like WordPress is adding advertisements to my blog. Maybe I should start paying to them for not to do so.

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I hope the ads are gone now.

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