**MATH SYMBOLS & NUMBER SETS**

SYMBOL |
MEANING |

angle A | |

triangle ABC | |

triangle ABC is congruent to triangle DEF | |

AB // CD | AB is parallel to CD |

AB is perpendicular to CD | |

triangle ABC is similar to triangle DEF | |

//gm | parallelogram |

A · B | A multiply B |

A is not equal to B | |

A < B | A is less than B |

A > B | A is greater than B |

A is less than or equal to B | |

A is greater than or equal to B | |

| a | | absolute value of a |

square root of a | |

cube root of a | |

nth root of a | |

therefore | |

as or since | |

such that | |

"a" is an element of set S, or "a" belongs to the set S | |

"a" is not an element of set S, or "a" doesn't belong to the set S | |

there exists, (crossed out -- there does not exist) | |

for all |

In mathematics, we group numbers into set or systems according to their quality (positive or negative) and their properties.

If we consider the evolution of numbers in human civilization, it's easy to see the progression.

What did cavemen need to count?

Rocks, caves, women -- whatever they collected. Well, they certainly didn't collect negative amounts so they didn't need negative numbers.

The first set of numbers we still deal with today (in childhood when we first start to count) are the **Natural Counting Numbers denoted N** -- whole numbers beginning at 1.

*N = *{1, 2, 3, 4, .......}

If you think about **Roman Numerals, there's no zero**, is there? There are **no negative numbers or fractions** either -- are there? **Roman Numerals are just the Natural Numbers in Roman rather than Arabic notation.**

Then, **we added the zero** once the concept had moved far enough West.

*N _{ }*0

The symbol ** N_{ }0** represents the natural numbers beginning at 0.

Many years ago I worked at a college where the math teachers would make a joke with "nought". Someone would write **y?** on a chalkboard -- and someone else would answer **y _{ 0} ?**

Once humans evolved enough to understand that the natural numbers would not serve all their needs -- if someone owes you something, you suddenly have a need for negative numbers -- the **integers** appeared on the scene.

*Z, the set of integers *includes all positive and negative whole numbers.

We're still stuck with whole numbers -- but at least now we have negative numbers to represent debt or loss.

** Z = {........ **–

Now that we've introduced the negative or left half of the number line, we can add and subtract at will. However, though we can multiply integers together without needing to invent a new set of numbers, we still can't divide every integer by any other one and get an answer in this system.

If we have to divide 6 by 3 we can get 2. But what if we need to know what half of 7 is. There is no answer for this question in the set of integers.

So, on to the rational numbers (those dreaded fractions).

*Q the set of rational numbers *is composed of all numbers

of the form
, where *a* and *b* are integers with ** ** **.**

Why ** Q** you ask? It stands for

The collection of all the number sets above is called the set of **Real Numbers **denoted ** R**.

It includes the **natural numbers, integers and rational numbers.**

The only other 2 sets of numbers are the **Irrational numbers** and the **Imaginary or Complex numbers**.

**Irrational numbers, **denoted ** I** are all those that cannot be expressed as a fraction because their decimal part is not repeating or terminating. The 2 most commonly known irrational numbers are and

As for the complex numbers** --** which assume that –1 has a square root called ** i **-- that's for folks who are bored with reality and have to invent things. We'll stick to reality for now. All of our work will be in the set

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(*all content **© MathRoom Learning Service; 2004 - *).