User:Dogmorfmocion/Sandbox

= Types of number =

Natural numbers
The natural numbers are the set (collection) of all non-negative whole numbers, thus starting with 0 (zero). Mathematicians represent this set using the letter N or $$\mathbb{N}$$ in Blackboard bold. This set is infinite in size, meaning that it has an unlimited amount of members. If you think of a natural number, no matter how large, one can add one or more natural numbers to it to make a larger one. Here are the first 10 natural numbers in decimal representation:

... 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ...

As described, this list could go on indefinitely. For instance, 541 is in $$\mathbb{N}$$; so is 99999, or even 1213849238. Typically, we use the natural numbers for counting things like objects ("I have 14 books"), or ordering ("James came 2nd in the race"). Sometimes you might see a mathematician or book exclude 0 from $$\mathbb{N}$$, but this is generally out of fashion although it is always prudent to check.

Integers
The integers are an expansion of the set of natural numbers, consisting of the naturals plus all their whole, negative equivalents. This set is written as Z or $$\mathbb{Z}$$. Like $$\mathbb{N}$$, the set is infinite in size; the negative numbers in the set extend to infinity as well. Here is an example of 10 integers:

... -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ...

Like a small Russian doll can fit into a larger one, the natural numbers can fit into $$\mathbb{Z}$$. They are known as a subset of $$\mathbb{Z}$$. The integers are useful for considering things like debt, where the amount an individual owes can be represented as a negative value.

Rational numbers
A rational number is any number that can be written as a ratio of two integers or a fraction. This set of numbers is written Q or $$\mathbb{Q}$$. This set is infinite in size and, as you might expect, $$\mathbb{Z}$$ is a subset of it. Here are rational numbers represented as fractions:

... -3/5, -1/4, 1/2, 1/1, 3/2 ...

These numbers can be represented as fractions in the example above or as decimal representations like the previous examples:

... -0.6, -0.25, 0.5, 1, 1.5 ...

There are some rational numbers that have an infinitely long decimal expansion. For instance, $$1/3 = 0.333... \,$$. The trailing dots in the decimal form tell us that the 3 recurs forever! We would never be able to write this number out in full as decimal representation. Every integer is in $$\mathbb{Q}$$ because it can be represented fractionally, divided by 1. For instance, $$45 = 45/1 \,$$, and $$-45 = -45/1 \,$$ (or even $$45/-1 \,$$). Generally, rational numbers are used to represent parts of a whole ("I have half an apple"). There is one restriction, however; we cannot divide by zero! Any fraction with a zero as the denominator is considered "undefined" and for all intents an purposes, we ignore it.

Real numbers
The real numbers, denoted by R or $$\mathbb{R}$$ contain all rational numbers plus a type called the irrational numbers. In the way that some rational numbers cannot be represented in decimal, the irrational numbers cannot be fully expressed in decimal OR as a fraction! There is no proper way to write these numbers out in full, so they are often approximated, or represented by a symbol. Here is an example of two famous irrational numbers:

π = 3.14159..., e = 2.71828...

These two numbers, known as constants (they are fixed values, like 3 or 7398) cannot be represented decimally (like $$\mathbb{Z}$$) or as a fraction (like $$\mathbb{Q}$$). In the example above, the values have been cut off 5 decimal points in; the values continue infinitely if we try to express them in full. You might argue that some fractions, like $$1/3$$ continue infinitely when expressed as a decimal - but that does not satisfy the criteria - an irrational number cannot be written properly as a fraction as $$1/3$$ can.

= Questions =

1. Place the following numbers into the most exclusive subset they belong to.

a) $$-130$$

b) $$1/5$$

c) $$24$$

d) $$-123923.451$$

e) $$9.0$$

f) $$-8/15$$

2. True or false?

a) $$\mathbb{Q}$$ is a subset of $$\mathbb{Z}$$.

b) $$\mathbb{N}$$ is a subset of $$\mathbb{R}$$.

c) $$\mathbb{Q}$$ contains numbers that cannot be fully expressed in decimal representation and as a fraction.

d) $$15/0 = 0 \,$$

e) $$0/1 = 0 \,$$