Whole numbers natural numbers integers and rational number

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As you could see in the last section, with its various number lines, there are a number of different ways to classify numbers. In fact, there are even more ways to classify numbers than last section displayed. This section will run through the most important and common classifications. You should memorize what each classification means.

Natural Numbers, Whole Numbers, Integers, and Rationals

Natural Numbers

The natural numbers, also called the counting numbers, are the numbers 1, 2, 3, 4, and so on. They are the positive numbers we use to count objects. Zero is not considered a "natural number."

Whole Numbers

The whole numbers are the numbers 0, 1, 2, 3, 4, and so on (the natural numbers and zero). Negative numbers are not considered "whole numbers." All natural numbers are whole numbers, but not all whole numbers are natural numbers since zero is a whole number but not a natural number.

Integers

The integers are ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ... -- all the whole numbers and their opposites (the positive whole numbers, the negative whole numbers, and zero). Fractions and decimals are not integers. All whole numbers are integers (and all natural numbers are integers), but not all integers are whole numbers or natural numbers. For example, -5 is an integer but not a whole number or a natural number.

Rational Numbers

The rational numbers include all the integers, plus all fractions, or terminating decimals and repeating decimals. Every rational number can be written as a fraction a/b, where a and b are integers. For example, 3 can be written as 3/1, -0.175 can be written as -7/40, and 1 1/6 can be written as 7/6. All natural numbers, whole numbers, and integers are rationals, but not all rational numbers are natural numbers, whole numbers, or integers.

We now have the following number classifications:
I. Natural Numbers
II. Whole Numbers
III. Integers
IV. Rationals

Numbers can fall into more than one classification. In fact, if a number falls into a category, it automatically falls into all the categories below that category. If a number is a whole number, for instance, it must also be an integer and a rational. If a number is an integer, it must also be a rational.

All Natural numbers, Whole numbers and Integers are rational numbers.

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Solution

The correct option is ATrueAll Natural numbers, Whole numbers and Integers are rational numbers because they can be written as pq form where q is not equal to zero

The Natural Numbers

The natural (or counting) numbers are 1,2,3,4,5, etc. There are infinitely many natural numbers. The set of natural numbers, {1,2 ,3,4,5,...}, is sometimes written N for short.

The whole numbers are the natural numbers together with 0.

(Note: a few textbooks disagree and say the natural numbers include 0.)

The sum of any two natural numbers is also a natural number (for example, 4 +2000=2004), and the product of any two natural numbers is a natural number (4×2000=8000). This is not true for subtraction and division, though.

The Integers

The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero.

{...,−5,− 4,−3,−2,−1,0,1,2,3,4,5,...}

The set of integers is sometimes written J or Z for short.

The sum, product, and difference of any two integers is also an integer. But this is not true for division... just try 1÷2.

The Rational Numbers

The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 13 and −11118 are both rational numbers. All the integers are included in the rational numbers, since any integer z can be written as the ratio z1.

All decimals which terminate are rational numbers (since 8.27 can be written as 827 100.) Decimals which have a repeating pattern after some point are also rationals: for example,

0.0833333....=112.

The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don't divide by 0).

The Irrational Numbers

An irrational number is a number that cannot be written as a ratio (or fraction).  In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers.

The first such equation to be studied was 2= x2. What number times itself equals 2?

2 is about 1.414, because 1.4142=1.999396, which is close to 2. But you'll never hit exactly by squaring a fraction (or terminating decimal). The square root of 2 is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern:

2=1.41421356237309...

Other famous irrational numbers are the golden ratio, a number with great importance to biology:

1+52=1.61803398874989...

π (pi), the ratio of the circumference of a circle to its diameter:

π=3.14159265358979...

and e, the most important number in calculus:

e=2.71828182845904...

Irrational numbers can be further subdivided into algebraic numbers, which are the solutions of some polynomial equation (like 2 and the golden ratio), and transcendental numbers, which are not the solutions of any polynomial equation. π and e are both transcendental.

The Real Numbers

The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers.  The real numbers are “all the numbers” on the number line.  There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers.  But, it can be proved that the infinity of the real numbers is a bigger infinity.

The "smaller", or countable infinity of the integers and rationals is sometimes called ℵ0(alef-naught), and the uncountable infinity of the reals is called ℵ1(alef-one).

There are even "bigger" infinities, but you should take a set theory class for that!

The Complex Numbers

The complex numbers are the set {a+ bi | a and b are real numbers}, where i is the imaginary unit, −1. (click here for more on imaginary numbers and operations with complex numbers).

The complex numbers include the set of real numbers.  The real numbers, in the complex system, are written in the form a+0i=a. a real number.

This set is sometimes written as C for short. The set of complex numbers is important because for any polynomialp(x) with real number coefficients, all the solutions of p(x)=0 will be in C.

Beyond...

There are even "bigger" sets of numbers used by mathematicians. The quaternions, discovered by William H. Hamilton in 1845, form a number system with three different imaginary units!

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