What are Real Numbers?

What are Real Numbers?

A real number is an element of the set R, which is the union of the set of rational numbers and the set of irrational numbers. For better understanding, let us see the different types of numbers.

Types of Numbers

We first learn about numbers in order to count.

Natural numbers: Counting numbers are called Natural numbers. These numbers are denoted by N = {1, 2, 3, .........}.

Whole numbers: The collection of natural numbers along with 0 is the collection of the Whole number and is denoted by W.

Integers: When doing subtraction, negative whole numbers were introduced. The set of positive and negative whole numbers is called the set of integers or the collection of natural numbers, their negatives along with the number zero are called Integers. This collection is denoted by Z.

Rational number: The numbers, which are obtained by dividing two integers, are called Rational numbers. Division by zero is not defined. Since every integer can be written as a fraction with 1 in the denominator, we say that the integers are a subset of the rational numbers.

Irrational Numbers: Not all numbers can be formed as a fraction. For example, the square root of 2 cannot be expressed as a fraction. These kinds of numbers are called irrational numbers. Other irrational numbers include pi.

Basic Properties of the Real Numbers

We can add, subtract, multiply and divide them (as long as we don't divide by zero). The order of addition and multiplication is unimportant, as there is a commutative property. A distributive property tells us how multiplication and addition interact with one another.

Given any two real numbers x and y, we know that one and only one of the following is true:

x = yx < y or x > y.

1. Addition

When adding real numbers with the same sign the sum will have the same sign as the numbers added.

2+3=5

−5+(−3)=−8

When adding real numbers with different signs you subtract the lesser absolute value from higher one and the sum will have the same sign as the number with the higher absolute value.

−5+3=−2

The additive commutative property tells us that the order in which you add the numbers does not change the sum.

x+y = y+x

And the additive associative property tells us that it also that the order in which we group three or more numbers does not affect the sum.

(x+y)+z=x+(y+z)

The additive identity property tells us that the sum of a number and 0 is always the number.

x+0=x

And the additive inverse property tells us that if you add a number with its opposite you will always get 0

x+(−x)=0

2. Subtraction

You know by now that

8−3=8+(−3)=5

This means that subtracting 3 from 8 is the same as adding negative 3 to 8. This is called the subtraction rule.

x-y=x+(-y)

3. Multiplication

The product of two real numbers with the same sign is always positive

2⋅3=6

(−3)⋅(−2)=6

If we multiply with an odd number of negative numbers the product will always be negative

3⋅(−2)⋅(−5)⋅1⋅(−2)=−60

And if we multiply with an even number of negative numbers the product will be positive

4⋅(−2)⋅(−5)=40

The multiplicative commutative property tells us that the product is not affected by the order in which you multiply the numbers

x.y=y.x

The same goes for the multiplicative associative property i.e. the product is not affected by how you group three or more numbers.

x(y⋅z)=(x⋅y)⋅z

The multiplicative identity property - if we multiply a number with 1 the product is always the number

x1=x

The multiplicative identity of 0 tells us that the product is always 0 when you multiply a number with 0

x0=0

The multiplicative property of -1 and this property tells us that a product of a number and -1 is the opposite of the number

x(1)=(−x)

4. Division

Reciprocals are numbers that when multiplied have the product 1.

4/5⋅5/4=1

These are also called multiplicative inverses. Zero does not have a multiplicative inverse since everything multiplied with 0 is 0.

The inverse property of multiplication:

x⋅(1/x)=(1/x)x=1,x≠0

It can also be written as

x/yy/x=1 where x,y≠0

The division rule - to divide a number x with a number y is the same as multiplying x with the multiplicative inverse of y

x/y=x1/y, y≠0

The quotient of two numbers with the same sign is always positive.

8/2=4

The quotient of two numbers with different signs is negative

−8/2=−4

The quotient of 0 and any nonzero real number is always 0

0/99=0

Since 0 does not have a multiplicative inverse you cannot divide a number by 0.

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