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Real Numbers

Real Numbers

Real numbers are values that can be expressed as an infinite decimal expansion. Real numbers include integers, natural numbers, and others we will talk about in the coming sections. Examples of real numbers are ¼, pi, 0.2, and 5.

Real numbers can be represented classically as a long infinite line that covers negative and positive numbers.

Number types and symbols

The numbers you use to count are known as whole numbers and are part of rational numbers. Rational numbers and whole numbers compose also the real numbers, but there are many more, and the list can be found below.

  • Natural numbers, with the symbol (N).

  • Whole numbers, with the symbol (W).

  • Integers with the symbol (Z).

  • Rational numbers with the symbol (Q).

  • Irrational numbers with the symbol (Q ').

Real Numbers Venn diagram showing real numbers StudySmarterVenn diagram of numbers

Types of real numbers

It is important to know that for any real number picked, it is either a rational number or an irrational number which are the two main groups of real numbers.

Rational numbers

Rational numbers are a type of real numbers that can be written as the ratio of two integers. They are expressed in the form p / q, where p and q are integers and not equal to 0. Examples of rational numbers are . The set of rational numbers is always denoted by Q.

Types of rational numbers

There are different types of rational numbers and these are

  • Integers, for example, -3, 5, and 4.

  • Fractions in the form p / q where p and q are integers, for example, ½.

  • Numbers that do not have infinite decimals, for example, ¼ of 0.25.

  • Numbers that have infinite decimals, for example, ⅓ of 0.333….

Irrational numbers

Irrational numbers are a type of real numbers that cannot be written as the ratio of two integers. They are numbers that cannot be expressed in the form p / q, where p and q are integers.

As mentioned earlier, real numbers consist of two groups – the rational and irrational numbers, expresses that irrational numbers can be obtained by subtracting rational numbers group (Q) from real numbers group (R). That leaves us with the irrational numbers group denoted by Q '.

Examples of irrational numbers

  • A common example of an irrational number is 𝜋 (pi). Pi is expressed as 3.14159265….

The decimal value never stops and does not have a repetitive pattern. The fractional value closest to pi is 22/7, so most often we take pi to be 22/7.

  • Another example of an irrational number is . the value of this is also 1.414213 ..., is another number with an infinite decimal.

Properties of real numbers

Just as it is with integers and natural numbers, the set of real numbers also has the closure property, commutative property, the associative property, and the distributive property.

  • Closure property

The product and sum of two real numbers is always a real number. The closure property is stated as; for all a, b ∈ R, a + b ∈ R, and ab ∈ R.

If a = 13 and b = 23.

then 13 + 23 = 36

so, 13 × 23 = 299

Where 36 and 299 are both real numbers.

  • Commutative property

The product and sum of two real numbers remain the same even after interchanging the order of the numbers. The commutative property is stated as; for all a, b ∈ R, a + b = b + a and a × b = b × a.

If a = 0.25 and b = 6

then 0.25 + 6 = 6 + 0.25

6.25 = 6.25

so 0.25 × 6 = 6 × 0.25

1.5 = 1.5

  • Associative property

The product or sum of any three real numbers remains the same even when the grouping of numbers is changed.

The associative property is stated as; for all a, b, c ∈ R, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.

If a = 0.5, b = 2 and c = 0.

Then 0.5 + (2 + 0) = (0.5 + 2) + 0

2.5 = 2.5

So 0.5 × (2 × 0) = (0.5 × 2) × 0

0 = 0

  • Distributive property

The distributive property of multiplication over addition is expressed as a × (b + c) = (a × b) + (a × c) and the distributive property of multiplication over subtraction is expressed as a × (b - c) = (a × b) - (a × c).

If a = 19, b = 8.11 and c = 2.

Then 19 × (8.11 + 2) = (19 × 8.11) + (19 × 2)

19 × 10.11 = 154.09 + 38

192.09 = 192.09

So 19 × (8.11 - 2) = (19 × 8.11) - (19 × 2)

19 × 6.11 = 154.09 - 38

116.09 = 116.09

Real Numbers - Key takeaways

  • Real numbers are values that can be expressed as an infinite decimal expansion.
  • The two types of real numbers are rational and irrational numbers.
  • R is the symbol notation for real numbers.
  • Whole numbers, natural numbers, rational numbers, and irrational numbers are all forms of real numbers.

Frequently Asked Questions about Real Numbers

Real numbers are values that can be expressed as an infinite decimal expansion.

Every real number picked is either a rational number or an irrational number. They include 9, 1.15, -6, 0, 0.666 ...

 It is the set of every number including negatives and decimals that exist on a number line. The set of real numbers is noted by the symbol R.

Irrational numbers are a type of real numbers.

Negative numbers are real numbers.

Final Real Numbers Quiz

Question

Values that can be expressed as an infinite decimal expansion are termed what?


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Answer

Real numbers

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Question

Which of the following is not a real number?

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Answer

None of the following

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Question

What are the types of real numbers?


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Answer

Rational and irrational numbers

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Question

What is the symbol notation for real numbers?


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Answer

R

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Question

What is not a real number?


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Answer

Imaginary numbers

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Question

A type of real number, written as the ratio of two integers is?


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Answer

Rational number

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Question

How are rational numbers expressed?


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Answer

p/q, where p and q are integers and not equal to 0.

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Question

Which of these is a rational number with infinite decimals?


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Answer

All the following

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Question

What is an irrational number?

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Answer

Irrational numbers are a type of real numbers that cannot be written as the ratio of two integers.


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Question

What is the symbol notation for irrational numbers?


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Answer

Q'

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Question

“The product and sum of two real numbers remain the same even after interchanging the order of the numbers”. What property of real numbers does this appropriately describe?


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Answer

Commutative property

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Question

Are there real numbers that aren’t either rational or irrational?


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Answer

No, there aren’t.

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Question

Are negative numbers real numbers?


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Answer

Negative numbers are real numbers.

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Question

Applying the distributive property of real numbers as a × (b + c) = (a × b) + (a × c), what will be the value of each equation if a = 66, b = -3 and c = 14


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Answer

66 ×  (-3 + 14) = (66 ×  (-3)) + (66 ×  14)

    726 = 726

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Question

Apply the associative to the next numbers as (a+b)+c and a+(b+c)  if a = 0.91, b = 12 and c = 0,  what will be the result of both?


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Answer

(0.91 +12) +0 = 0.91 + (12 + 0)

        12.91 = 12.91

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