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These decimals either stop or repeat.

What do these examples tell us?

Every rational number can be written both as a ratio of integers , ( p q , where p and q are integers and q 0 ) , and as a decimal that either stops or repeats.

Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:

Fractions Integers
Number 4 5 7 8 13 4 20 3 −2 −1 0 1 2 3
Ratio of Integers 4 5 7 8 13 4 20 3 2 1 1 1 0 1 1 1 2 1 3 1
Decimal Form 0.8 −0.875 3.25 −6. 6 −2.0 −1.0 0.0 1.0 2.0 3.0

Rational number

A rational number is a number of the form p q , where p and q are integers and q 0 .

Its decimal form stops or repeats.

Are there any decimals that do not stop or repeat? Yes!

The number π (the Greek letter pi , pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.

π = 3.141592654 . . .

We can even create a decimal pattern that does not stop or repeat, such as

2.01001000100001…

Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational.

Irrational number

An irrational number    is a number that cannot be written as the ratio of two integers.

Its decimal form does not stop and does not repeat.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

Rational or irrational?

If the decimal form of a number

  • repeats or stops , the number is rational .
  • does not repeat and does not stop , the number is irrational .

Given the numbers 0.58 3 , 0.47 , 3.605551275 . . . list the rational numbers irrational numbers.

Solution


Look for decimals that repeat or stop. The 3 repeats in 0.58 3 . The decimal 0.47 stops after the 7 . So 0.58 3 and 0.47 are rational.


Look for decimals that neither stop nor repeat. 3.605551275 has no repeating block of digits and it does not stop. So 3.605551275 is irrational.

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For the given numbers list the rational numbers irrational numbers: 0.29 , 0.81 6 , 2.515115111 .

0.29 , 0.81 6 2.515115111

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For the given numbers list the rational numbers irrational numbers: 2.6 3 , 0.125 , 0.418302

2.6 3 , 0.125 0.418302

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For each number given, identify whether it is rational or irrational: 36 44 .

  1. Recognize that 36 is a perfect square, since 6 2 = 36 . So 36 = 6 , therefore 36 is rational.
  2. Remember that 6 2 = 36 and 7 2 = 49 , so 44 is not a perfect square. Therefore, the decimal form of 44 will never repeat and never stop, so 44 is irrational.
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For each number given, identify whether it is rational or irrational: 81 17 .

rational irrational

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For each number given, identify whether it is rational or irrational: 116 121 .

irrational rational

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We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real number     s .

Real number

A real number is a number that is either rational or irrational.

All the numbers we use in elementary algebra are real numbers. [link] illustrates how the number sets we’ve discussed in this section fit together.

Questions & Answers

What is the lcm of 340
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15x
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15x
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1y
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find the equation whose roots are 1 and 2
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(x - 2)(x -1)=0 so equation is x^2-x+2=0
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NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
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(×+1)(×-4) = x^2-3×-4
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Quadratic equations involving factorization
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a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
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a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
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a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
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Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon?
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(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
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4x-7y=8 2x-7y=1 what is the answer?
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During two years in college, a student earned $9,500. The second year, she earned $500 more than twice the amount she earned the first year.
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9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
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Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
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Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.
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The sum of two numbers is 155. The difference is 23. Find the numbers
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The sum of two numbers is 155. Their difference is 23. Find the numbers
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Joy is preparing 20 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution?
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Practice Key Terms 6

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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