(√5+7) -(3+√5)
What is the answer...
Is it Rational or Irrational number
Solve and explain
Answers
Step-by-step explanation:
A rational number is a number that can be written in the form
p
q
pq, where
p
p and
q
q are integers and
q
≠
o
q≠o.
All fractions, both positive and negative, are rational numbers. A few examples are
4
5
,
−
7
8
,
13
4
,
and
−
20
3
45,−78,134,and−203
Each numerator and each denominator is an integer.
We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.
Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.
3
=
3
1
−
8
=
−
8
1
0
=
0
1
3=31−8=−810=01
Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.
What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers. We’ve already seen that integers are rational numbers. The integer
−
8
−8 could be written as the decimal
−
8.0
−8.0. So, clearly, some decimals are rational.
Think about the decimal
7.3
7.3. Can we write it as a ratio of two integers? Because
7.3
7.3 means
7
3
10
7310, we can write it as an improper fraction,
73
10
7310. So
7.3
7.3 is the ratio of the integers
73
73 and
10
10. It is a rational number.
In general, any decimal that ends after a number of digits such as
7.3
7.3 or
−
1.2684
−1.2684 is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.
EXAMPLE
Write each as the ratio of two integers:
1.
−
15
−15
2.
6.81
6.81
3.
−
3
6
7
−367
Solution:
1.
−
15
−15
Write the integer as a fraction with denominator 1.
−
15
1
−151
2.
6.81
6.81
Write the decimal as a mixed number.
6
81
100
681100
Then convert it to an improper fraction.
681
100
681100
3.
−
3
6
7
−367
Convert the mixed number to an improper fraction.
−
27
7
−277
TRY IT
Let’s look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number, since
a
=
a
1
a=a1 for any integer,
a
a. We can also change any integer to a decimal by adding a decimal point and a zero.
Integer
−
2
,
−
1
,
0
,
1
,
2
,
3
−2,−1,0,1,2,3
Decimal
−
2.0
,
−
1.0
,
0.0
,
1.0
,
2.0
,
3.0
−2.0,−1.0,0.0,1.0,2.0,3.0
These decimal numbers stop.
We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we just considered.
Ratio of Integers
4
5
,
7
8
,
13
4
,
20
3
45,78,134,203
Decimal Forms
0.8
,
−
0.875
,
3.25
,
−
6.666
…
,
−
6.
¯¯¯¯¯¯
66
0.8,−0.875,3.25,−6.666…,−6.66¯
These decimals either stop or repeat.
What do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.
Rational Numbers
Fractions Integers
Number
4
5
,
−
7
8
,
13
4
,
−
20
3
45,−78,134,−203
−
2
,
−
1
,
0
,
1
,
2
,
3
−2,−1,0,1,2,3
Ratio of Integer
4
5
,
−
7
8
,
13
4
,
−
20
3
45,−78,134,−203
−
2
1
,
−
1
1
,
0
1
,
1
1
,
2
1
,
3
1
−21,−11,01,11,21,31
Decimal number
0.8
,
−
0.875
,
3.25
,
−
6.
¯¯¯
6
0.8,−0.875,3.25,−6.6¯
−
2.0
,
−
1.0
,
0.0
,
1.0
,
2.0
,
3.0
−2.0,−1.0,0.0,1.0,2.0,3.0
Irrational Numbers
Are there any decimals that do not stop or repeat? Yes. The number
π
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…….
π=3.141592654…….
Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. For example,
√
5
=
2.236067978…..
5=2.236067978…..
A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number.
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.
If the decimal form of a number
stops or repeats, the number is rational.
does not stop and does not repeat, the number is irrational.
EXAMPLE
Identify each of the following as rational or irrational:
1.
0.58
¯¯¯
3
0.583¯
2.
0.475
0.475
3.
3.605551275
…
3.605551275…
Show Solution
TRY IT
Let’s think about square roots now. Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.
EXAMPLE
Identify each of the following as rational or irrational:
1.
√
36
36
2.
√
44
44
Show Solution
TRY IT
In the following video we show more examples of how to determine whether a number is irrational or rational.