which of the following is an irrational number
a.0
b.0.3
c.3/4
d.√55
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which of the following is an irrational number
a.0
b.0.3
c.3/4
d.√55
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Answer:
RATIONAL AND IRRATIONAL
NUMBERS
CALCULUS IS A THEORY OF MEASUREMENT. The necessary numbers are the rationals and irrationals. But let us start at the beginning.
The following numbers of arithmetic are the counting-numbers or, as they are called, the natural numbers:
1, 2, 3, 4, and so on.
(At any rate, those are their Arabic numerals.)
If we include 0, we have the whole numbers:
0, 1, 2, 3, and so on.
And if we include their algebraic negatives, we have the integers:
0, ±1, ±2, ±3, and so on.
± ("plus or minus") is called the double sign.
The following are the square numbers, or the perfect squares:
1 4 9 16 25 36 49 64, and so on.
And these are their roots:
1 2 3 4 5 6 7 8, and so on.
The student is no doubt familiar with the radical sign: radical.
radical = 5.
"The square root of 25 is 5."
Rational and irrational numbers
1. What is a rational number?
A rational number is simply a number of arithmetic: a whole number, a fraction, a mixed number, or a decimal; together with its negative image.
2. Which of the following numbers are rational?
1 −1 0 2
3 − 2
3 5½ −5½ 6.085 −6.085 3.1415926535897
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All of them. All decimals are rational. That long one is an approximation to π, which, as we shall see, is not equal to any decimal. For if it were, it would be rational.
3. Why the name rational?
The natural numbers of arithmetic have a ratio to one another. Every number of arithmetic can be written as a fraction, which has the same ratio to 1 as the numerator has to the denominator.
a
b : 1 = a : b.
A number that has the same ratio to 1 as two natural numbers—whose relationship we can always name—we say is rational.
Ratio is the language of arithmetic. We will see that language cannot express the relationship of an irrational number to 1.
Upon extending this to the negative numbers of algebra:
4. A rational number can always be written in what form?
As a fraction a
b , where a and b are integers (b not equal to 0).
That is the formal definition of a rational number.
Finally, we can in principle (by Euclid VI, 9) place any rational number exactly on the number line.
Rational, irrational numbers
We can say that we truly know a rational number.
WE ARE ABOUT TO SEE that the square root of a number that is not a perfect square—√2, √3, √46—is not a rational number. It is not a number of arithmetic. Let us consider √2 ("Square root of 2"). 7-5 is close because
7
5 · 7
5 = 49
25
—which is almost 2.
To see that there is no rational number whose square is 2, suppose there were. Obviously, it is not a whole number. It will be in the form of a fraction in lowest terms. But the square of a fraction in lowest terms is also in lowest terms.
squares
No new factors are introduced and the denominator will never divide into the numerator to give 2—or any whole number.
There is no rational number whose square is 2 or any number that is not a perfect square. We say therefore that Square root of 2 is an irrational number.
By recalling the Pythagorean theorem, we can see that irrational Rational, irrational numbersnumbers are necessary. For if the sides of an isosceles right triangle are called 1, then we will have 12 + 12 = 2, so that the hypotenuse is Square root of 2 . There really is a length that logically deserves the name, "Square root of 2 ." Inasmuch as numbers name the lengths of lines, then Square root of 2 is a number.
5. Which natural numbers have rational square roots?
Only the square roots of the square numbers; that is, the square roots of the perfect squares.
Square root of 1 = 1 Rational
Square root of 2 Irrational
Square root of 3 Irrational
Square root of 4 = 2 Rational
Square root of 5, Square root of 6, Square root of 7, Square root of 8 Irrational
Square root of 9 = 3 Rational
And so on.
This follows from the same proof that Square root of 2 is irrational.
Not every length, then, can be named by a rational number. Pythagoras realized that in the 6th century B.C. Rational, irrational numbersHe realized that the relationship of the diagonal of a square to the side was not as two natural numbers—which we can always name. That relationship, he said, was without a name. For if we ask, "What relationship has the diagonal to the side?"—we cannot say. Nowadays, of course, we call it "Square root of 2. Square root of 2." But the idea of an irrational number had not yet occurred. It was not until many centuries after Pythagoras that the radical sign was created.
6. Say the name of each number.
a) Square root of 3 "Square root of 3."
b) Rational, irrational numbers "Square root of 5."
c) Rational, irrational numbers "2." This is a rational—nameable—number.
d) Rational, irrational numbers "Square root of 3/5."
e) Rational, irrational numbers "2/3."
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