BEGIN by recalling that a variable is a symbol that takes on values. A value is a number.

Thus, if x is a variable, then x might have the value 2, or −3, or 5.2, and so on.

Next, the following are called the integers:

0, ±1, ±2, ±3, and so on.

± (“plus or minus”) is called the double sign.

And the following are called the square numbers, or the perfect squares:

1 4 9 16 25 49 64 . . .

For, these are the numbers 1· 1, 2· 2, 3· 3, 4· 4, and so on.

Rational and irrational numbers

1. What are the rational numbers?

They are the numbers of arithmetic: The whole numbers, fractions, mixed numbers, and decimals; together with their negative images.

2. Which of the following numbers are rational?

1 −6 3½ − 2

3 0 7.38609

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All of them!

3. A rational number can always be written in what form?

As a fraction a

b , where a and b are integers (b 0).

When a and b are positive, that is, when they are natural numbers, then we can always name their ratio. Hence the term, rational number.

At this point, the student might wonder, What is a number that is not rational?

An example of such a number is (“Square root of 2”). is not a number of arithmetic: there is no whole number, no fraction, and no decimal whose square is 2. (1.414 is close, because (1.414)² = 1.999396 — which is almost 2.)

But to prove that there is no rational number whose square is 2, then

suppose there were. Then we could express it as a fraction m

n in lowest

terms. That is, suppose

m

n · m

n = m· m

n· n = 2.

But that is impossible. Since m

n is in lowest terms, then m and n have

no common divisors except 1. Therefore, m· m and n· n also have no common divisors — they are relatively prime — and it will be impossible to divide n· n into m· m and get 2

There is no rational number whose square is 2. Therefore we call an irrational number.

By recalling the Pythagorean theorem, we can see that these irrational numbers 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 . There really is a length that logically deserves the name, “.” Insofar as numbers name the lengths of lines, then is a number.

4. Which numbers have rational square roots?

Only the square roots of the square numbers.

= 1 Rational

Irrational

Irrational

= 2 Rational

, , , Irrational

= 3 Rational

And so on.

Only the square roots of square numbers are rational.

The existence of these irrationals was first realized by Pythagoras in the 6th century B.C. He called them “unnameable” or “speechless” numbers. For, if we ask, “In the isosceles right triangle, what ratio has the hypotenuse to the side?” — we cannot say. We can name it only as “Square root of 2.”

5. Say the name of each number.

a) “Square root of 3.” b) “Square root of 5.”

c) “2.” This is a rational — nameable — number.

d) “Square root of 3/5.” e) “2/3.”

The decimal representation of irrationals

When a rational number is expressed as a decimal, then either the decimal will terminate or there will be a predictable pattern of digits. But when an irrational number is expressed as a decimal, then, clearly, the decimal cannot terminate — for if it did, the number would be rational

Moreover, there will not be a predictable pattern of digits. For example,

1.4142135623730950488016887242097

Now, with rational numbers you sometimes see

1

11 = .090909. . .

Here, the three dots (ellipsis) mean, “What you see is a decimal approximation. We could continue the approximation for as many decimal places as we please, according to the indicated pattern.”

But if we write ellipsis for an irrational number,

= 1.41421356237. . . ,

then here the three dots mean, “What you see is a rational approximation. We could continue the approximation for as many decimal places as we please according to the rule or method for calculating the next digit.” (Not the subject of these Topics.)

On the other hand, if we just see

2.562373095048801. . .

then we might get the idea that this is an irrational number, but we would certainly have no idea which irrational number it is

Real numbers

5. What is a real number?

Any number that you would expect to find on the number line. It is a number required to label any point on the number line. It is a number whose absolute value names the distance of any point from 0.

6. What are the two main categories of real numbers?

Rational and irrational.

Problem. We have categorized numbers as real, rational, irrational, and integer. Name all the categories to which each of the following belongs.

3 Real, rational, integer. −3 Real, rational, integer.

−½ Real, rational. Real, irrational.

5¾ Real, rational. − 11/2 Real, rational.

1.732 Real, rational. 6.920920920. . . Real, rational.

2.984682057691. . . Real. And let us assume that it is irrational, that is, that the digits do not repeat.

2.984682057691 Real, rational. A terminating decimal is rational.

7. What is a real variable?

A variable whose values are real numbers.

Calculus is the study of functions of a real variable.

Problem. Let x be a real variable, and let 3 < x < 4. Name five values that x might have.

Numbers that can be written in the form p/q where p and q are integers are known as rational numbers. Included in these are the integers (take q = 1). Thus for example the rationals include {0, 5/2, -18, -4/3, 27/5}. We unually write rational numbers in their lowest terms, for example 8/10 is usually written 4/5. We commonly write rationals in decimal form, so that 1/4 is the same as 0.25, 13/8 = 1.625 and 4/5 = 0.8. Some rationals, however, when written in decimal form don’t stop a few places after the decimal. For example 1/3 = 0.333…, 10/11 = 0.909090… and 3/13 = 0.230769230769… When you write a rational number in decimal form you obtain either a decimal that stops after a finite number of terms, or a pattern that repeats as in the latter three examples.

Not all numbers are integers or rational numbers. If you construct a decimal that does not terminate and does not repeat it is not a rational number. For example 0.102003000400005… Numbers that are not rational are called irrational. The most famous of these is the ratio of the circumference of a circle to its diameter, called . is approximately 3.14159265358979323846; no matter how far we take the decimal expansion, it never repeats. Another irrational number is which is approximately 1.4142135623731.

Here is a way you can construct your own irrational number, or at least start because you will never finish. Start with 0 and then toss a coin. If you get a head write 0.1 and if you get a tail write 0.0. Now toss the coin again and write the second decimal place as a 1 if you tosses a head and 0 if you tosses a tail. So now you have either 0.10, 0.11, 0.00 or 0.01. Toss the coin again to get the third place after the decimal. Repeat forever.

In the days of Pythagoras (570-490 BC) the beauty of the rational numbers was revered. Pythagoras refused to accept the existence of the irrational numbers and it is said that one of his students, Hippasus, was sentenced to death for his refusal to deny their existence.

Well, rational numbers will listen to you and help you come to a solution and irrational numbers get angry very easily and rarely listen to others. 🙂

LOL, just kidding

Rational numbers are the numbers of arithmetic: The whole numbers, fractions, mixed numbers, and decimals; together with their negative images.

In mathematics, a rational number is a number which can be expressed as a ratio of two integers.

In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Almost all real numbers are irrational, in a sense which is defined more precisely below.

We did this in maths last week….

Basically, rational numbers can be written as fractions (including recurring decimals)

Irrational numbers cant be written as fractions, e.g pie, and the square root of 2. Irrational numbers are normally the square roots of prime numbers.

Hope this helps =)

just learned this in class today…irrational numbers are numbers whos decimals are infinate and non repeating…meaning they never end and the samenumber sequence never repeats….such as pi

rational numbers are any other number whole numbers …or decimal numbers with repeating deciamls, or decimal sequences, or that are finite ( with a definate end)

for example, 1 devided by 2 is .5, it has a definate end, 2 devided by 3 is .6666, its a repeating decimal, and 22 devided bny 7 is 3.142857142857, it has a repeating decimal sequence

Irrational numbers have no logical value. They are abstractions. For instance the square root of negative one. There is no number to represent this value, so we have invented the idea of an irrational number to represent something our traditional arabic numeral system cannot represent.