# Is 7 0 7 0 or infinite

### What are irrational numbers?

Like rational numbers, you cannot represent irrational numbers as fractions, periodic or terminating numbers.

you are non-periodic and infinite.

Examples:

\$\$ sqrt (2) = 1.414213562 ... \$\$

\$\$1,41441444144441444441…\$\$

Non-square roots are always irrational numbers.

You can already pull some roots

• \$\$ sqrt (9) = 3 \$\$
• \$\$ sqrt (0.16) = 0.4 \$\$, since \$\$ 0.4 * 0.4 = 0.16 \$\$
• \$\$ sqrt (4/9) = 2/3 \$\$, since
\$\$ 2 * 2 = 4 \$\$ and \$\$ 3 * 3 = 9 \$\$

The square numbers \$\$ 1, 4, 9, 16, 25,… \$\$ will help you with this

Note: Square numbers are always natural numbers.

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### Nest irrational numbers in an interval

With interval nesting, you can save irrational numbers as decimal number without using the root key of your calculator.

Example: \$\$ sqrt (2) \$\$

### Step 1: Find the first interval.

Between which natural numbers is \$\$ sqrt (2) \$\$?

• Try the square numbers \$\$ 1 \$\$, \$\$ 4 \$\$, \$\$ 9 \$\$ and \$\$ sqrt (2) ^ 2 \$\$.
• Since \$\$ 1 ^ 2 = 1le2le2 ^ 2 = 4 \$\$, \$\$ sqrt (2) \$\$ lies between \$\$ 1 \$\$ and \$\$ 2 \$\$.
• Always choose that smallest interval, in which the value \$\$ 2 \$\$ is also present. So not \$\$ [1; 9] \$\$, but just \$\$ [1; 2] \$\$.

interval

An interval is a set of numbers between two numbers.

The closed interval \$\$ [2; 5] = {x in QQ | -2lexle5} \$\$ contains the \$\$ - 2 \$\$ and the \$\$ 5 \$\$ and all rational numbers in between.

### Select the interval nesting more closely

Note: Calculation steps marked in blue are calculated with the calculator.

### Step 2: Box the interval further.

• Add a digit after the decimal point.
• Use the calculator to find out between which of the numbers \$\$ (1,1) ^ 2, (1,2) ^ 2, (1,3) ^ 2,…, (1,9) ^ 2 \$\$ the number \$\$ 2 \$\$ lies.
• \$\$ 1,4lesqrt (2) le1,5 \$\$, because \$\$ (1,4) ^ 2 = 1.96 \$\$\$\$ le2le \$\$\$\$ (1.5) ^ 2 = 2.25 \$\$

### 3rd step: Two decimal places

• Use the calculator to calculate between which of the numbers \$\$ (1.41) ^ 2, (1.42) ^ 2, (1.43) ^ 2,…, (1.49) ^ 2 \$\$ the number \$\$ 2 \$\$ lies.
• \$\$ 1.41lesqrt (2) le1.42 \$\$,
because \$\$ (1.41) ^ 2 = 1.9881 \$\$\$\$ le2le \$\$\$\$ (1.42) ^ 2 = 2.0164 \$\$

### 4th step: three decimal places

• Use the calculator to calculate between which of the numbers \$\$ (1.411) ^ 2, (1.412) ^ 2, (1.413) ^ 2,…, (1.419) ^ 2 \$\$ the number \$\$ 2 \$\$ lies.
• \$\$ 1,414lesqrt (2) le1,415 \$\$,
because \$\$ (1.414) ^ 2 = 1.999396 \$\$\$\$ le2le \$\$\$\$ (1.415) ^ 2 = 2.002225 \$\$

So you can nest \$\$ sqrt (2) \$\$ more and more precisely and get one Approximate value.

### II. Assumption: \$\$ sqrt (2) \$\$ is rational (is a shortened fraction)

To show: There is a contradiction.

### Preliminary considerations:

• If you multiply a number \$\$ n \$\$ by \$\$ 2 \$\$, the result is an even number \$\$ (2 * n) \$\$.
• If the square of a number is even, so is the number itself. Example: 64 is straight and 8 is too.
• Fractions can be shortened if the numerator and denominator have a common factor.

In this evidence process, you show an assertion by assuming the opposite of the assertion and that leads to a contradiction.

Procedure:

I. Claim

II. Assumption to the contrary of assertion

IV. Assumption wrong, assertion holds

Already around 300 BC The mathematician showed Euclidthat \$\$ sqrt (2) \$\$ is an irrational number. He also carried out a contradiction proof.

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### Proof by contradiction: \$\$ sqrt (2) \$\$ is irrational

Proof step Explanations
1) \$\$ sqrt (2) = p / q \$\$ According to the claim, \$\$ sqrt (2) \$\$ can be represented as an abbreviated fraction
(Have \$\$ p \$\$ and \$\$ q \$\$ none common divisor).
2) \$\$ 2 = p ^ 2 / q ^ 2 \$\$ Squar both sides of the equation.
3) \$\$ 2 * q ^ 2 = p ^ 2 \$\$ Transform the equation according to \$\$ p \$\$.
4) \$\$ p ^ 2 \$\$
is just
This follows from the representation of \$\$ p \$\$.
5) \$\$ p \$\$
is just
This follows from the second preliminary consideration.
6) \$\$ p = 2 * n \$\$ \$\$ p \$\$ is even, i.e. double of any number \$\$ n \$\$.
7) \$\$ p ^ 2 = 4 * n ^ 2 \$\$ Squar both sides of the equation.

### Proof by contradiction: \$\$ sqrt (2) \$\$ is irrational

Proof stepExplanation
8) \$\$ 4 * n ^ 2 = 2 * q ^ 2 \$\$Equalization of \$\$ p ^ 2 = 4 * n ^ 2 \$\$ and \$\$ p ^ 2 = 2 * q ^ 2 \$\$.
9) \$\$ 2 * n ^ 2 = q ^ 2 \$\$ Division by 2.
10) \$\$ q ^ 2 \$\$
is just
This follows from the representation of \$\$ q ^ 2 \$\$.
11) \$\$ q \$\$ is even This follows from the second preliminary consideration.
12) \$\$ q = 2 * m \$\$ \$\$ q \$\$ is even, i.e. double of any number \$\$ m \$\$.
13) \$\$ sqrt (2) = p / q = (2 * n) / (2 * m) \$\$ \$\$ p \$\$ and \$\$ q \$\$ are even and both divisible by \$\$ 2 \$\$.

### III. That is a contradiction to assumption.

\$\$ p \$\$ and \$\$ q \$\$ have a common factor. So \$\$ sqrt (2) \$\$ is not an abbreviated fraction after all.

### IV. The assumption is wrong, the claim is valid.

This proves: \$\$ sqrt (2) \$\$ is irrational.