4th December 2019

artofproblemsolving

14

# Is the square root of 2 a rational or irrational number?

Proof that the

**square root of 2**is**irrational**. Assume is**rational**, i.e. it can be expressed as a**rational**fraction of the form , where and are two relatively prime integers. Now, since , we have , or . Since is even, must be even, and since is even, so is .Is the square root of 2 A whole number?

A proof that the

**square root of 2**is irrational. Let's suppose √**2**is a rational**number**. Then we can write it √**2**= a/b where a, b are**whole numbers**, b not zero. We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction.1

## Is the square root of six a rational number?

Now this is the contradiction: if a is even and b is even, then they have a common divisor (2). Then our initial assumption must be false, so the

**square root of 6**cannot be**rational**. There you have it: a**rational**proof of irrationality.2

## Is the square root of 11 a rational or irrational number?

Sal proves that the square root of any prime number must be an irrational number. For example, because of this proof we can quickly determine that √

**3**, √**5**, √**7**, or √11 are irrational numbers. Created by Sal Khan. Proofs concerning irrational numbers.3

## Is the square root of 2 A rational numbers?

A proof that the square

**root**of**2**is irrational. Let's suppose √**2**is a**rational number**. Then we can write it √**2**= a/b where a, b are whole**numbers**, b not zero. We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction.4

## Is the square root of 5 a rational or irrational number?

1. This irrationality proof for the

**square root of 5**uses Fermat's method of infinite descent: Suppose that √**5**is rational, and express it in lowest possible terms (i.e., as a fully reduced fraction) as mn for natural**numbers**m and n. Then √**5**can be expressed in lower terms as 5n − 2mm − 2n, which is a contradiction.5

## Can a square root be a rational number?

So it

**could**not have been made by squaring a**rational number**! This means that the value that was squared to make 2 (ie the**square root**of 2) cannot be a**rational number**. In other words, the**square root**of 2 is irrational.6

## Is the square root of eight a rational number?

At 4:16 khan explains how [A x (

**square root of 8**) is irrational. But since 'A' is a variable, couldn't 'A' be written as a principle**root**, like the**square root of 8**. Because then you would just multiply the two**roots**, making the**square root**of 64, which is a**rational number**.7

## Is the square root of an irrational number?

Real

**numbers**have two categories: rational and**irrational**. If a**square root**is not a perfect**square**, then it is considered an**irrational number**. These**numbers**cannot be written as a fraction because the decimal does not end (non-terminating) and does not repeat a pattern (non-repeating).8

## What is a square root of 2?

The square

**root**of**2**, or the (1/**2**)th power of**2**, written in mathematics as √**2**or**2**1⁄**2**, is the positive algebraic number that, when multiplied by itself, gives the number**2**. Technically, it is called the principal square**root**of**2**, to distinguish it from the negative number with the same property.9

## Is the number 50 rational or irrational?

Answer :

**50**is a**rational**number because it can be expressed as the quotient of two integers:**50**÷ 1. Related Links :**Rational**and**Irrational**Numbers.10

## Is the square root of 10 a rational number?

The

**square root of 10**is a single**number**. That**number**is irrational, which means that its decimal expansion goes on forever without repeating, and cannot be expressed as a fraction. All integer**square roots**that are not perfect squares of other integers will be irrational.11

## Is the number 0 rational or irrational?

Any number which doesn't fulfill the above conditions is irrational. It can be represented as a ratio of

**two**integers as well as ratio of itself and an irrational number such that zero is not dividend in any case. People say that is rational because it is an integer.12

## Is the square root of 12 a rational number?

Yes, absolutely. There is no

**rational number**whose**square**equals**12**. The principal**square root of 12**is 2√3 which is approximately 3.46 . Since**12**is not a**square**, its**square root**is irrational.13

## What is value of Root 5?

Table of Squares and Square Roots

NUMBER | SQUARE | SQUARE ROOT |
---|---|---|

2 | 4 | 1.414 |

3 | 9 | 1.732 |

4 | 16 | 2.000 |

5 | 25 | 2.236 |

14

## Is the square root of 49 a rational number?

EXPLANATION: The

**square root**of 36 is 6, which is an integer, and therefore**rational**. (T/F): The**square root of 49**is**an irrational number**. False. EXPLANATION: The**square root of 49**is 7, which is an integer, and therefore**rational**.15

## Is the square root of a rational number?

Alternatively, an irrational number is any number that is not rational. It is a number that cannot be written as a ratio of

**two**integers (or cannot be expressed as a fraction). For example, the square root of**2**is an irrational number because it cannot be written as a ratio of**two**integers.16

## Is the number 2 a rational number?

In mathematics, a

**rational number**is any**number**that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a**rational number**. Irrational**numbers**include √**2**, π, e, and φ.17

## Is the number a irrational number?

**Irrational Number**. An

**irrational number**is a

**number**that cannot be expressed as a fraction for any integers and .

**Irrational numbers**have decimal expansions that neither terminate nor become periodic. Every transcendental

**number is irrational**.

18

## Is the square root of 17 a rational or irrational number?

Prove that the

**square root of 17**is irrational. Here is a proof that the**root**of 2 is irrational, taken from the notes of a course I taught last semester. We use what is called a proof by contradiction. where p and q are relatively prime integers, i.e. that cannot be**rational**and thus must be irrational.19

## What are numbers with square roots that are whole numbers?

Squaring, which we learned about in a previous lesson (exponents), has an inverse too, called "finding the square root." Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers: 1,

**4**,**9**,**16**,**25**,**36**,**49**,**64**,**81**,**100**…20

## Is square root of 2 a real number?

The

**real numbers**include all the rational**numbers**, such as the integer −5 and the fraction 4/3, and all the irrational**numbers**, such as √**2**(1.41421356, the**square root of 2**, an irrational algebraic**number**). Included within the irrationals are the transcendental**numbers**, such as π (3.14159265).