Which one of the following is not true?

Foundation

Mathematics Foundation

Real Number Properties

Question

If a is a rational number and

\sqrt{b}

is an irrational number then

a \sqrt{b}

is irrational number

The square root of every positive integer is always irrational

Every surd is an irrational number

\sqrt{2}

is an irrational number

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(a) All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction. An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers:

π = 3.141592 \dots

\sqrt{2} = 1.414213 \dots

Therefore,

\sqrt{2}

is an irrational number.

(b) Let us take a rational number

a = \frac{2}{1}

and an irrational number

b = \sqrt{2}

, then their product can be determined as:

a \times b = 2 \times \sqrt{2} = 2 \sqrt{2}

which is also an irrational number.

Therefore, if

a

is a rational number and

\sqrt{b}

is an irrational number than

a \sqrt{b}

is an irrational number.

(c) A surd is an irrational root of a rational number. So we know that surds are always irrational and they are always roots.

For example,

\sqrt{2}

is a surd since

2

is rational and

\sqrt{2}

is irrational.

Surds are numbers left in root form

\sqrt{}

to express its exact value. It has an infinite number of non-recurring decimals.

Therefore, every surd is an irrational number.

(d) Let us take a positive integer

4

, now square root of

4

will be:

\sqrt{4} = 2

which is not an irrational number

Hence, the square root of every positive integer is not always irrational.