**Properties of Numbers**

The following is a handy list of tips that you can remember about numbers (think about each one of these):

- The number line goes on till infinity in both directions, which is indicated by the arrows.
- The numbers on the number line are indicated by their respective signs, which shows that the line includes both positive and negative numbers.
- The integer zero is neutral and neither positive or negative.
- The number in the halfway of 1 and 2 is 1.5 and in the half way of -1 and -2 is -1.5.

- Between two rational numbers we can always find a rational number.
- Numbers, which are terminating and non-recurring are rational numbers. Similarly numbers, which are non-terminating and recurring are rational numbers.
- Set of natural numbers is contained in set of integers which is contained in set ofrational numbers which is further contained in set of real numbers, which is further contained in set of complex numbers.

- Addition as well as product of two real numbers is a real number.
- Two real numbers can be added or multiplied in either order i.e. Addition and multiplication of real numbers is commutative.
- Two real numbers cannot be subtracted or divided in either order i.e. Subtraction and Division of real numbers is not commutative.
- In set of real numbers we don’t define square root of negative numbers.

Rather we define set of complex numbers for this purpose. Any complex number z ={ a + ib , where a and b belongs to set of real numbers and i= }

**Properties of Zero**

- a x 0 =0 always for any real number a.
- a +0 = a always for any real number a.
- a – 0 = a always and 0 – a = -a for any real number a.
- 0/a =0 when “a” is a non-zero real number.
- a/0 is not defined i.e. we don’t define division by zero.
- a
^{0}=1for any non-zero real number a. - 0
^{0}is not defined.

0.45 ————terminating decimal is rational.

o.33333…. = 1/3 (can be expressed as a fraction). Hence all recurring decimals are rational numbers.

0.30402378….. —-cannot be expressed as a fraction, hence non-terminating and non -recurring decimals are irrational.

Correction is required as 0^0 is 1.

Dear Jyoti,

0^0 is not equal to 1. It is undefined.

Numbers which are Non – terminating and recurring numbers are irrational numbers.

These numbers can be represented in a p/q format where q is non zero so they will be categorized as rational.

Correction required ***Numbers, which are terminating and recurring are rational numbers. Similarly numbers, which are non-terminating and non-recurring are irrational numbers.***

Read carefully what is written : Non-terminating and recurring (both condition at the same time is implied) are rational numbers.

And what you have stated is also correct: i.e, numbers, which are non-terminating and non-recurring are irrational numbers

Nope Saloni 🙂

In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat (Non terminating & non recurring).

Hope it helps 🙂

Not terminating and recurring are irrational numbers right?

Non Terminating and Non recurring decimals are irrational numbers.

No they are rational, eg 1/3=0.333333…..