"On the outside its full of leaves, but on the inside its bare and empty"

- SETH

Grade 9 Number Systems

Natural numbers The counting numbers 1, 2, 3 … are called natural numbers. The set of natural numbers is denoted by N. N = {1, 2, 3, …} Whole numbers If we include zero to the set of natural numbers, then we get the set of whole numbers. The set of whole numbers is denoted by W. W = {0, 1, 2, …} Integers The collection of numbers … –3, –2, –1, 0, 1, 2, 3 … is called integers. This collection is denoted by Z, or I. Z = {…, –3, –2, –1, 0, 1, 2, 3, …}

Rational numbers Rational numbers are those which can be expressed in the form integers and q Note: 1.

12 12 3 4 , where the HCF of 4 and 5 is 1 15 15 3 5 12 4 and are equivalent rational numbers (or fractions) 15 5 a Thus, every rational number ‘x ’can be expressed as x , where a, b are integers b

p , where p, q are q

0.

Example: , , , etc.

1 3 6 2 4 9

such that the HCF of a and b = 1 and b

0.

2. Every natural number is a rational number. 3. Every whole number is a rational number. [Since every whole number W can be expressed as 4. Every integer is a rational number.

There are infinitely many rational numbers between any two given rational numbers.

W ]. 1

Example:

Find 5 rational numbers Solution:

3 5 and . 8 12

3 3 3 9 9 6 54 8 8 3 24 24 6 144 5 5 2 10 10 6 60 12 12 2 24 24 6 144

It can be observed that:

54 55 56 57 58 59 60 144 144 144 144 144 144 144 3 55 7 19 29 59 5 8 144 18 48 72 144 12 55 7 19 29 59 3 5 . Thus, , , , and are 5 rational numbers between and 144 18 48 72 144 8 12

Irrational Numbers Irrational numbers are those which cannot be expressed in the form are integers and q Example: 0.

p , where p, q q

2, 7, 14, 0.0202202220.......

There are infinitely many irrational numbers. Real Numbers The collection of all rational numbers and irrational numbers is called real numbers. So, a real number is either rational or irrational. Note: Every real number is represented by a unique point on the number line (and vice versa). So, the number...

Natural numbers The counting numbers 1, 2, 3 … are called natural numbers. The set of natural numbers is denoted by N. N = {1, 2, 3, …} Whole numbers If we include zero to the set of natural numbers, then we get the set of whole numbers. The set of whole numbers is denoted by W. W = {0, 1, 2, …} Integers The collection of numbers … –3, –2, –1, 0, 1, 2, 3 … is called integers. This collection is denoted by Z, or I. Z = {…, –3, –2, –1, 0, 1, 2, 3, …}

Rational numbers Rational numbers are those which can be expressed in the form integers and q Note: 1.

12 12 3 4 , where the HCF of 4 and 5 is 1 15 15 3 5 12 4 and are equivalent rational numbers (or fractions) 15 5 a Thus, every rational number ‘x ’can be expressed as x , where a, b are integers b

p , where p, q are q

0.

Example: , , , etc.

1 3 6 2 4 9

such that the HCF of a and b = 1 and b

0.

2. Every natural number is a rational number. 3. Every whole number is a rational number. [Since every whole number W can be expressed as 4. Every integer is a rational number.

There are infinitely many rational numbers between any two given rational numbers.

W ]. 1

Example:

Find 5 rational numbers Solution:

3 5 and . 8 12

3 3 3 9 9 6 54 8 8 3 24 24 6 144 5 5 2 10 10 6 60 12 12 2 24 24 6 144

It can be observed that:

54 55 56 57 58 59 60 144 144 144 144 144 144 144 3 55 7 19 29 59 5 8 144 18 48 72 144 12 55 7 19 29 59 3 5 . Thus, , , , and are 5 rational numbers between and 144 18 48 72 144 8 12

Irrational Numbers Irrational numbers are those which cannot be expressed in the form are integers and q Example: 0.

p , where p, q q

2, 7, 14, 0.0202202220.......

There are infinitely many irrational numbers. Real Numbers The collection of all rational numbers and irrational numbers is called real numbers. So, a real number is either rational or irrational. Note: Every real number is represented by a unique point on the number line (and vice versa). So, the number...

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