Types of Numbers (Junior Cert Mathematics): Revision Notes

Types of Numbers

Natural Numbers $(\mathbb{N})$:

• What Are They? Positive whole numbers.

• Examples:$1,2,3,4…$ Integers $(\mathbb{Z})$:

• What Are They? Whole numbers that can be positive, negative, or zero.

• Examples:$-3, -2, -1, 0, 1, 2, 3,$.. Real Numbers ($\mathbb{R}$):

• What Are They? All numbers on the number line, including whole numbers, fractions, and decimals.

• Examples:$-3, -1.4, 0.2, 6, \frac{7}{2}, \sqrt{8}, \dots$ Rational Numbers $(\mathbb{Q})$:

• What Are They? Numbers that can be written as a fraction (where both the top and bottom are whole numbers).

• Examples: $-5, 3, \frac{1}{2}, -\frac{9}{4}, \dots$ Irrational Numbers:

• What Are They? Numbers that cannot be written as a simple fraction. Their decimal form goes on forever without repeating.

• Examples: $\sqrt{3}, \sqrt{2}, \pi, \dots$ Prime Numbers:

• What Are They? Numbers greater than 1 that have only two factors: 1 and the number itself.

• Examples: $2, 3, 5, 7, 11, 13, 17, \dots$ Composite Numbers:

• What Are They? Numbers greater than 1 that are not prime (they have more than two factors).

• Examples: $6, 9, 15, 20, \dots$

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Extra Detail

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1. Natural Numbers

Key Points:

1. Natural Numbers: The counting numbers 1, 2, 3, 4, 5, and so on are called natural numbers.
2. No End: Natural numbers go on forever, meaning there is no largest natural number.
3. No Zero**:** Zero is not a natural number. Natural numbers start from 1.

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Types of Natural Numbers:

4. Even Natural Numbers: These are natural numbers that can be divided by 2 without leaving a remainder. Examples: {2, 4, 6, 8, 10, ...}

5. Odd Natural Numbers: These are natural numbers that cannot be divided by 2 without leaving a remainder. Examples: {1, 3, 5, 7, 9, ...}

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Consecutive Natural Numbers:

• Consecutive natural numbers follow one another in order.
• For example, 6, 7, 8, 9 are consecutive natural numbers.
• Even numbers can also be consecutive, such as 2, 4, 6, 8.

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2. Integers

Key Points:

6. Integers: The set of integers includes all whole numbers, both positive and negative, as well as zero.

• Positive Integers: {1, 2, 3, 4, 5, ...}
• Negative Integers: {..., -5, -4, -3, -2, -1}
• Zero: 0 is considered an integer, but it is neither positive nor negative.

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Adding and Subtracting Integers:

3. Real Numbers (R)

Key Points:

7. Real numbers include natural numbers, integers, fractions, and decimals.
8. Every point on the number line corresponds to a real number. Examples:

• 3 (a natural number)
• 2 (an integer)
• 0.75 (a rational number)
• $\sqrt{2}$ (an irrational number)

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4. Rational Numbers (Q)

Key Points:

9. Rational numbers include integers, fractions, and finite or repeating decimals.
10. Rational numbers can always be expressed as a simple fraction. Examples:

• $\frac{1}{2}$ (a fraction)
• 4 (an integer, which can be written as $\frac{4}{1}$)
• 0.75 (a decimal, which can be written as $\frac{3}{4}$)

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5. Irrational Numbers

Key Points:

11. Irrational numbers are numbers that cannot be written as a fraction where both the top number (numerator) and the bottom number (denominator) are whole numbers.
12. They have unique, non-repeating decimal patterns. For example, the number $\sqrt{2}$ is an irrational number because there's no fraction of whole numbers that equals $\sqrt{2}$ .

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6. Prime Numbers

Key Points:

13. Prime numbers have exactly two distinct factors: 1 and the number itself.
14. The number 2 is the smallest and the only even prime number. Examples: 2, 3, 5, 7, 11, 13

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7. Composite Numbers

Key Points:

15. Composite numbers have more than two factors.
16. All even numbers greater than 2 are composite because they can be divided by 2. Examples:

• 4 (factors are 1, 2, 4)
• 6 (factors are 1, 2, 3, 6)
• 9 (factors are 1, 3, 9)

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