Strand 3 — Number

Number Systems

1st Year · 2nd Year · 3rd Year (Junior Cert)

✓By the end of this lesson students will be able to classify numbers as natural numbers, integers, rational numbers, or real numbers.
✓By the end of this lesson students will be able to understand the relationships and hierarchy between different number sets (N, Z, Q, R).
✓By the end of this lesson students will be able to express very large or very small numbers using scientific notation.
✓By the end of this lesson students will be able to perform calculations involving numbers in scientific notation.
✓By the end of this lesson students will be able to identify irrational numbers, including surds, and understand their place within the real number system.

Key concepts

Natural Numbers (N)

These are the counting numbers, starting from 1. They are positive whole numbers.

N = {1, 2, 3, 4, ...}

Integers (Z)

These include all natural numbers, their negative counterparts, and zero. They are whole numbers, whether positive, negative, or zero.

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational Numbers (Q)

Any number that can be expressed as a fraction a/b, where 'a' and 'b' are integers and 'b' is not zero. This includes all integers, terminating decimals, and recurring (repeating) decimals.

Q = {a/b | a ∈ Z, b ∈ Z, b ≠ 0}

Irrational Numbers

Numbers that cannot be expressed as a fraction a/b. Their decimal representation is non-terminating and non-recurring. Examples include π (pi) and the square roots of non-perfect squares.

Surds

A surd is an irrational number that is the root of an integer (e.g., square root, cube root) that cannot be simplified to a whole number. For example, √2, √3, √5 are surds. √4 is not a surd because it simplifies to 2. Basic simplification involves extracting perfect square factors from under the root sign. More advanced manipulation of surds is typically covered at Leaving Cert Higher Level.

Real Numbers (R)

This set includes all rational and irrational numbers. Essentially, any number that can be placed on a number line.

R = Q ∪ {Irrational Numbers}

Scientific Notation

A way of writing very large or very small numbers concisely. A number in scientific notation is written as a × 10ⁿ, where 'a' is a number between 1 and 10 (1 ≤ a < 10) and 'n' is an integer.

a × 10ⁿ (where 1 ≤ a < 10 and n ∈ Z)

Key facts to remember

1The number sets are nested: Natural Numbers (N) ⊂ Integers (Z) ⊂ Rational Numbers (Q) ⊂ Real Numbers (R).
2Natural numbers are positive whole numbers {1, 2, 3, ...}.
3Integers include positive and negative whole numbers and zero {..., -2, -1, 0, 1, 2, ...}.
4Rational numbers can be written as a fraction a/b (where b ≠ 0), including terminating and recurring decimals.
5Irrational numbers cannot be written as a fraction; their decimal expansions are non-terminating and non-recurring.
6Real numbers comprise all rational and irrational numbers.
7A surd is an irrational root of an integer (e.g., √2, √5).
8Scientific notation expresses a number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.

Worked examples

Example 1

Classify the following numbers into the smallest possible set from N, Z, Q, R: -5, 0.75, √7, 10, 2/3, -√9.

I-5: This is a negative whole number. It belongs to the set of Integers (Z). Since all integers are rational and real, Z is the smallest set.

II0.75: This is a terminating decimal, which can be written as the fraction 3/4. It belongs to the set of Rational Numbers (Q).

III√7: Since 7 is not a perfect square, √7 is an irrational number. All irrational numbers are real numbers. R (specifically, Irrational) is the smallest set.

IV10: This is a positive whole number. It belongs to the set of Natural Numbers (N). Since all natural numbers are integers, rational, and real, N is the smallest set.

V2/3: This is a fraction. It belongs to the set of Rational Numbers (Q).

VI-√9: First, simplify √9 to 3. So, -√9 = -3. This is a negative whole number. It belongs to the set of Integers (Z).

Answer

-5 ∈ Z, 0.75 ∈ Q, √7 ∈ R (Irrational), 10 ∈ N, 2/3 ∈ Q, -√9 ∈ Z

When classifying, always look for the most specific set the number belongs to.

Example 2

Express the following numbers in scientific notation: (a) 345,000,000 (b) 0.000000067

I(a) For 345,000,000:

II - Move the decimal point to the left until there is only one non-zero digit before it: 3.45

III - Count how many places the decimal point moved: 8 places.

IV - Since the original number was large (greater than 1), the power of 10 is positive.

V - Result: 3.45 × 10⁸

VI(b) For 0.000000067:

VII - Move the decimal point to the right until there is only one non-zero digit before it: 6.7

VIII - Count how many places the decimal point moved: 8 places.

9 - Since the original number was small (less than 1), the power of 10 is negative.

10 - Result: 6.7 × 10⁻⁸

Answer

(a) 3.45 × 10⁸, (b) 6.7 × 10⁻⁸

The 'a' part of a × 10ⁿ must always be between 1 and 10 (1 ≤ a < 10).

Example 3

Simplify the surd √72.

IFind the largest perfect square factor of 72. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The perfect square factors are 1, 4, 9, 36. The largest is 36.

IIRewrite √72 as √(36 × 2).

IIIUse the property √(ab) = √a × √b: √36 × √2.

IVCalculate the square root of the perfect square: 6 × √2.

VThe simplified surd is 6√2.

Answer

6√2

This type of simplification is an introduction to surds. More complex operations are typically covered at Leaving Cert Higher Level.

Common mistakes

✗Including 0 or negative numbers in the set of Natural Numbers (N).
✗Forgetting that all integers are also rational numbers.
✗Incorrectly determining the sign of the exponent 'n' in scientific notation (positive for large numbers, negative for small numbers).
✗Not ensuring that the 'a' part of scientific notation (a × 10ⁿ) is between 1 and 10.
✗Confusing surds with square roots of perfect squares (e.g., thinking √9 is a surd).

Exam tips

★Memorise the definitions and symbols for N, Z, Q, and R, and understand their relationships.
★Practice converting numbers to and from scientific notation, paying close attention to the decimal point placement and the sign of the exponent.
★When classifying numbers, always identify the *smallest* set the number belongs to.
★For surds, remember that only roots of non-perfect squares are irrational (surds); roots of perfect squares are rational.

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