Number Systems

A number is called a rational number if it can be written in the form p/q, where p and q are integers and q ≠ 0. Examples: 3/4, −5, 0 and 22/7 are all rational numbers. Every integer is rational because any integer n can be written as n/1.

A number that cannot be written in the form p/q (p, q integers, q ≠ 0) is called an irrational number. Examples: √2, √3, √5 and π are irrational.

Every rational number has a decimal expansion that either terminates (ends), such as 1/8 = 0.125, or is non-terminating recurring (a block of digits repeats forever), such as 1/3 = 0.333... Every irrational number has a decimal expansion that is non-terminating and non-recurring — it goes on forever with no repeating pattern, such as √2 = 1.41421356...

Worked Example: Insert a rational number between 1/4 and 1/2.

• Write both numbers with the same denominator: 1/4 = 2/8 and 1/2 = 4/8.
• Any fraction between 2/8 and 4/8 works, for instance 3/8.
• So 3/8 is a rational number between 1/4 and 1/2 (infinitely many such numbers exist).

Rational numbers and irrational numbers together are called real numbers. Every real number, rational or irrational, corresponds to exactly one point on the number line, and every point on the number line corresponds to exactly one real number.

Locating √2 on the number line: Mark point O at 0 and point A at 1. At A, draw AB perpendicular to the number line with AB = 1 unit, and join OB. By the Pythagoras theorem, OB² = OA² + AB² = 1² + 1² = 2, so OB = √2. With O as centre and OB as radius, draw an arc cutting the number line at point P. Then OP = OB = √2, so P represents √2.

This idea extends to locate √x for any positive real number x:

1. Mark a point A on the number line and mark B such that AB = x units.
2. From B, mark a point C such that BC = 1 unit.
3. Find the midpoint O of AC and draw a semicircle with centre O and radius OC.
4. Draw a line through B perpendicular to AC, meeting the semicircle at D. Then BD = √x.
5. With B as centre and BD as radius, draw an arc cutting the number line at E. The point E represents √x.

Successive magnification is the process of visualising a decimal number on the number line by repeatedly zooming into smaller and smaller intervals, as if viewed through a magnifying glass. For example, to locate 5.37: first zoom into the portion between 5 and 6; divide it into 10 equal parts and zoom into the part between 5.3 and 5.4; divide that part into 10 equal parts again to pinpoint 5.37. Repeating this process any number of times lets us locate any real number, including irrational numbers, as precisely as we like.

When rational and irrational numbers are combined by addition, subtraction, multiplication and division, the following results hold:

• Rational ± Irrational = Irrational (e.g., 3 + √2 is irrational).
• Non-zero Rational × Irrational = Irrational, and non-zero Rational ÷ Irrational = Irrational (e.g., 5√3 is irrational).
• Irrational ± Irrational or Irrational × Irrational can be either rational or irrational depending on the numbers involved. For example, √2 + (−√2) = 0 (rational), but √2 + √3 is irrational; also √5 × √5 = 5 (rational), but √5 × √3 = √15 (irrational).

For positive real numbers a and b, the following identities are very useful for simplifying expressions with square roots:

1. √a × √b = √(ab)
2. √a ÷ √b = √(a/b)
3. (√a + √b)(√a − √b) = a − b
4. (a + √b)(a − √b) = a² − b
5. (√a + √b)² = a + 2√(ab) + b

Worked Example: Simplify (5 + √7)(5 − √7).

• Using identity (a + √b)(a − √b) = a² − b with a = 5, b = 7:
• (5 + √7)(5 − √7) = 5² − 7 = 25 − 7 = 18.

Worked Example: Simplify (√3 + √7)².

• Using (√a + √b)² = a + 2√(ab) + b with a = 3, b = 7:
• (√3 + √7)² = 3 + 2√21 + 7 = 10 + 2√21.

For a real number a > 0 and positive integers p, q, the expression a^p/q is defined as the positive q-th root of a^p: a^p/q = (a^p)^1/q. In particular, a^1/n is the positive n-th root of a, so a^1/2 = √a and a^1/3 is the cube root of a.

The familiar laws of exponents for integers extend to rational exponents, for a, b > 0:

• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m−n
• (a^m)ⁿ = a^mn
• a^m × b^m = (ab)^m
• a⁰ = 1

Worked Example: Simplify 2^2/3 × 2^1/3.

• Using a^m × aⁿ = a^m+n: 2^2/3 × 2^1/3 = 2¹ = 2.

Worked Example: Evaluate 64^−2/3.

• 64^1/3 = 4, since 4³ = 64.
• So 64^2/3 = (64^1/3)² = 4² = 16.
• Therefore 64^−2/3 = 1/64^2/3 = 1/16.

Rationalising the denominator means converting a fraction whose denominator contains a surd (an irrational square root) into an equivalent fraction with a rational denominator, by multiplying the numerator and denominator by a suitable expression called the rationalising factor.

• The rationalising factor of √a is √a, since √a × √a = a.
• The rationalising factor of a + √b is a − √b (its conjugate), since (a + √b)(a − √b) = a² − b.
• The rationalising factor of √a + √b is √a − √b, since (√a + √b)(√a − √b) = a − b.

Worked Example: Rationalise the denominator of 1/(3 + √2).

• Multiply numerator and denominator by the conjugate (3 − √2):
• 1/(3 + √2) × (3 − √2)/(3 − √2) = (3 − √2)/(3² − (√2)²) = (3 − √2)/(9 − 2) = (3 − √2)/7.

Worked Example: Rationalise the denominator of 1/(√7 − √5).

• Multiply numerator and denominator by (√7 + √5):
• 1/(√7 − √5) × (√7 + √5)/(√7 + √5) = (√7 + √5)/((√7)² − (√5)²) = (√7 + √5)/(7 − 5) = (√7 + √5)/2.