Maths-Chapter 1 The Number Systems

Natural numbers

N = {1, 2, 3, …}

Whole numbers

W = {0, 1, 2, 3, …}

Integers

Z = {…, -2, -1, 0, 1, 2, …}

Rational numbers

Numbers that can be written as p/q where p and q are integers and q ≠ 0

Irrational numbers

Numbers that cannot be written as p/q form

Real numbers

Set of all rational and irrational numbers

Z comes from

Zahlen (German word meaning "to count")

Rational from

Ratio or quotient

Equivalent rational numbers

E.g., 1/2 = 2/4 = 10/20 = 25/50

Rational number in lowest terms

p and q have no common factors other than 1

Is zero a rational number?

Yes, because 0 = 0/1

Are integers rational?

Yes, every integer can be written as m/1

Is every rational number an integer?

No, e.g., 3/5 is not an integer

Between any two rational numbers

There are infinitely many rational numbers

Definition of irrational number

Cannot be expressed as p/q

Examples of irrational numbers

√2, √3, √5, π, 0.101101110…

Decimal expansion of rational number

Terminating or non-terminating repeating

Decimal expansion of irrational number

Non-terminating non-repeating

√2 on number line

Diagonal of a square of side 1 unit

√3 on number line

Pythagorean construction using √2 and 1 unit

Real number line

Every point corresponds to a unique real number

π (pi)

Irrational number ≠ 22/7

Decimal expansion of 10/3

3.3… (non-terminating repeating)

Decimal expansion of 7/8

0.875 (terminating)

Decimal expansion of 1/7

0.142857… (non-terminating repeating)

Repeating block of 1/7

142857

0.333… as rational

1/3

1.272727… as rational

14/11

0.2353535… as rational

233/990

Non-terminating repeating numbers

Rational

Non-terminating non-repeating numbers

Irrational

Example of irrational between 1/7 and 2/7

0.150150015000…

Operations on real numbers

Closed under addition, subtraction, multiplication, division (except division by 0)

Sum of rational + irrational

Irrational

Product of non-zero rational and irrational

Irrational

Sum or product of two irrationals

May be rational or irrational

Rationalisation

Process of making the denominator rational

Identity: √a × √b

= √(ab)

Identity: √a / √b

= √(a/b)

Identity: (√a + √b)(√a - √b)

= a - b

Identity: (a + b)²

= a² + 2ab + b²

(5 + √7) + 2√5

= 5 + 2√5 + √7

(5 + √5)(5 - √5)

= 25 - 5 = 20

(3 + √7)²

= 9 + 6√7 + 7 = 16 + 6√7

(11 - √7)(11 + √7)

= 121 - 7 = 114

1/√2 rationalised

√2/2

1/(2 + √3) rationalised

(2 - √3)/1 = 2 - √3

5/(√3 - 5) rationalised

Multiply by (√3 + 5)/(√3 + 5)

1/(√7 + √3 + 2) rationalised

Multiply by (√7 + √3 - 2)/(√7 + √3 - 2)

Laws of exponents

am . an = am+n

Power of a power law

(am)n = amn

Division of exponents

am / an = am−n

Zero exponent

a⁰ = 1

Negative exponent

a^(-n) = 1/aⁿ

Fractional exponent

ⁿ√a = a^(1/n)

(2¹/³ × 2²/³)

= 2¹ = 2

(3⁴)^1/5

= 3⁴⁄⁵

(5^1/3)/(5^1/5)

= 5^(2/15)

(13^1/5)(17^1/5)

= (13 × 17)^1/5 = 221^1/5