REAL NUMBERS 1

Since ancient times, natural numbers have been used by all civilisations. The idea of zero and negative numbers took longer to understand. Integers emerged in the late 7th century in India. From this, 9th century Arabic mathematics merged with the decimal-positional numeral system.

Fractions were not used in the way they are today until around the 14th century.

Irrational numbers were discovered by the Pythagoreans around the 5th century BCE. However, for almost 2 000 years they were treated as geometric magnitudes.

The concept of real number has emerged over time through the study of functions. A German mathematician, Cantor, changed the way we thought about rational and irrational numbers in 1871. It was then that we started to think of these numbers as forming a single set, with their own characteristics.

Use what you have learned to solve problems

The number π in ancient times

The number π is an irrational number. However, over the centuries it has been given many different rational values. Some of these include:

Ancient Egypt (approx. 20th century BCE)

Ancient Babylon (approx. 20th century BCE)

Archimedes (3rd century BCE)

Ptolemy (2nd century)

Liu Hiu (5th century)

The organisation of different types of numbers

3,16

25/8

22/7

377/120

355/113

The sets of numbers that we know and have a clear structure are:

• Natural numbers, N

• If we include their opposites (negatives), we get the set of integers, Z

• If we include fractional numbers as well, we get the rational numbers, Q

• If we also include the irrational numbers, do we get a clearly structured set?

❚ Solve

1 a) Write three natural numbers and three integers that are not natural numbers. b) Write three rational numbers that are not integers and three irrational numbers. c)In your notebook, organise the numbers from your previous answers in a diagram like the one at the beginning of this exercise.

2 Given that the value of π is 3.14159265359… give the margin of error in each of the approximations from the previous page.

For example: 377 120 = 1 3,141666… 3,141592… The margin of error is less than 1 ten thousandth: error < 0.0001