 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum  # Unit 3

## How Big is Infinity? Ancient Mathematicians

Throughout the ages, the notion of infinity has been a source of mystery and paradox, a philosophical question to ponder. As a mathematical concept, infinity is at the heart of calculus, the notion of irrational numbers — and even measurement. This unit explores how mathematics attempts to understand infinity, including the creative and intriguing work of Georg Cantor, who initiated the study of infinity as a number, and the role of infinity in standardized measurement.

## Unit Goals

• Ideas of infinity come to light when considering number and geometry, the worlds of the discrete and the continuous.
• Incommensurability is the idea that there is no measurement unit that fits into some two quantities a whole number of times.
• Incommensurability led to the discovery of irrational numbers.
• Irrational numbers have decimal expansions that never end and never repeat.
• Two sets are the same size if their elements can be put into one-to-one correspondence with one another.
• The size of a set is its cardinality.
• There is more than one type of infinity.
• The sets of rational and real numbers are examples of two different sizes of infinity.
• To properly describe the different sizes of infinity, a new definition of number is required.
• Given a set of any size, one can create a larger set by taking the subsets of the original set.

# Video Transcript It takes courage to push beyond the boundaries of understanding, to both explore and explain the boundlessness of the infinite. Numbers and counting are real — intrinsic to our everyday life. But acknowledging their existence ties us to the existence of the infinitude.

# Textbook Despite its nebulous reality, the concept of infinity has long teased at mathematicians' minds. Around 500 BC it manifested in the form of incommensurable quantities, a concept akin to heresy in the view of many, particularly the followers of Pythagoras.