 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.