# Learning and using our base ten place value number system: theoretical perspectives and twenty-first century uses

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## Abstract

This paper makes a proposal, from the perspective of a research mathematician interested in mathematics education, for broadening and deepening whole number arithmetic instruction, to make it more relevant for the twenty-first century, in particular, to enable students to deal with large numbers, arguably an essential skill for modern citizenship. It suggests that, rather than being content with accurate calculation for numbers with few digits, arithmetic instruction should develop understanding of the overall structure of the base ten place value system (including decimal fractions), and develop comfort with large (and small) numbers that are known only approximately. This would include emphasizing the role of the constituent parts (here called *place value parts*) into which place value notation decomposes any decimal number. Beyond this, understanding large numbers, and how numbers function in the real world, entails approximation. The notion of *relative error* (and its variant, *percent error*) is presented as a useful way to think about error. Relative error behaves in manageable ways under addition and multiplication, and any real number can be well approximated (i.e., with small relative error) by a base ten number with a small number of (non-zero) place value parts. Examples are given of using approximate computation with large numbers to clarify interesting questions.

## Keywords

Place value arithmetic Five stages of place value Place value parts Relative error Approximate calculation## References

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