How high can you count on your fingers? (Spoiler: much higher than 10) - James Tanton
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Let’s Begin…
How high can you count on your fingers? It seems
like a question with an obvious answer. After all, most of us have ten
fingers -- or to be more precise, eight fingers and two thumbs. This
gives us a total of ten digits on our two hands, which we use to count
to ten. But is that really as high as we can go? James Tanton investigates.
Additional Resources for you to Explore
Many cultures throughout history have used versions of positional notation to represent numbers, in various bases too. The Sumerians of around 3000 BCE and the Babylonians of 2000 BCE used a base-60 notational system, some cultures followed a base-20 notational system, others base 12, and modern western culture uses base 10. These base numbers might well have been chosen from counting on fingers and toes (with base 60 being a combination of base-10 and base-12 thinking).
Many historical units of weights and measures come in groups of twelve (there are twelve inches in a foot, for instance) most likely because it is easier to work with common fractions – halves, thirds, quarters – when one measures in groups of twelve. Modern western culture has vestiges of base-12 thinking in it: there are special names for the numbers 1 through 12 before moving onto a systematic naming system for 13 onwards, and people still talk of dozens and grosses. (Western culture has vestiges of base-20 thinking too. How does the Gettysburg address begin? How do you say 87 in French?)
Mathematically, positional thinking is an astonishingly powerful tool. Adding 468 and 379 is just a matter of adding 4 hundreds and 3 hundreds to get 7 hundreds, adding 6 tens and 7 tens to get 13 tens, and adding 8 ones and 9 ones to get 17 ones. We can say the answer is thus “seven hundred and thirteen-ty seventeen,” which is mathematically valid and correct. (The suffix “ty” in English is short for ten.) As society has agreed to write only single digits in each notational position, it prefers to regard 17 ones as 1 ten and 7 ones, and 13 tens as 1 hundred and 3 tens. This shows that the answer can also be regarded as 8 hundreds, 4 tens, and 7 ones, that is, as the number 847. This rewriting of counts of ones, tens, hundreds, and so forth, is called “carrying” in the standard long addition algorithm.
To see the absolute beauty and power of positional notation thinking throughout arithmetic, algebra, and beyond, explore the topic of Exploding Dots found here.
A BOLD AND AUDACIOUS PLAN
A new organization called the Global Math Project aims to engage students and teachers around the world in thinking and talking about the same appealing piece of mathematics during a series of annual Global Math Weeks. Inspired by the work of code.org, which makes coding accessible for millions of students across the globe, the Global Math Project will share the inherent joy, wonder, relevance, and meaning of mathematics with students everywhere and create a forum for the global celebration of creative mathematical thinking.
During the first Global Math Week, beginning 10.10.2017, one million students will experience the joy and wonder and astounding depth of place-value through the story of Exploding Dots. This mathematical story is regularly described by students, teachers, and parents alike as “mind blowing.” The Global Math Project website will serve as a portal to interactive content, follow-up materials, comments from participants, and public events at locations around the world.
One can learn all about the Global Math Project and learn how to sign up for Global Math Weeks at the website. Join in the fun!
Many historical units of weights and measures come in groups of twelve (there are twelve inches in a foot, for instance) most likely because it is easier to work with common fractions – halves, thirds, quarters – when one measures in groups of twelve. Modern western culture has vestiges of base-12 thinking in it: there are special names for the numbers 1 through 12 before moving onto a systematic naming system for 13 onwards, and people still talk of dozens and grosses. (Western culture has vestiges of base-20 thinking too. How does the Gettysburg address begin? How do you say 87 in French?)
Mathematically, positional thinking is an astonishingly powerful tool. Adding 468 and 379 is just a matter of adding 4 hundreds and 3 hundreds to get 7 hundreds, adding 6 tens and 7 tens to get 13 tens, and adding 8 ones and 9 ones to get 17 ones. We can say the answer is thus “seven hundred and thirteen-ty seventeen,” which is mathematically valid and correct. (The suffix “ty” in English is short for ten.) As society has agreed to write only single digits in each notational position, it prefers to regard 17 ones as 1 ten and 7 ones, and 13 tens as 1 hundred and 3 tens. This shows that the answer can also be regarded as 8 hundreds, 4 tens, and 7 ones, that is, as the number 847. This rewriting of counts of ones, tens, hundreds, and so forth, is called “carrying” in the standard long addition algorithm.
To see the absolute beauty and power of positional notation thinking throughout arithmetic, algebra, and beyond, explore the topic of Exploding Dots found here.
A BOLD AND AUDACIOUS PLAN
A new organization called the Global Math Project aims to engage students and teachers around the world in thinking and talking about the same appealing piece of mathematics during a series of annual Global Math Weeks. Inspired by the work of code.org, which makes coding accessible for millions of students across the globe, the Global Math Project will share the inherent joy, wonder, relevance, and meaning of mathematics with students everywhere and create a forum for the global celebration of creative mathematical thinking.
During the first Global Math Week, beginning 10.10.2017, one million students will experience the joy and wonder and astounding depth of place-value through the story of Exploding Dots. This mathematical story is regularly described by students, teachers, and parents alike as “mind blowing.” The Global Math Project website will serve as a portal to interactive content, follow-up materials, comments from participants, and public events at locations around the world.
One can learn all about the Global Math Project and learn how to sign up for Global Math Weeks at the website. Join in the fun!
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