ClassX Video Lessons How high can you count on your fingers? (Spoiler: much higher than 10) – James Tanton
When you think about counting on your fingers, you probably assume you can only count up to ten. After all, most of us have ten fingers, or more accurately, eight fingers and two thumbs. This is why the numbers we use every day are called digits, just like our fingers.
But did you know you can count much higher than ten using just your fingers? In some cultures, people count up to twelve on one hand. How do they do it? Each finger has three sections, and you can use your thumb to point to each section. This way, you can count up to twelve on one hand. If you use your other hand to keep track of how many times you reach twelve, you can count up to 60!
Want to go even higher? Use the sections on your second hand to count twelve groups of twelve, reaching up to 144. But we can push this even further. Each finger has three sections and three creases, giving us six parts to count. This means you can count up to 24 on each hand. Using your other hand to track groups of 24, you can count all the way up to 576!
Is there a way to count even higher? Let’s think about something different. One of the coolest things in math is positional notation. This is where the position of a number gives it a different value, like in the number 999. Even though the number 9 is repeated, each one represents a different value.
We can use this idea with our fingers too. Instead of counting sections, let’s use each finger in two positions: up and down. This is perfect for a counting system called binary, which uses powers of two. In binary, each position doubles the value of the previous one. So, your fingers can represent values of 1, 2, 4, 8, and so on, up to 512.
With this method, you can count any number up to 1,023 using all ten fingers. For example, the number 7 is 4 + 2 + 1, so you would raise those three fingers. The number 250 is 128 + 64 + 32 + 16 + 8 + 2, so you would raise the corresponding fingers.
Can we go higher than 1,023? It depends on how flexible your fingers are! If you can bend each finger halfway, you have three positions: down, half-bent, and raised. This lets you count using a base-three system, reaching up to 59,048. If you can bend your fingers into four different positions, you can count even higher. The limit is up to your own flexibility and creativity.
Even with just two positions per finger, we’re already counting pretty efficiently. In fact, computers work in a similar way. They use tiny switches that can be either on or off, just like our fingers can be up or down. This base-two system allows computers to perform billions of calculations using just 1’s and 0’s.
So, next time you think about counting on your fingers, remember that you can go way beyond ten. With some creativity and a bit of math, your fingers can help you count to some pretty big numbers!
Try counting up to 60 using just one hand. Use your thumb to point to each of the three sections on your four fingers. Once you reach 12, use your other hand to keep track of how many times you’ve counted to 12. See how quickly you can reach 60!
Practice counting in binary using your fingers. Start with one hand and assign each finger a value: 1, 2, 4, 8, and 16. Raise the fingers that add up to a number you choose. For example, to show the number 5, raise the fingers for 4 and 1. Challenge a friend to guess the number you’re showing!
Experiment with counting in a base-three system by bending your fingers into three positions: down, half-bent, and raised. Try counting from 0 to 26 using this method. How high can you go with your finger flexibility?
In groups, create a relay race where each team member must count to a certain number using finger counting methods discussed in the article. The first team to complete the relay wins. This will help you practice different counting systems and work as a team!
Learn how computers use binary by simulating a simple calculation. Use your fingers to represent binary numbers and perform basic operations like addition. For example, add the binary numbers 101 (5) and 110 (6) using your fingers to see how computers perform calculations.
Here’s a sanitized version of the transcript:
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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. It’s no coincidence that the ten symbols we use in our modern numbering system are called digits as well.
But that’s not the only way to count. In some places, it’s customary to go up to twelve on just one hand. How? Well, each finger is divided into three sections, and we have a natural pointer to indicate each one, the thumb. That gives us an easy way to count to twelve on one hand. And if we want to count higher, we can use the digits on our other hand to keep track of each time we get to twelve, up to five groups of twelve, or 60.
Better yet, let’s use the sections on the second hand to count twelve groups of twelve, up to 144. That’s a pretty big improvement, but we can go higher by finding more countable parts on each hand. For example, each finger has three sections and three creases for a total of six things to count. Now we’re up to 24 on each hand, and using our other hand to mark groups of 24 gets us all the way to 576.
Can we go any higher? It looks like we’ve reached the limit of how many different finger parts we can count with any precision. So let’s think of something different. One of our greatest mathematical inventions is the system of positional notation, where the placement of symbols allows for different magnitudes of value, as in the number 999. Even though the same symbol is used three times, each position indicates a different order of magnitude.
So we can use positional value on our fingers to beat our previous record. Let’s forget about finger sections for a moment and look at the simplest case of having just two options per finger: up and down. This won’t allow us to represent powers of ten, but it’s perfect for the counting system that uses powers of two, otherwise known as binary. In binary, each position has double the value of the previous one, so we can assign our fingers values of one, two, four, eight, all the way up to 512.
Any positive integer, up to a certain limit, can be expressed as a sum of these numbers. For example, the number seven is 4 + 2 + 1, so we can represent it by having just these three fingers raised. Meanwhile, 250 is 128 + 64 + 32 + 16 + 8 + 2. How high can we go now? That would be the number with all ten fingers raised, or 1,023.
Is it possible to go even higher? It depends on how dexterous you feel. If you can bend each finger just halfway, that gives us three different states: down, half bent, and raised. Now, we can count using a base-three positional system, up to 59,048. And if you can bend your fingers into four different states or more, you can get even higher. That limit is up to you and your own flexibility and ingenuity.
Even with our fingers in just two possible states, we’re already working pretty efficiently. In fact, our computers are based on the same principle. Each microchip consists of tiny electrical switches that can be either on or off, meaning that base-two is the default way they represent numbers. Just as we can use this system to count past 1,000 using only our fingers, computers can perform billions of operations just by counting off 1’s and 0’s.
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This version maintains the original content while ensuring clarity and readability.
Counting – The action of finding the total number of items in a group by adding them one by one. – In math class, we practiced counting the number of squares in a grid.
Fingers – Parts of the hand used to represent numbers in basic arithmetic operations. – When learning to add, many students use their fingers to help them count.
Digits – Numerical symbols ranging from 0 to 9 used to represent numbers. – The number 345 is made up of three digits: 3, 4, and 5.
Positional – Relating to the position of digits in a number, which determines their value. – In the number 582, the positional value of 5 is in the hundreds place.
Notation – A system of symbols used to represent numbers or quantities. – Scientific notation is used to express very large or very small numbers in a compact form.
Binary – A number system that uses only two digits, 0 and 1, to represent values. – Computers use binary code to process data, as it is efficient for electronic circuits.
Values – The numerical worth or magnitude of a number or expression. – In the equation x + 3 = 7, the value of x is 4.
Systems – Methods or sets of rules used for organizing numbers or data. – The decimal and binary systems are two different ways to represent numbers.
Computers – Electronic devices that process data and perform calculations at high speed. – Computers can solve complex mathematical problems much faster than humans.
Calculations – Mathematical processes of finding an answer by using numbers and operations. – We used a calculator to perform the calculations needed for our math homework.
