What If We Counted Differently?
College Mathematics | Math for Teachers | Part I | Math for Teachers I
Other Bases
We use base ten so naturally that it can feel like the only possible way to write numbers. But it isn't. Our number system is built around groups of ten because we use a base-ten place-value system. We could build a place-value system around groups of five, six, eight, two, or almost any other whole-number base.
Exploring another base gives us a chance to see place value from a new perspective. Instead of relying on familiar patterns, we have to ask:
What does each place represent?
When do we regroup?
Which digits are available?
What does a written number actually mean?
In this lesson, we will step outside base ten and discover which parts of place value are fundamental—and which parts depend specifically on the base we are using.
What You'll Learn
By the end of this lesson, you should be able to:
Explain what the base of a number system tells us.
Describe how a positional number system is organized.
Identify the digits available in a given base.
Explain how regrouping works in bases other than ten.
Determine the place values in another base.
Interpret numbers written in bases such as base five or base six.
Represent a quantity in a different base.
Explain why the written numeral 10 has different values in different bases.
Compare the structure of base ten with other positional systems.
Explain why working in another base can strengthen understanding of place value and arithmetic.
Start Here: Why Ten?
Think about the number 347. We automatically interpret it as 3 hundreds + 4 tens + 7 ones. But why are the places ones, tens, hundreds, thousands? Why do we regroup after ten ones? Why does the digit to the left become one ten? The answer is not that place value has to work this way. It is because we use base ten. If we chose a different grouping size, the same place-value idea could create a very different-looking number system.
What Is a Base?
The base tells us how many units of one place are needed to make one unit of the next place. In base ten:
10 ones = 1 ten
10 tens = 1 hundred
10 hundreds = 1 thousand
So each place is worth ten times as much as the place immediately to its right. That is why we call it base ten.
The Base Determines the Grouping Size
Suppose instead that we used base five. Then we would regroup whenever we collected five units. So:
5 ones = 1 group of five
5 groups of five = 1 group of twenty-five
The places would represent 1, 5, 25, 125, ... rather than 1, 10, 100, 1,000, .… The place-value idea stays the same. The grouping size changes.
Which Digits Can We Use?
Our base-ten system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Notice that the largest digit is one less than the base. That pattern continues in other bases.
Base Five
Available digits: 0, 1, 2, 3, 4. There is no digit 5. Once we have five ones, we regroup them as one unit in the next place.
Base Six
Available digits: 0, 1, 2, 3, 4, 5. Once we have six units in a place, we regroup.
This is exactly the same reason we do not have a single digit representing “ten” in base ten. Once we reach ten ones, we write 10 which means one ten and zero ones.
What Does 10 Mean?
This is one of the most important ideas in the lesson. The numeral 10 does not always represent the quantity ten. It means: 1 group of the base + 0 ones. So, in Base Ten 10₁₀ means 1 ten + 0 ones so its value is ten. In Base Five 10₅ means 1 five + 0 ones so its value is five. In Base Six 10₆ means 1 six + 0 ones so its value is six. The written symbols look the same. Their values depend on the base.
Exploring Other Bases
Let's explore base six. The available digits are 0, 1, 2, 3, 4, 5. After 5, we cannot write a single digit 6. Instead, six ones regroup as 10₆. So counting in base six begins 0, 1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 20.… This may look strange at first. That's useful. When the notation stops feeling automatic, we are forced to think about what the places actually mean.
Building Numbers in Base Six
Consider 243₆. What does this mean? The base-six places are
ones
sixes
thirty-sixes
because 6² = 36. So 243₆ means
2 groups of 36
4 groups of 6
3 ones
Its value in base ten is 2 × 36 + 4 × 6 + 3 which gives 72 + 24 + 3 = 99. So 243₆ = 99₁₀.
Try It
Consider 324₅. The base-five places are 25, 5, 1. So 324₅ means 3 × 25 + 2 × 5 + 4 × 1. Calculate:
3 × 25 = ______
2 × 5 = ______
4 × 1 = ______
Building a Quantity in Another Base
Suppose we want to represent the base-ten quantity 17 in base five. Start by grouping in fives. 17 contains 3 groups of 5 with 2 ones remaining. So in base five, 17₁₀ = 32₅ because 3 × 5 + 2 = 17. This is not changing the quantity. We are changing the representation.
Regrouping Works the Same Way
In base ten, 10 ones ↔ 1 ten. In base six, 6 ones ↔ 1 six. In base five, 5 ones ↔ 1 five. The fundamental idea is the same. When we collect enough units to match the base, we trade them for one unit in the next place. This is what place-value regrouping really means.
Why Learn Other Bases?
You might reasonably ask “Why should future elementary teachers learn a number system they probably will not teach directly?” Because working in another base lets us experience place value as learners again. Base ten is so familiar that many of its rules feel obvious. In another base, questions that usually disappear into habit become visible. For example:
Why do we regroup?
Why are only certain digits allowed?
What does a place represent?
Why does carrying work?
Why does borrowing work?
What does 10 actually mean?
Those are exactly the kinds of questions children face while learning base ten.
Becoming a Beginner Again
Imagine learning base six for the first time. You may need to stop and think about something as simple as “What comes after 5?” You may make mistakes. You may write an illegal digit. You may forget when to regroup. You may understand a procedure one moment and feel confused the next. That experience is valuable for future teachers. It can help us remember that ideas that feel automatic to an expert may require significant reasoning for a beginner.
A Teacher's View: Seeing Base Ten More Clearly
Suppose a child writes 4 + 7 = 11 and struggles to understand why regrouping is needed. As an adult, you may see the regrouping immediately. But if you solve a similar problem in base six, the process may suddenly require conscious thought. For example 4₆ + 3₆. There are seven base-ten units altogether. But base six only allows the digits 0 through 5. So six of those units must be regrouped. The answer is 11₆. That experience can help us understand why regrouping is a place-value idea, not merely a written algorithm.
Symbols and Quantities Are Different
One of the deeper ideas in this lesson is that a number and the symbols used to represent that number are not the same thing. For example the quantity we call ten in base ten can be represented as 10₁₀. But in base five, the same quantity is 20₅ because 2 groups of five + 0 ones = ten. Different numeral. Same quantity. This distinction between a quantity and its representation appears throughout mathematics.
Student Thinking: What Do You Notice?
Situation A
A student working in base six writes 6₆.
Think About:
Is this a valid numeral in base six?
What digits are available?
What should happen when we reach six units?
Situation B
A student says “10 always means ten.”
Think About:
Why does that statement seem reasonable?
Why is it not true across different bases?
How would you explain what 10 actually represents?
Situation C
A student writes 23₅ and says “That means twenty-three.”
Think About:
What is the student doing?
What does the 2 represent in base five?
What is the actual base-ten value of 23₅?
Situation D
A student converts 14₁₀ into base five and writes 24₅.
Think About:
Is the student correct?
How could you verify the answer?
What decomposition of 14 does the numeral 24₅ represent?
Common Misconceptions
“Base ten is just how numbers work.”
Base ten is one possible place-value system. The structure of positional notation can be built around other bases.
“10 always means ten.”
The value of 10 depends on the base. It represents one group of the base and zero ones.
“Every base uses the digits 0 through 9.”
A base uses digits from zero through one less than the base. For example, base six uses 0, 1, 2, 3, 4, 5.
“Changing the base changes the quantity.”
Changing the base changes how a quantity is represented. The quantity itself can remain the same.
“Other bases are just conversion exercises.”
Conversion can be useful, but the deeper purpose is to understand:
grouping,
place value,
regrouping,
notation,
and the relationship between symbols and quantities.
“If another base feels confusing, I must not understand place value.”
In fact, some confusion is useful here. The unfamiliar system forces you to examine ideas that base ten has made automatic.
Check Your Understanding
Try these without looking back.
What does the base of a number system tell us?
Which digits are available in base five?
Which digits are available in base six?
What does 10₆ represent?
What are the first three place values in base six?
What is the base-ten value of 23₅?
Why is 6₆ not a valid base-six numeral?
How would you represent the base-ten quantity 8 in base five?
What stays the same when moving from base ten to another positional base?
Why might learning another base be particularly useful for future teachers?
Teaching Challenge
Choose either base five or base six.
Part 1: Identify the Digits
The digits available are:
Part 2: Identify the First Four Places
First:
Second:
Third:
Fourth:
Part 3: Build a Number
Choose a valid three-digit number in your base. Explain the value of each digit. Convert the number to base ten.
Part 4: Anticipate a Misconception
Write one mistake a beginner might make. What underlying idea might still be developing? What question or model could help?
Reflect
Working in another base can feel surprisingly uncomfortable. That discomfort is part of the point. Which parts of base-ten place value did you realize you had been taking for granted? What became harder when the familiar digits and regrouping points changed? Now think from a teaching perspective. How might struggling with another base help you better understand a child who is learning base ten for the first time? And finally, what parts of place value are truly fundamental, regardless of which base we use?