Recently in math class we have been working with decimals. This is one of the areas in math where a solid foundation in place value will take you a long way. If for instance you want to know the product of 15 and 3.2 it’s nice to know that 10 x 3.2 is simply a decimal move and 5 x 3.2 is just half again. I am always surprised to find that place value is not well understood by 5^{th} grade. I guess I need to stop being surprised. But perhaps place value is just a little more complicated than we give it credit for, and maybe there is just not enough emphasis in the earlier grades.

Let’s look at the base-ten place value table:

Thousands | Hundreds | Tens | Ones |
Tenths | Hundredths |

1000 | 100 | 10 | 1 |
1/10 | 1/100 |

10^{3} |
10^{2} |
10^{1} |
10^{0} |
10^{-1} |
10^{-2} |

The reason for the ease of computations with 10 and powers of 10 is that ten is the base. This is why when you multiply 10 times anything “you just add a zero.” I try to teach my students to use the more mathematically accurate phrasing of “you just move the decimal,” but it’s no more challenging. The ones, or units, are the center of the place value system – all other places are exponentially larger or smaller than the base to the zero power.

*Exponent review:*

*All numbers to the zero power are equal to 1**Negative exponents are just powers of base (ten in this case) in the denominator.*

To understand place value, I find that it is helpful to examine what life is like in other bases. Let’s look at a base-five place value table like the one above.

One hundred twenty-fives | Twenty-fives | Fives | Ones |
Fifths | Twenty-fifths |

1000 | 100 | 10 | 1 |
1/10 | 1/100 |

5^{3} |
5^{2} |
5^{1} |
5^{0} |
5^{-1} |
5^{-2} |

In base 5 there is no numeral 6, or 5 for that matter. There are only 0, 1, 2, 3, and 4. Once you get to 5 the notation is 10 or more clearly written, 10_{five}. The number 12_{ten} is 22_{five}, and the number 78_{ten} is 303_{five}.

When we want to multiply any number in base five times five all we have to do is move the decimal! 11_{5} (which is 6 to us) times 10_{5} (which is 5) is 110_{five} (which is 30 to us). The product is still true regardless of the base. Likewise 22.2_{five} (which is 12 and two-fifths) times 5 is 222_{five} (12.4 * 5 = 62).

The place value system is dense, but when we understand the places and bases of our number system everything makes a whole lot more sense (or perhaps cents or pence if you live on the other side of the pond fence).

You write

*******

This is why when you multiply 10 times anything “you just add a zero.” I try to teach my students to use the more mathematically accurate phrasing of “you just move the decimal,” but it’s no more challenging.

*******

Even more mathematically accurate is to consider the framework as fixed, and then the decimal point is fixed. So multiplying by 10 has the effect of moving all the digits one place to the left, and if the number was a whole number then a zero is to be put in the units (ones) position.

and

Once you start with different bases you are forced to explain that 10 is not ten, it’s the way we write ten in the decimal system. Which is a very good thing ( the forcing !). Letting Kindergarten kids think that ten is 10 sticks in the mind for years and seriously gets in the way of understanding place value.

Beyond the acrobatics of operating in base 5, I am wondering if there is any real application. As an elementary teacher, it will take a long time to explain and re-explain and re-re-explain this to my students. Why not just stay in base ten land?

I think that kids better understand base-ten after leaving and coming back from another base. I really appreciated Howard’s comment above – “10” is not “ten” unless you are in the base-ten number system.

Place value is complex and I don’t think you can build a real understanding of it without going into that complexity. Sometimes when I go over this in my class it feels like we are decoding the number system – it’s like a game…

That being said, I don’t dwell for very long. A little mini lesson, a few gasps from those excited students who are getting it and we’ll come back to it another day.

Base 5, not much use. However, when communicating with computers at a low level numbers are represented in the hexadecimal system, base sixteen using 0 to 9 and then A to F. Sixteen is then written 10, being one sixteen and no ones.

The combination %20 pops up now and again in URLs. The % says “base sixteen” and the 20 is thirtytwo (two sixteens), which is the character code for the space character.

Inside the computer everything is either ON or OFF, and the base 2 system is the description.

Thanks for this Spencer. My students have been exploring the same ideas. I have a resources to offer and a mathematical question. Resource first: Eames’ 1970 Power of Ten video makes a huge impact on students’ capacity to understand what it means to shrink or grow by a power of ten. It is old, but not dated at all! It is free and easy to find on YouTube. Now my question: Why is any x^0 always one? In your example this seems to be true even in other bases.

I love the Eames’ Power of Ten – I make a point of showing it to my students as well.

Great question about x^0. Many texts seem to “define” that x^0 equals 1 for all bases where x <> 0 and we are meant to just accept it. I think this is pretty much where I was, but your question made me look into it further as I knew that was not going to be an adequate justification.

Watch what happens when we multiply numbers with like bases.

Keep in mind the following property of exponents:

x^m * x^n = x^m+n

So

x^2 * x^-1 = x^1 or (x*x) * (1/x) = x (which is x^1)

Likewise

x^1 * x^-1 = x^0 or x * (1/x) = x/x = 1, so x^0 = 1

I’m not clear on the case where zero is the base as 0^0 would be undefined as would any negative exponent. Anyway – what is the point of base-zero?