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 5th 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:
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.
- 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|
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, 10five. The number 12ten is 22five, and the number 78ten is 303five.
When we want to multiply any number in base five times five all we have to do is move the decimal! 115 (which is 6 to us) times 105 (which is 5) is 110five (which is 30 to us). The product is still true regardless of the base. Likewise 22.2five (which is 12 and two-fifths) times 5 is 222five (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).