20061208
01.b. Number Systems: Decimal
So let's start with that numeral system that you're the most familiar with, that you've been working with since before kindergarten - decimal. "Decimal" comes from the latin "decimus" which means a tenth. Which brings us to the actual crux of the matter; that decimal is a base 10 number system. It's called "base 10" because are ten different values, zero through nine, that any of the numeral places can hold. We could create a 'value table' to describe or dissect the values of a numeral in a given place in the number, even though you don't think of it that way.
You know this; of course you do. There's the ones position, there's the tens, the hundreds, the thousands and so on. If we look at a number like 1536. One thousand five hundred and thirty six. We tend to evaluate it working from right to left, from the least significant value to the greater values. We have six in the ones position, three in the tens, five in the hundreds, and one in thousands. And this all seems like second nature to us right now, because it is. Because this is our native number system. Most people speculate that this is because we have ten digits on our hands, ten fingers (counting our thumbs).
But it's important for us to hang onto these ideas that we don't even think about in decimal, these ideas of
• position (place) value
• base value (range of valid numbers for each position)
• place significance (least significant & most significant values of placement)
because it's all going to get pretty kooky when we start to do some mathematics on other base systems. Actually, it's going to start to get a little kooky right now if we look a little deeper into the way we understand base 10, the way we construct the value table we used a moment ago.
What we think of as the ones place might better be called the zero-eth position. Huh? Stay with me. In terms of number sets, you may recall the "natural numbers", the counting numbers. Negative numbers are not part of the set of natural numbers. Historically there has been some issue about whether this set includes zero or not, because zero was a rather advanced abstraction. Many ancient societies, including the Romans, didn't have a numeral for zero. It isn't naturally where we begin to count. We tend to think that if there aren't any, we can't count them. But if we think of our value table as an inventory sheet, then we can begin to appreciate that zero is, indeed, a natural number. It's a number that is less than one and so it is the first counting number.
Back to the the least significant digit. We call it the ones place because 1 is what we multiply by that positional numeral in order to get the number value. But why exactly is that? If we think about it, why is the first position in a base 10 system not ten? Well, it's because the place value is the result of ten (the base) to the zero-eth power. 100 = 1. We'll be working with exponents but this is a part that you may not remember and will have to memorize. Anything to the power of zero equals one. It just does. Therefore, the least significant decimal digit is ones. Six times one is six. The next place significance value is the tens because ten to the power of one equals ten. Three time ten equals thirty. Ten to the power of two (ten times ten) is a hundred and five times a hundred is five hundred. Ten to the power of three (ten times ten times ten) is a thousand and so on.
Next ==>
You know this; of course you do. There's the ones position, there's the tens, the hundreds, the thousands and so on. If we look at a number like 1536. One thousand five hundred and thirty six. We tend to evaluate it working from right to left, from the least significant value to the greater values. We have six in the ones position, three in the tens, five in the hundreds, and one in thousands. And this all seems like second nature to us right now, because it is. Because this is our native number system. Most people speculate that this is because we have ten digits on our hands, ten fingers (counting our thumbs).
But it's important for us to hang onto these ideas that we don't even think about in decimal, these ideas of
• position (place) value
• base value (range of valid numbers for each position)
• place significance (least significant & most significant values of placement)
because it's all going to get pretty kooky when we start to do some mathematics on other base systems. Actually, it's going to start to get a little kooky right now if we look a little deeper into the way we understand base 10, the way we construct the value table we used a moment ago.
What we think of as the ones place might better be called the zero-eth position. Huh? Stay with me. In terms of number sets, you may recall the "natural numbers", the counting numbers. Negative numbers are not part of the set of natural numbers. Historically there has been some issue about whether this set includes zero or not, because zero was a rather advanced abstraction. Many ancient societies, including the Romans, didn't have a numeral for zero. It isn't naturally where we begin to count. We tend to think that if there aren't any, we can't count them. But if we think of our value table as an inventory sheet, then we can begin to appreciate that zero is, indeed, a natural number. It's a number that is less than one and so it is the first counting number.
Back to the the least significant digit. We call it the ones place because 1 is what we multiply by that positional numeral in order to get the number value. But why exactly is that? If we think about it, why is the first position in a base 10 system not ten? Well, it's because the place value is the result of ten (the base) to the zero-eth power. 100 = 1. We'll be working with exponents but this is a part that you may not remember and will have to memorize. Anything to the power of zero equals one. It just does. Therefore, the least significant decimal digit is ones. Six times one is six. The next place significance value is the tens because ten to the power of one equals ten. Three time ten equals thirty. Ten to the power of two (ten times ten) is a hundred and five times a hundred is five hundred. Ten to the power of three (ten times ten times ten) is a thousand and so on.
Next ==>
// posted by Mikki Adams @ 8.12.06 
