Elementary School Mathematics/Number System and Place Value

The Number System and Place Value.

Different Number Systems -- Bases

Understanding What a Number System Is

Our normal number system (the decimal system), consists of 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). "Deci-" is the Latin prefix for ten. Thus, "decimal" denotes the number system that counts in tens. We can count from 0 to 9, but when we reach 9 and try to increment, we need to add another place in the number. In the decimal system, the number after 9 is 10. Each place is the number of the base to an incremented power. This may sound confusing, but try this example: 1234. 4*10⁰ + 3*10¹ + 2*10² + 1*10³ = 1234. The first digit (4) is equal to four 1's (1 = 10⁰). The second (3) is equal to three 10's (10 = 10¹). The third (2) is equal to two 100's (100 = 10²), and the fourth digit is equal to one 1000 (1000 = 10³). When writing in different systems, care must be taken to ensure that the reader isn't confused as to which system is currently being used. Thus, you will periodically see N₁₀ to denote a number in the decimal system, N₂ to denote a number in the binary system, N₈ for octal, or N₁₆ for hexadecimal. These (binary, octal, decimal, and hexadecimal) are the most commonly used systems, although you can theoretically have a system with any number as a base.

The Binary System

Computers use base two for their number system, meaning they only use the digits 0 and 1. This corresponds to whether an electric current is on or off, and the presence (or lack) of the current is what runs the applications on your computer. Starting with 0, the next binary value is 1, just like the decimal system. However, after 1, there is no such thing as a 2 in binary. If you realize that 2₁₀ = 1*2¹, you've taken the first step to understanding binary. 2₁₀ = 1*2¹ + 0*2⁰. Or, 2₁₀ = 10₂.

The Decimal System

The first number represents the ones, the second number going left represents the tens, and the then third number going left represents the hundreds. For example the number 123 is broken down into three different places, the 3 is in the one's column, the 2 is in the ten's column, and the 1 is in the hundreds column. So we take the left most digit and read it as one hundred and twenty three, we can break it down via addition as 100 + 20 + 3 = 123. Going farther than that will require a comma after the hundreds place, and before the thousands place.

Numbers can become quite large or small, we shall example the thousands, ten thousands, hundred thousands, millions, ten millions, hundred millions, billions, ten billions, hundred billions. You can see how the number system expands on itself and the farther to the left it goes the larger the number. Let us use the number 123,000 and notice that there are zeroes in the one's, ten's, and hundred's places so we can ignore them for now. A zero by itself is worth nothing, but put numbers to the left of that zero and the value will increase. 123,000 has the 3 in the thousand's place, the 2 in the ten thousand's place, and the 1 in the hundred thousand's place. Notice that we placed a comma before the 3 and after the last zero to the left. This is the way to separate the classes of numbers to make them easier to read. The 123,000 number is read as one hundred and twenty three thousand.

123,000,000 is another example and the 3 is in the millions, the 2 in the ten millions, and the 1 in the hundred millions. There is a second comma right before the millions mark, and another one right before the thousands mark. The student will note that every three spaces to the left a comma is added to make the number easier to read. The 123,000,000 is read as one hundred and twenty three million. The 3 is in the million's place, the 2 is in the ten million's place, and the 1 is in the hundred million's place.

The pattern repeats itself with 123,000,000,000 as you can see by now six numbers with a number in front of it goes into the millions, nine numbers with a number in front of it goes into the billions. So if there was 1,000,000 numbers the six zeroes before the 1 shows that it is one million and the 1,000,000,000 shows that it is one billion. The student should now learn the pattern of our number system. After the billion is the trillion 1,000,000,000,000 or twelve zeroes. These numbers are usually so high that they can record a nation's national debt and spending. The 123,000,000,000 is one hundred and twenty three billion, and if you guessed right 123,000,000,000,000 is one hundred and twenty three trillion.

Each space going left has a place value. The first digit is ones, and then the second digit is ten, and then the third digit is hundreds. Each value goes up 10 or rather multiplied by ten so 10 times 10 makes 100. 20 times 10 makes 200. 11 times 10 makes 110. That is each digit can hold 0 to 9, but after passing up the 9 or higher the value goes up to the next position. You cannot, for example place a two digit number in the one's place or above, you would have to place the left digit in the ten's place and the digit to the right of the left digit to the one's place. No place can hold a number higher than 9, the student has to learn later to carry the one in addition and subtraction in dealing with moving values from one place to another for addition and subtraction, which goes beyond the scope of this article for now.

So far we have covered positive Integer numbers. Later on in this article we will cover Rational numbers and other types. To do so means we will be learning what the decimal is, and what the numbers to the right of the decimal means. All whole numbers we covered so far are to the left of the decimal so 123 is really 123.00 and if we had to add in one-half or 1 over 2 to 123 we would get 1 over 2 is 0.50 and added to 123 equals 123.50 in which 0.50 is a number less than one but greater than zero.