“Although counting to small numbers is universal in human cultures, counting to large numbers requires a system to keep track. Our Hindu–Arabic numeral system is based on two ideas (Wu, 2011). First, there are only ten symbols called “digits” (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Second, all possible counting numbers are created by using those ten digits in different places—the concept of “place value.” Any number, then, is the sum of the products of each “face” (digit) and its “place”; for example, 1,926 is 1 thousand, 9 hundreds, 2 tens, and 6 ones.” (Clements & Sarama, 2014) Many students are familiar with Base 10 Numeration through the base ten blocks found in many elementary classrooms, this collection of units, rods, flats and cubes helps form students understanding of the base ten system. Students learn that ten units make a rod, ten rods form a flat and ten flats make a cube. Depending on the age and understanding of the students each piece might represent a different value, for most students’ units represent 1, rods 10, flats 100 and cubes 1 000. These tangible blocks allow students to visualize the base ten number system. One of the key features of the base ten numeral system is that it is positional, a numbers placement indicates it’s value, this allows the same ten digits to be used to produce an infinite number of numbers. Positional notation allows a small set of digits to be used rather than creating a new symbol for increasing magnitude.
The base ten number system rose to common usage after being spread from India by middle eastern traders as early as the fifth century. But the Hindu-Arabic numerals were not the only base ten numeration system in use, the Egyptians and Aegean among others used base ten systems, however they did not use positional notation, meaning they needed to create a new symbol for each increasing magnitude. The Chinese, Aztecs and Babylonians began using positional notation without zero. The Hindu-Arabic numeral system introduced using zero as a place holder which increased the accuracy of numerical communication. Base ten was likely chosen as humans have ten fingers. Base ten can also be divided by two, five and ten to represent common fractions and divisions.
Value of each decade
Students are first introduced to using tens and ones in their primary years, developing their comfort with the first two decades to the left of the decimal. As numbers move along the decades to the left of the decimal point their value increases by a power of ten, or it is ten times bigger than the number directly to its right. On the left of the decimal decades are grouped into periods with each period having a repeating pattern. Starting with the units the places go ones, tens, hundreds before going to the next grouping in the thousands, thousands, ten thousand, hundred thousand before repeating in the millions, billions and so on.
For the number 987 654 321. 345 678
| Period | Millions | Thousands | Units | Decimal | ||||||||||||
| Place | Hundred million | Ten million | Million | Hundred thousand | Ten thousand | Thousands | Hundreds | tens | ones | . | Tenths | Hundredths | Thousandths | Ten thousandths | Hundred thousandths | millionths |
| Value | 100 000 000 | 10 000 000 | 1 000 000 | 100 000 | 10 000 | 1 000 | 100 | 10 | 1 | 0.1 | 0.01 | 0.001 | 0.000 1 | 0.000 01 | 0.000 001 | |
| Example | 900 000 000 | 80 000 000 | 7 000 000 | 600 000 | 50 000 | 4 000 | 300 | 20 | 1 | 0.3 | 0.04 | 0.005 | 0.000 6 | 0.000 07 | 0.000 008 | |
Increasing and decreasing powers
The base ten place value chart is built on powers of ten or ten multiplied by itself a specified number of times. Powers of ten become important as students become stronger mathematicians and begin to use math in other subject areas to represent key ideas. Powers of ten are used in biology and chemistry to represent the size of groups and in statistics to communicate about large groups of people. In chemistry Avogadro’s Number 6.022 × 10²³ is used to communicate about how many atoms are in a mole. 6.022 × 10²³ is much easier to read than 602 200 000 000 000 000 000 000. By relying on an understanding of the powers used to move left and right along the place value chart students can communicate about and understand very larger and very small numbers. As discussed above each decade is one power of ten or ten times greater than the period to its right. Students can move left or to increasing powers along the place value chart by multiplying by tens. Similarly, it is possible to represent decreasing powers by dividing by 10 or using a negative exponent with the ten to represent smaller decimals. The table below demonstrates a few powers of ten and how they are calculated, notice for the positive exponents that the number of places to the left of the decimal is the same as the power, in contrast when working with negative exponents the number of places the right of the decimal matches the exponent. In this form a positive power of ten indicates a whole number while a negative power of ten indicates a decimal.
| Power | Symbol | Equation | Number | Place |
| -4 | 10-4 | 1/ (10x10x10 x10) | 0.000 1 | Ten thousandths |
| -3 | 10-3 | 1/ (10x10x10) | 0.001 | Thousandths |
| -2 | 10-2 | 1/(10×10) | 0.01 | Hundredths |
| -1 | 10-1 | 1/10 | 0.1 | Tenths |
| 0 | 100 | 1 | Ones | |
| 1 | 101 | 10 | 10 | Tens |
| 2 | 102 | 10×10 | 100 | Hundreds |
| 3 | 103 | 10x10x10 | 1 000 | Thousands |
| 4 | 104 | 10x10x10 x10 | 10 000 | Ten thousand |
| 5 | 105 | 10x10x10 x10x10 | 100 000 | Hundred thousand |
| 6 | 106 | 10x10x10 x10 x10x10 | 1 000 000 | Millions |