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A2. Numeration: Exponents, Place Value, and Scientific Notation Numeration includes the process of naming numbers in both written and oral form. It encompasses ideas of exponents, scientific notation, and place value, as well as the basic operations and facts about numbers. As a simple example we write the numeral “352” and say “three hundred fifty two”. We use the exponent form as a shortcut method for writing repeated multiplication. In exponent form there are two important features, the base and the exponent. For natural number exponent (such as 1, 2, 3, …), the exponent tells us how many times the base is multiplied by itself. As an example 103 tells us we multiply the base, 10, together three times, where 3 is the exponent. A number with an exponent is often said to be "raised to the power" of. Exponent 103 = 10 x 10 x 10 = 1,000 “10 cubed” “10 raised to the 3rd power” Base As seen in the above example, our decimal number system uses exponent form to name numbers. In this numeration scheme, the base is 10 and the exponent is an integer (positive whole numbers, negative, whole numbers, and 0). The table below shows different powers of ten that are used in our conventional use of place value. Place Value Convention Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Exponential Form 103 = 10 x 10 x 10 = 1,000 102 = 10 x 10 = 100 101 = 10 100 = 1 1 = 0.1 10-1 = 10 1 = 0.01 10-2 = 100 1 = 0.001 10-3 = 1000 To practice using exponent form, visit the Exponent Learning Object. This study guide will discuss exponents in more detail in Module IV: Algebraic Concepts. In our decimal number system, the value of each digit depends on its position or place in the number. In the number 352.19, the 3 is in the hundreds place and the 1 is in the tenths place. The concept of place value can be extended indefinitely on either side of the decimal point as illustrated in the figure below. The value of each place to the left is 10 times the place to its right. Place Value 10 times larger 10 times smaller Every number can, therefore, be decomposed into the sum of the values of each place value. This decomposition is called the expanded form of a number. The table below illustrates this concept with rational numbers. Standard Form 352 634.79 Expanded Form 3 x 100 + 5 x 10 + 2 x 1 = 3 x 102 + 5 x 101 + 2 x 100 3 x 100 + 5 x 10 + 2 x 1 + 7 x 0.1 + 9 x 0.01 = 3 x 102 + 5 x 101 + 2 x 100 + 7 x 10-1 + 9 x 10-2 For very large or very small whole numbers, the convention is to use scientific notation. As of July 2008, the national debt was estimated at $9.54 trillion dollars. This can be expressed in place value as 9,540,000,000,000. Another way to express this number is in scientific notation as 9.54 x 1012 (as we moved the decimal place 12 places to the left). Mathematically, a number is in scientific notation if it is in the form a x 10 , where a is b a number from 1 up to but less than 10 (1 ≤ a < 10) and b is any integer. Therefore, we use positive exponents to represent very large numbers and negative exponents to express very small numbers. The table below gives you two more examples. Example Speed of Light Weight of a single M & M Standard Form 300,000 km/s 0.0019 pounds Scientific Notation 3.0 x 105 km/s 1.9 x 10-3 pounds To solve story problems relating to place value and scientific notation visit the Place Value Learning Object.