Download 10 = 10 x 10 x 10 = 1000

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.