## Learning Math: Number and Operations

# Place Value

## Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.

**In This Session**

**Base Two Numbers**

Part A:

Part A:

**Exponents and Logarithms**

Part B:

Part B:

**Place-Value Representation in Base Ten and Base Four**

Part C:

Part C:

Homework

Homework

In Session 2, while exploring the significance of the number 0, we mentioned its role in a place-value system. In this session, we will strengthen our understanding of place value by looking at systems based on numbers other than 10.

**LEARNING OBJECTIVES**

In this session, you will do the following:

- Interpret the value of numbers in base systems other than base ten
- Translate number values from one base system to another
- Relate base two numbers to circuitry and Boolean algebra
- Understand the rules of exponents and how they relate to logarithms

### Key Terms

**Previously Introduced:**

*e*

*e* is a transcendental number with the decimal approximation e = 2.7183. It is the base of natural logarithms. The value of *e* is found by taking the limit of (1 + 1/n)^{n} as n approaches infinity. This number arises in many applications — for example, in calculus as a function whose value and slope are everywhere equal, and in compound interest as a base when interest is computed continuously.

**New in This Session:**

base

The base of a number system is the number representing the value of each place in a representation. For example, “base ten” tells us that each digit in a number is some value of 10. In base ten, the number 1,234 represents four different values of 10: (1 • 10^{3}) + (2 • 10^{2}) + (3 • 10^{1}) + (4 • 10^{0}). Meanwhile, 1,234 in base five represents (1 • 5^{3}) + (2 • 5^{2}) + (3 • 5^{1}) + (4 • 5^{0}), and so on. These representations may appear identical, but if you perform the calculations, you’ll see that 1,234 in base ten is a different number from 1,234 in base five.

**exponent**

An exponent is a superscript number that indicates repeated multiplication of the base number or variable. It is also referred to as the power to which the base number or variable is raised.

**laws of exponents**

The laws of exponents are rules regarding simplification of expressions involving exponents. One such rule is that multiplying exponential expressions with the same base is equivalent to adding the exponents, so, for example, x^{3} • x^{4} = x^{7}. Another rule is that dividing exponential expressions with the same base is equivalent to subtracting the exponents, so, for example, x^{3} / x^{4} = x^{-1}.

**logarithm**

A logarithm is an exponent. The notation log_{2} 8 = 3 states that 2 is the base, 3 is the exponent, and 8 is the result. Logarithms can simplify complex exponentiation and multiplication problems numerically by using the laws of exponents to convert the more complicated operations into addition and subtraction. Most calculators are programmed with the LOG key (to perform logarithms to base ten) and the LN key (to perform logarithms to base *e*, approximately 2.718).

**scientific notation
**

A number is written in scientific notation when it is in the form F • 10^{E}, where the decimal F has exactly one non-zero digit to the left of the decimal point and E is an integer. Any real number can be written in scientific notation. For example, the number 23,831 is written as 2.3831 • 10^{4}, and the number 0.00123 is written as 1.23 • 10^{-3}.