Understanding Decimal Numbers

Study Notes

Decimal Numbers

Decimal numbers are used primarily in base ten systems, where each digit represents a multiple of a power of ten. For instance, the number 87.29 is written using four digits: (8 \times 10^{1}) + (7 \times 10^0) + (2 \times 10^{-1}) + (9 \times 10^{-2}). In other bases, like binary, octal, or hexadecimal, decimal numbers would need to follow different rules. However, our everyday math uses only integers and decimals.

The decimal point separates the integer part from the fractional part of a number. It's important to know how many places behind it we can count since it determines the precision of measurements. For example, if you have a ruler with a decimal place resolution of one tenth of an inch, the maximum possible value you could measure exactly is 1.00 inches; any larger value will need to involve a nonzero number to its right.

Decimal numbers can also have a fractional part expressed by a negative exponent indicating the denominator of the fraction, such as (7 \times 10^{-1}) or (2 \times 10^{-2}). This means that we need to consider both positive and negative powers of ten when working with decimal numbers.

The beauty of decimals lies in their simplicity. We don't have to learn any new symbols or rules; we just use the same ones we already know from integers with some slight adjustments. For instance, if you want to divide ten by two point zero, which is an infinite decimal, you can just write it as [ \frac{10}{2.0} = 5 ] instead of using fractions like [\frac{\overline{10}}{2} = \frac{10}{\overline{2}}= 5].

Description

Learn about decimal numbers and how the decimal point separates the integer and fractional parts. Explore the concept of negative exponents and the precision of measurements with decimal values. Understand the simplicity of decimals and how they relate to fractions.