Decimals: Understanding Decimal Representation and Operations

Study Notes

Decimal

A decimal is any number expressed using an integral part followed by one or more digits after a point. Decimals can represent fractions of a whole unit or measure, allowing us to express very precise quantities when working with measurements such as length, time, mass, volume, and temperature. In general, decimals have three main components: the whole part, the fractional part, and the digit immediately following the decimal point.

Decimal Places

The place value of a digit in a decimal number refers to its position relative to the rightmost digit. The first place value column from the right has a place value equal to 10^(-n), where n is the number of digits to the left of it. For example, if we consider a two-digit decimal number like 17.98, the place values of each digit are shown below:

Place Value	Digit
10^-0	8
10^-1	9
10^-2	7
10^-3	1

In this pattern, the place value increases by a factor of ten for each additional digit to the left of the current digit.

Significant Figures

When writing down the decimal representation of a number, we only need to include significant figures. A significant figure represents the smallest measurement that could realistically be made under normal circumstances. It may be accompanied by a power of ten indicating how many orders of magnitude smaller or larger the quantity is. For example, a ruler might show measurements accurate to within 0.1 inches, so you would record your result as 4.9 instead of 4.99.

Converting Decimals to Fractions

To convert a decimal to a fraction, locate the decimal point. If there's an integer part before the decimal point, move the decimal point to the right until the integer part is gone. Then, make the remaining numbers 00, put them on top of each other, draw lines through all the zeros except for the ones next to the non-zero digits. This gives you your fraction form.

For example, let's convert the decimal number 3.75 to a fraction:

            _______________
        3 . 7  5 = 375/1000
              --
              7

So, the conversion is ( \frac{375}{1000} ).

Operations with Decimals

Operations with decimals follow the same rules as operations with integers. However, aligning the decimal points requires some attention to detail. When adding or subtracting decimals, line up the decimal points; the smaller number should be moved to the right under the larger number, with zeros added as necessary to fill out the columns on the right.

Addition Example

Let's say we want to add these two decimals:

     1.025 + 1.45
    ---------
    +-----+
    1.475

Therefore, the sum is 1.475.

Subtraction Example

Now, let's subtract the smaller decimal number from the larger one:

   1.025 - 1.45
   ---------
    -+-+
    0.575

So, the difference is 0.575.

Decimal Rounding

Decimal rounding is the process of approximating a decimal number by truncating or rounding it to a specified number of decimal places. The rules for rounding depend on the context and the requirements of the problem at hand.

Integer Part Rounding

If the digit to the right of the rounding point is less than 5, we can simply truncate that digit. For example, if we want to round the decimal number 1.4982 to two decimal places, we get 1.498.

Fractional Part Rounding

If the digit to the right of the rounding point is greater than or equal to 5, we need to consider the next digit to decide whether to add 1 to the digit before it. For instance, if we want to round the decimal number 1.4982 to two decimal places, we first look at the second digit from the left: 8. Since 8 is greater than 5, we add 1 to the previous digit: 1. 49 (\rightarrow) 1.50. Now, since there's a zero after the 0, we don't need to change anything else. So, the rounded value is 1.50.

Decimal arithmetic is used extensively in everyday life, from calculating tips and change to working with measurements in science and engineering. By understanding the basics of decimal representation, operations, rounding rules, and conversion to fractions, we can effectively work with decimals in various contexts.

Description

This quiz covers the basics of decimals, including their components, decimal places, significant figures, conversion to fractions, operations, and rounding. Learn how to work with decimals effectively in various contexts.