LESSON 7
DECIMALS
Expressing Parts of a Whole in Base Ten
What is a Decimal?
A decimal is a way of writing numbers that are not whole, using a decimal point to separate the whole number part from the fractional part. Every digit to the right of the decimal point represents a fraction with a denominator that is a power of ten.
For example, 3.75 means 3 wholes, 7 tenths, and 5 hundredths. Decimals and fractions represent the same values — 0.5 and 1/2 are exactly equal. Decimals are especially useful in money, measurement, and science.
Decimal Place Value
Each position in a decimal number has a specific value. Moving left from the decimal point: ones, tens, hundreds. Moving right: tenths, hundredths, thousandths, and so on. Understanding place value is the key to reading and writing decimals correctly.
Comparing Decimals
To compare decimals, line up the decimal points and compare digit by digit from left to right. Add zeros to the right of shorter decimals if needed — this does not change their value. The first position where the digits differ tells you which number is greater.
Adding and Subtracting Decimals
To add or subtract decimals, always line up the decimal points first. Fill in any missing places with zeros so the columns align. Then add or subtract exactly as you would with whole numbers, keeping the decimal point in the same position in the answer.
Multiplying and Dividing Decimals
To multiply decimals, ignore the decimal points and multiply as whole numbers. Then count the total decimal places in both factors and place the decimal point that many places from the right in the answer. To divide by a decimal, move the decimal point in both numbers to make the divisor a whole number.
Key Facts
The decimal system is based on the number 10 and is called the base-10 system. It was developed in ancient India and spread through Arabic mathematicians.
Adding zeros to the right of a decimal does not change its value — 0.5, 0.50, and 0.500 are all equal.
Money is one of the most common uses of decimals — $3.75 means 3 dollars, 7 dimes, and 5 pennies.
A repeating decimal like 0.333... (one third) goes on forever. It can be written as 0.3̄ or as the fraction 1/3.