decimals

2010

Whole numbersThe position of a digit within a number indicates

the value of the digit. The further the digit is to the left in a number, the larger the place value.

Each place to the left of another has a value which is 10 times larger. Each place to the right has a value which is 1/10 of the previous position.

Decimal partsWhole numbers have units as their smallest place value. It is possible to keep going to smaller values than units. To show values smaller than units, a decimal point, or dot, is placed after the units.

We now want to extend this idea to the right of the units place. Write a period to the right of the units place. This is called the decimal point. Each digit to the right of that decimal point will represent a fraction whose denominator is a power of 10. The first place to the right of the decimal point is the tenths place.

Decimal partsThe value of the positions to the left and right

of the decimal point are shown in the table below.

DefinitionA decimal fraction is a fraction whose denominator is 10, 100, 1000, or some other power of 10.

Reading or Writing Decimals in Words

Step 1 Read the digits to the left of the decimal point as a whole number.

Step 2 Read the decimal point as the word “and.”

Step 3 Read the digits to the right of the decimal point as a whole number followed by the place value of the rightmost digit.

Representing Decimals on the Number Line

We can represent decimals on a number line as rational numbers. We divide each unit of the number line into tenths, hundredths, or thousandths, and then we find the image point of the decimal.

Comparing and Ordering Decimals

1. Make the number of decimal places the same for each number, by adding zeros.

2. Ignore the decimal point.3. The greatest number is the greatest decimal.

ADDITION OF DECIMALS

1. Arrange the decimals vertically with the decimal points lined up.

2. Add extra zeros if necessary so that each addend has the same number of decimal places.

3. Add the digits with the same place value from right to left, like natural numbers.

4. Insert the decimal point directly below the decimal points of the addends.

SUBTRACTION OF DECIMALS1. Write the decimals vertically and line up

the decimal points.2. Add extra zeros if necessary, so that each

number has the same number of decimal places.

3. Subtract the digits with the same place value from right to left, as for natural numbers. Borrow if necessary.

4. Insert the decimal point directly below the decimal points of the numbers.

MULTIPLICATION OF DECIMALS

1. Ignore the decimal points and multiply the decimals as whole numbers.

2. Add the number of decimal places in the two decimals. The result is the number of decimal places in the product.

3. Insert the decimal point in the product so that it has the correct number of decimal places.

DIVISION OF DECIMALS

Dividing a Whole Number by a Whole Number

To divide a whole number by another whole number, where the quotient is not a natural number, divide as in division of whole numbers and then place the decimal point and the required zeros to dividend.

DIVISION OF DECIMALS

Dividing a Decimal by a Whole Number

1. Ignore the decimal point and divide the decimal like a whole number.

2. Add zeros to the dividend as necessary.3. Insert the decimal point in the quotient so

that it has the correct number of decimal places.

DIVISION OF DECIMALSDividing a Decimal by a Decimal Number

1. Make the divisor into a whole number by moving the decimal point to the right. Count how many places the decimal point moves.

2. Move the decimal point in the dividend the same number of places to the right. Add zeros to the right of the dividend if necessary, before moving the decimal point.

3. Follow the steps for dividing a decimal by a whole number.

DIVISION OF DECIMALSDividing a Whole Number by a Decimal

Number

1. Make the divisor into a whole number by moving the decimal point to the right. Count how many places the decimal point moves.

2. Perform the division. Add zeros to the right of the dividend if necessary.

3. Move the decimal point in the quotient the same number of places to the right.

RELATIONSHIP BETWEEN FRACTIONS AND DECIMAL NUMBERS

Terminating Decimals

When we divide the numerator of a fraction by the denominator, we obtain a decimal. If the decimal has a finite number of decimal places, we call it a terminating decimal.

To convert a terminating decimal to a fraction, follow the steps.

1. Ignore the decimal point and write the number.

2. Draw the fraction bar.3. Write power of 10 to the denominator.

The number of 0’s is as many as the decimal digits.

4. Simplify the fraction.

Repeating Decimals

If a decimal has a fractional part whose digits repeat endlessly, we call it a repeating decimal.

RELATIONSHIP BETWEEN FRACTIONS AND DECIMAL NUMBERS

To convert a repeating decimal to a fraction, follow the steps.

1. Write the number without the decimal point.

2. Subtract the non-repeating part from this number.

3. Write the difference as the numerator of the fraction.

4. Write the denominator as a sequence of 9’s and zeros. The number of 9’s is as many as the number of repeating digits of the fraction part. The number of zeros is as many as the number of non-repeating digits of the fraction part.

5. Simplify the fraction.