Hello there, and welcome to today's mathematics lesson! Today, we are going to explore decimal fractions. By the end of this lesson, you will understand what decimal fractions are, how to read and write them, how to convert between decimals and ordinary fractions, and how to perform all four operations—addition, subtraction, multiplication, and division—with decimal numbers.
Let us begin with the fundamental question: What exactly is a decimal fraction?
A decimal fraction, or simply a decimal, is a fraction whose denominator can be expressed as ten or some higher power of ten, such as one hundred, one thousand, and so on. In other words, any fraction that can be written with a denominator of ten, one hundred, one thousand, ten thousand, or any power of ten, qualifies as a decimal fraction.
For example, seven-tenths, written as 7/10, is a decimal fraction. So is thirteen-hundredths, or 13/100, and eight hundred fifty-one thousandths, or 851/1000.
Now, here is something interesting. Even fractions that do not immediately show powers of ten in their denominators can be decimal fractions. Take three-fifths, or 3/5. If we multiply both numerator and denominator by two, we get 6/10. Since this equals six-tenths, three-fifths is indeed a decimal fraction.
Similarly, seven-eighths, or 7/8, becomes 875/1000 when we multiply numerator and denominator by one hundred twenty-five. Since one thousand is ten cubed, seven-eighths is also a decimal fraction.
To write decimal fractions more compactly, we use a special notation. Instead of writing the denominator, we place a dot called the decimal point in the proper position.
For instance, three-tenths becomes 0.3. Two hundred thirteen-hundredths becomes 2.13. Seven-thousandths becomes 0.07. And fifty-nine ten-thousandths becomes 0.0059.
When there is no whole number to the left of the decimal point, we generally write a zero there. So instead of writing point seven two, we write 0.72. Similarly, point zero zero four becomes 0.004.
Every decimal number has two parts. In 2.4, for example, the two is called the integral part, and point four is the decimal part. Together, they mean two plus four-tenths.
Here is an important property: adding zeros after the decimal part does not change the value. Three point five is exactly the same as three point five zero, or three point five zero zero, and so on.
Now, let us learn how to read decimal numbers correctly.
The integral part is read as a whole number, and the decimal part is read digit by digit. So 21.45 is read as twenty-one point four-five. 152.639 is read as one hundred fifty-two point six-three-nine. And 0.08 can be read as point zero-eight or zero-point zero-eight.
Understanding place value is crucial. To the left of the decimal point, we have units, tens, hundreds, thousands, and so on. To the right, the first place is tenths, the second is hundredths, the third is thousandths, and the fourth is ten-thousandths.
In the number 5.46, the five sits in the units place, the four in the tenths place, and the six in the hundredths place.
Let us now see how to convert a decimal number into an ordinary fraction.
Remove the decimal point from the numerator. In the denominator, write one followed by as many zeros as there are digits in the decimal part. Then simplify to lowest terms.
For example, 0.27 becomes 27/100. Three point six four becomes 364/100, which simplifies to 91/25, that is three and sixteen twenty-fifths, 3 16/25.
Five point seven five zero becomes 5750/1000, which reduces to 23/4, that is five and three-quarters, 5 3/4.
Converting fractions to decimals works differently depending on the denominator.
When the denominator is already ten, one hundred, one thousand, or similar, simply count from the right of the numerator and place the decimal point after as many digits as there are zeros in the denominator.
Two hundred fifty-nine over ten becomes 25.9. Two hundred fifty-nine over one hundred becomes 2.59. Two hundred fifty-nine over one thousand becomes 0.259.
When the denominator is not a power of ten, we must divide. Divide normally, and place the decimal point in the quotient immediately after completing the division of the units digit. If needed, add zeros to continue the division.
Fifteen divided by four equals 3.75. Twenty-seven divided by five equals 5.4.
The number of digits after the decimal point is called the number of decimal places.
Decimals with the same number of decimal places are called like decimals. Three point nine, eight point seven, and twelve point nine are all like decimals—they each have one decimal place.
Decimals with different numbers of decimal places are called unlike decimals. Five point nine seven has two decimal places, while forty-two point zero three eight has three, so they are unlike decimals.
Remember, we can always add zeros at the end to convert unlike decimals into like decimals. Zero point nine, five point three two, seventeen point zero five four, and two hundred thirty-two point four seven seven zero three are unlike decimals. But 0.90000, 5.32000, 17.05400, and 232.47703 are like decimals, all having five decimal places.
Now we come to operations with decimals, beginning with addition.
Write the numbers so that all decimal points fall in the same vertical line. Digits with the same place value must be directly above one another. Add from the right, just as with whole numbers. Fill empty places with zeros if needed. In your answer, place the decimal point directly below the other decimal points.
Let us add three point nine two, eleven point zero five seven, and two hundred thirty-six point eight four. First, convert to like decimals: three point nine two zero, eleven point zero five seven, and two hundred thirty-six point eight four zero. Adding these gives two hundred fifty-one point eight one seven.
Whole numbers can be written as decimals by placing a decimal point and zeros after the units digit. Fifteen becomes fifteen point zero, or fifteen point zero zero, and so on.
Subtraction follows similar rules.
Align the decimal points vertically. Place digits with the same place value one below another. Fill empty places with zeros to create like decimals. Subtract from the right as usual. Place the decimal point in your answer directly under the others.
Subtract five point two seven from thirteen point eight nine. Align them: thirteen point eight nine minus five point two seven. Nine minus seven is two. Eight minus two is six. Bring down the decimal point. Three minus five—we cannot do this, so we borrow one from the tens place, making thirteen minus five equals eight. The answer is 8.62.
For 0.283 subtracted from two, write two as 2.000. Align the decimal points. Starting from the right: zero minus three requires borrowing, giving ten minus three equals seven. Nine minus eight equals one. Nine minus two equals seven. One minus zero equals one. The result is 1.717.
When evaluating expressions with mixed operations, convert to like decimals first, then proceed. Eleven point three four nine minus five point five seven plus nine point two eight minus twelve point six becomes, in like decimals: eleven point three four nine minus five point five seven zero plus nine point two eight zero minus twelve point six zero zero. Group the positive terms: eleven point three four nine plus nine point two eight zero equals twenty point six two nine. Group the negative terms: five point five seven zero plus twelve point six zero zero equals eighteen point one seven zero. Finally, twenty point six two nine minus eighteen point one seven zero equals two point four five nine.
Multiplication of decimals has several cases.
First, multiplying by ten, one hundred, one thousand, and so on. Simply shift the decimal point to the right by as many places as there are zeros in the multiplier.
3.2985 times ten equals 32.985. Times one hundred: 329.85. Times one thousand: 3298.5.
When multiplying a decimal by a whole number, multiply normally ignoring the decimal point. Then, in your answer, count from the right as many places as there are decimal places in the original decimal.
0.3 times six equals 1.8. 0.26 times eighteen equals 4.68.
For multiplying two decimals together, multiply the numbers normally, ignoring both decimal points. In your final answer, count from the right a number of places equal to the sum of decimal places in both factors.
32.5 times 0.07: the first number has one decimal place, the second has two, so the product needs three decimal places. The answer is 2.275.
0.2 times 0.0004 needs one plus four, or five decimal places: 0.00008.
Division of decimals also has several cases.
Dividing by ten, one hundred, one thousand: shift the decimal point to the left by as many places as there are zeros.
623.42 divided by ten equals 62.342, shifting one place left. Divided by one hundred: 6.2342, shifting two places left. Divided by ten thousand: 0.062342, shifting four places left.
When dividing by a whole number, divide normally and place the decimal point in the quotient immediately after crossing the decimal point in the dividend.
16.952 divided by eight: eight goes into sixteen twice. Place the decimal point after the two. Eight goes into nine once, remainder one. Bring down five, making fifteen. Eight goes into fifteen once, remainder seven. Bring down two, making seventy-two. Eight goes into seventy-two nine times. The answer is 2.119. 0.945 divided by nine: nine does not go into zero, so we write zero and place the decimal point. Nine goes into nine once. Nine does not go into four, so we write zero. Bring down five, making forty-five. Nine goes into forty-five five times. The answer is 0.105.
For dividing by another decimal, first convert the divisor to a whole number by shifting both decimal points equally. Then divide as in the previous case.
4.8 divided by 0.8: shift both points one place right to get forty-eight divided by eight, which equals six.
5.625 divided by 1.25: shift both points two places right to get 562.5 divided by 125, which equals 4.5.
If the division does not terminate, you may add zeros to the dividend and continue as far as needed. This is valid because adding zeros at the end of a decimal does not change its value.
Finally, let us distinguish between terminating and non-terminating decimals.
When division ends with no remainder, the quotient is a terminating decimal. 31.76 divided by four equals 7.94 exactly, a terminating decimal.
When the remainder never becomes zero, no matter how long you continue, the quotient is a non-terminating decimal. 13.78 divided by seven equals 1.9685..., and the digits continue indefinitely. We indicate this by writing three dots after several digits to show the division continues forever.
Let us recap the key points from today's lesson.
First, a decimal fraction is a fraction whose denominator can be expressed as ten or a higher power of ten, and we write it using a decimal point instead of showing the denominator.
Second, place value extends to the right of the decimal point as tenths, hundredths, thousandths, and so on.
Third, to convert between decimals and fractions, use the place value to determine the denominator, then simplify.
Fourth, for addition and subtraction, always align decimal points vertically and work with like decimals.
Fifth, for multiplication, count total decimal places in factors to place the decimal in the product.
And sixth, for division, you may need to shift decimal points to make the divisor a whole number; some divisions produce terminating decimals while others produce non-terminating decimals.
That brings us to the end of our lesson on decimal fractions. I hope you now feel confident working with these numbers in all their forms. Remember, decimals are simply another way of writing fractions, and with practice, operations with them become second nature. Keep practicing, stay curious, and I will see you in the next lesson!