2. The word “Decimal” really means “based
on 10” (From latin decima: a tenth part).
We sometimes say “decimal” when we
mean anythingt do with our number
system, but a “decimal number” usually
means there is a decimal point.
3. 3
DECIMAL NUMBER – a number with a decimal point and decimal
places.
In a numeral, the value of each digit is ten times less than the
digit to its immediate left, then the value of the first digit after the
decimal point is 1/10 or 0.1 as much as that of the ones place.
That is, if the place value of the digits from left to right is
…hundreds, tens, ones, then the place value after the decimal
point (immediate right of the ones place) is tenths, (1/10 of 1).
Now we continue with smaller and smaller values, from tenths, to
hundredths, and so on.
4. 4
14.295
EXAMPLE:
Decimal Point
Units
Tens
1/10 (Tenths)
1/100 (Hundredths)
1/1000 (Thousandths)
10x Bigger
10x Smaller
The digit to the left of
the decimal point is a
whole number, 4 for
example. As we move
further to the left,
every digit place gets
10 times bigger. As
we move further right,
every digit place gets
10 times smaller (one
tenths as big).
The value of a digit in a numeral is the product of that digit and its place value. For
instance, the value of 1 in 14.925 is 1 x 10 = 10, while that of 9 is 9 x 0.01 – 0.09
5. Diagram showing the classifications of decimal.
5
Decimal
Non-Terminating
Repeating Non-repeating
Terminating
6. All rational numbers can be written as either terminating
or repeating decimals. Examples of terminating decimals
are 0.5, 2.84 and 0.875 while examples of repeating
decimals are 0.333… and 0.727272… Non-terminating
decimals that do not repeat, such as 1.1010010001…, are
called irrational numbers.
Examples of irrational numbers are 0.1372 , 𝜋, and 2.
6
8. To write a terminating decimal as a fraction, use the
name for the last decimal place in the number as the
denominator.
a. 0.8 =
8
10
Read as 8-tenths. Write then as the
denominator
=
4
5
Simplify
b. 0.25 =
25
100
Read as 25-hundredths. Write
hundred as the denominator.
=
1
4
Simplify.
8
The simpler
rule is to
write the
decimal as a
fraction the
way it is read.
10. Example 1:
Write 0.333… as a fraction
Solution:
𝑥 = 0.333 … Let x represent the number.
10𝑥 = 3.333 … Multiply by 10 because 1 digit repeats
10𝑥 = 3.333 …
− 𝑥=0.333…
9𝑥=3
Subtract to eliminate the decimal places.
Divide each side by 9. Simplify
𝑥 =
3
9
=
1
3
𝐴𝑛𝑠𝑤𝑒𝑟: 0.333 … =
1
3
10
11. Example 2:
Write 0.090909… as a fraction
Solution:
𝑥 = 0.090909 … Let x represent the number.
100𝑥 = 9.090909 … Multiply by 100 because 2 digit repeats
100𝑥 = 9.090909 …
− 𝑥= 0.090909…
99𝑥= 9
Subtract to eliminate the repeating decimal.
𝑥 =
9
99
=
1
11
Divide each side by 99. Simplify.
𝐴𝑛𝑠𝑤𝑒𝑟: 0.090909 … =
1
11
11
12. Example 3:
Write 0.2333… as a fraction
Solution:
𝑥 = 0.2333 Let x represent the number.
10𝑥 = 2.333 … Multiply by 10.
100𝑥 = 23.333 … Multiply by 100.
100𝑥 = 23.333 …
−10𝑥= 2.333
90𝑥=21
Subtract.
𝑥 =
21
90
=
7
30
Divide each side by 90. Simplify.
𝐴𝑛𝑠𝑤𝑒𝑟: 0.2333 … =
7
30
12
14. To add or subtract decimals:
1. If it is necessary, write the numerals in column so that
the decimal points are directly under each other. Zeros
may be annexed to the numerals naming decimal
fractions so that the addends may have the same
number of decimal places.
0.39 + 4.6 0.39 8.4 − 2.76 8.40
+4.6 +2.76
14
15. 2. Add or subtract as in the addition of whole
numbers
0.39 8.40
+4.6 −2.76
4 99 5 64
15
16. 3. Place the decimal point in the sum or
difference directly under the decimal points in the
addends or subtrahend and minuend.
0.39 8.40
+4.6 −2.76
4.99 5.64
16
18. To multiply decimals:
1. Write the given numerals in column and multiply as in the multiplication of
whole numbers. The decimal point in the multiplier does not necessarily
have to be under the decimal point in the multiplicand.
2. Find the total numbers of decimal places in the factors and point off in the
product, counting from right to left, as many decimal places as there are
in the factors.
0.83 2 decimal places
× 2.4 1 decimal place
332
166
1.992 3 decimal places
18
19. 3. When the product contains fewer figures/digits than
the required number of decimal places, prefix as many
zeros as are necessary.
0.25 2 decimal places
× 0.03 2 decimal places
0.0075 4 decimal places
19
21. To divide decimals:
1. If the divisor is a whole number,
a. Divide as in division of whole numbers.
b. Place the decimal point in the quotient directly
above the decimal point in the dividend.
2.9
5 14.5
10
45
45
0
21
22. c. Divide as in the division of whole numbers and place
the decimal point in the quotient directly above the new
place of the decimal point in the dividend.
23
4 92
8
12
12
d. When the dividend contains fewer decimal places than
required, annex as many zeros as are necessary to a
decimal dividend.
22
23. 2. If the divisor is a decimal,
a. Make the divisor a whole number by multiplying
it by a power of 10 or by simply moving the
decimal point to the right of the last figure.
b. Move the decimal point in the dividend to the
right as many places as you moved the decimal
point in the divisor.
0.4 9.2 4 92
23
Multiply 0.4 by 10 to
make it a whole
number. Do the same
with the dividend.
