A decimal number contains a decimal point. The decimal point shows where the fractional part of a number begins. To the left of the decimal point, we have the whole number part, and to the right we have the fractional part, made up of tenths, hundredths, thousandths, and so on.
A recurring decimal is a decimal which repeats the last digit, or last few digits, forever. For example, “one third” written as a decimal is $0.3333333\dotso$. The notation for a recurring number is a dot above the number. We can write a dot above the first $3$ to more neatly show that it is recurring: $0.\dot{3}$. If a group of numbers is repeated, we write one dot where the numbers begin to repeat, and one on the last number that is repeated. For example, $\frac{452}{555}$ can be written as $0.8144144144144\dotso$, but is better written as $0.8\dot{1}4\dot{4}$.
An irrational number cannot be written as a fraction of whole numbers. When written in decimal form, the digits go on forever and do not repeat. Examples of such numbers are $\pi = 3.1415926\dotso$ or $\sqrt{2} = 1.414213\dotso$.
When multiplying a decimal number by $10$, the digits stay the same but the decimal place moves one to the right. Division is the opposite of multiplication, so the decimal point moves to the left. For multiplying and dividing by $100$, move the decimal point two places; for $1,\!000$ move the decimal point three places; and so on.
The fractional part of a decimal can be thought of as the sum of fractions, with each place digit representing a fraction over a different denominator. For example, $0.25$ has a $2$ in the tenths column and $5$ in the hundredths, which is the same as $\frac{2}{10} + \frac{5}{100} = \frac{25}{100} = \frac{1}{4}$. Alternatively, look at the final digits column and write the fractional part as a whole number over that digit's denominator. For example, $0.331$ is equal to $3$ tenths, $3$ hundredths and $1$ thousandths or $331$ thousandths, i.e. $\frac{331}{1000}$.
Write the following decimals as fractions, reduced to simplest form:
a) $0.8$
b) $0.275$
a) $0.8 = \dfrac{8}{10} = \dfrac{4}{5}$.
b) $\dfrac{2}{10}+\dfrac{7}{100}+\dfrac{5}{1000} = \dfrac{200}{1000}+\dfrac{70}{1000}+\dfrac{5}{1000} = \dfrac{275}{1000}= \dfrac{11}{40}$.
Rounding means reducing the number of decimal places (d.p.) a number has but keeping its value close to what it was. The purpose is to make the number simpler. Suppose we want to round a number to $2$ decimal places (or $2$d.p.). To do this, we look only at the next digit to the right (the third decimal digit). If this is:
Round $251.6731$ to:
a) $3$ decimal places
b) $2$ decimal places
c) $1$ decimal place.
a) The fourth decimal digit is $1$. Since $1\lt 5$, we leave the third decimal digit alone: $251.6731$ to $3$d.p. is $2.673$.
b) The third decimal digit is $3$. Since $3\lt 5$, we leave the second decimal digit alone: $251.6731$ to $2$d.p. is $2.67$.
b) The second decimal digit is $7$. Since $7\gt 5$, we increase the first decimal digit by one: $251.6731$ to $1$d.p. is $2.7$.
Sometimes we are asked to round a number to a given number of “significant figures” (s.f.) or “significant digits”. Here are the basic rules for significant digits:
1) All nonzero digits are significant. 2) All zeroes between significant digits are significant. 3) All zeroes which are both to the right of the decimal point and to the right of all non-zero significant digits are significant.
The rules for rounding are the same. If the next number is:
Round $702.019$,396 to
a) Five significant digits
b) Four significant digits
c) One significant digit
a) $702.02$
b) $702.0$
c) $7$ or $70$
Round $0.097265$ to four, three, and two significant digits:
a) Four significant digits
b) Three significant digits
c) Two significant digits
a) $0.09127$
b) $0.0913$
c) $0.091$
a) Round $4.0032$ to 3 d.p.
b) Round $1.1674$ to 2 s.f.
c) Round $0.00123$ to 3 d.p.
d) Round $0.00123$ to 3 s.f.
a) \[4.003\]
b) \[1.2\]
c) \[0.001\]
d) \[0.00123\]
Test yourself: Rounding and estimating