### Decimals and Rounding (Economics)

#### Definition

A decimal number contains a decimal point. We use decimals to write fractions as a single number.

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. Tenths are $10$ times larger than hundredths, hundredths are $10$ times larger than thousandths...

For instance, $92.67$ is made up of $9$ tens, $2$ ones, $6$ tenths and $7$ hundredths. Let's look at this decimal number in a place-value chart to better understand how decimals work. 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 $\sqrt{2} = 1.414213\dotso$.

To add or subtract decimals, write the numbers with one beneath the other and the decimal points lined up (put zeros on the end of numbers if necessary so that the numbers are the same length). Then add the numbers column by column from right to left.

##### Examples

a) Add $0.055$ to $1.07$. Solution: \begin{align} &0.055\\ +&1.070\\ &\overline{1.125} \end{align}

b) Subtract $0.035$ from $4.0729$. Solution: \begin{align} &4.0729\\ +&0.0350\\ &\overline{4.0379} \end{align}

#### Multiplication and Division

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.

##### Examples
• $16.21 \times 10 =162.1$
• $1.09 \times 100 = 109$
• $0.74 \div 10 = 0.074$

To multiply a decimal number by another decimal number, multiply the numbers as normal (ignoring the decimal points) and then put the decimal point in the answer-it will have as many decimal places as the two original numbers combined.

###### Example 1

Multiply $1.6$ by $0.12$.

###### Solution

First multiply without decimal points: $16\times 12=192$ $1.6$ has $1$ decimal place and $0.12$ has $2$ decimal places, so the answer has 3 decimal places: $1.6\times 0.12=0.192$

#### Dividing a Decimal Number by a Whole Number

To divide a decimal number by a whole number first do the calculation ignoring the decimal point, then put the decimal point in the same spot as the number being divided.

###### Example 1

Divide $0.124$ by $4$.

###### Solution

First divide $0.124$ by $4$ ignoring the decimal point. $124\div 4=31$ $0.124$ has $3$ decimal places and $4$ has no decimal places, so the answer has $3$ decimal places: $0.124\div 4=0.031$

#### Decimals and Fractions

##### Converting Decimals into Fractions

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 $1$ thousandth or $331$ thousandths, i.e. $\frac{331}{1000}$.

##### Converting Fractions into Decimals

To convert a fraction into a decimal divide the numerator by the denominator.

###### Worked Example

Write the following decimals as fractions, reduced to simplest form:

a) $0.8$

b) $0.275$

###### Solution

a) The final digit of the decimal number $0.8$ is in the tenths column so we have: $0.8 = \dfrac{8}{10} = \dfrac{4}{5}.$

b) $\dfrac{2}{10}+\dfrac{7}{100}+\dfrac{5}{1000} = \dfrac{200}{1000}+\dfrac{70}{1000}+\dfrac{5}{1000} = \dfrac{275}{1000}= \dfrac{11}{40}.$

###### Worked Example

Convert the following fractions into their decimal equivalents:

a) $\frac{35}{10}$.

b) $\frac{11}{20}$.

###### Solution

a) Dividing the numerator by the denominator gives: $\frac{35}{100}=0.35$

b) Dividing the numerator by the denominator gives: $\frac{35}{100}=0.55$

#### Rounding

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:

• $\geq 5$ we increase the second decimal digit by $1$.
• $\lt 5$ we leave the second decimal digit as it is.
###### Worked Example

Round $251.6731$ to:

a) $3$ decimal places

b) $2$ decimal places

c) $1$ decimal place.

###### Solution

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 $2.7$.

#### Significant Figures

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:

• Equal to or greater than $5$, we round up.
• $4$ or less, we don't round up.
###### Worked Example 1

Round $702.019$ to

a) Five significant digits

b) Four significant digits

c) One significant digit

###### Solution

a) $702.02$

b) $702.0$

c) $700$

###### Worked Example 2

Round $0.097065$ to four, three, and two significant digits:

a) Four significant digits

b) Three significant digits

c) Two significant digits

###### Solution

a) $0.09707$

b) $0.0971$

c) $0.097$

###### Worked Example

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.

###### Solution

a) $4.003$

b) $1.2$

c) $0.001$

d) $0.00123$

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