\color{cyan}{\textnormal{Division}} is essentially repeated subtraction.
Division answers: “What if I take away some number of things a specific number of times from a starting quantity?”
Once again, this sounds complicated, so let’s look at an example. Say you had 100 cupcakes and wanted to figure out how many cupcakes each person could have if the party size was 20. You could represent this via subtraction, where you repeatedly subtract 20 from 100, and count how many times you performed that operation until you reach 0.
100 - 20 = 80 80 - 20 = 60 60 - 20 = 40 40 - 20 = 20 20 - 20 = 0
You performed this operation 5 times, which means each person could have 5 cupcakes! Maybe you could save some money and order 40 cupcakes instead. Thanks math!
Now, Division is the inversion operation to multiplication. That is, if you multiply a number by 5, then divide by 5, you end up with the same value:
100 * 5 = 500 500 / 5 = 100
Therefore, you can see that, with these arithmetic operations, you can model a wide range of quantifiable real-life situations.
Like subtraction, the order of the terms is important. Division refers to their operands as \color{cyan}{\textnormal{divisor}} (the first term) and \color{cyan}{\textnormal{dividend}} (the second term). The result is known as the \color{cyan}{\textnormal{quotient}}.
\text{divisor} / \text{dividend} = \text{quotient} \text{divisor} \div \text{dividend} = \text{quotient} \frac{divisor}{dividend} = \text{quotient}
The final equation utilizes a fraction. See more on Fractions.
Fractions also refer to their top and bottom values differently. In a fraction, you typically see the top (divisor) being referred to as the \color{cyan}{\textnormal{numerator}} and the bottom being referred to as the \color{cyan}{\textnormal{denominator}}. This is a common case, and can be seen below:
\frac{\text{numerator}}{\text{denominator}} = \text{quotient}
Division also has alternative symbols. While the / is most common, you may also see: \div, which is a symbol which draws from the idea of a fraction.
There are many different conventions, properties, characteristics, and miscellaneous good-to-knows here. This chapter discusses these topics.
\color{cyan}{\textnormal{Fractions}} represent two numbers divided by each other. Fractions are ways to represent Rational Numbers. The name “rational” comes from the word “ratio”; to which fractions are closely related.
A \color{cyan}{\textnormal{ratio}} is simply a means of expressing how many times a given number contains another number. For instance, if you had 15 apples and 3 oranges, you would say there is a ratio of apples to oranges of 5:1. In other words, for every 1 orange, there are 5 apples!
Extra Credit: Ratio Notation
Ratios are typically expressed via the colon notation. For the apple & oranges example, it would look like this:
5:1
Ratios are almost always expressed in whole numbers. That is, if you had 2.5 apples for every 1 orange, it would follow convention (and be easier to understand) if it was represented as:
5:2
Rather than 2.5:1, although both are legal and valid.
Now, a fraction is expressed as two values: a numerator and a denominator. The numerator is the “first number” in the division (that is, the divisor) and the denominator is the “second number” in the division (that is, the dividend).
In other words, you could express a simple division problem, say: 20 \div 5 as a fract:
\frac{20}{5}
Note that this can be simplified to a whole number: 4.
In this fraction, 20 represents the numerator, while 5 represents the denominator. The resulting representation (the entire expression) is called a fraction. The general form for a fraction is:
\frac{\text{numerator}}{\text{denominator}}
Fractions are simply another means of representing division and/or rational numbers.
There are two main types of fractions: Mixed and Improper.
\color{cyan}{\textnormal{Mixed Fractions}} represent the ratio with only the decimal part as a fraction. That is, if you had the value of 2.4, you could represent this as a mixed fraction:
2.4 = 2\frac{2}{5}
The reason it is called “mixed” is because it consists of a whole number (2) and a fraction (\frac{2}{5}).
The other type of fraction is an \color{cyan}{\textnormal{Improper Fraction}}, which is a fraction such that the numerator is larger than the denominator. You could represent 2.4 as an improper fraction as well:
2.4 = \frac{12}{5}
Notice that the numerator is smaller than the denominator. This is why it is called “improper”, since convention dictates that the numerator is smaller for most fractions.
You may see fractions written as a slanted form, like this:
^1 / _4
This is generally frowned upon, since it is harder to read and understand when in the context of an expression:
^1 / _4 (3 + 2)
Is the (3+2) on the bottom or multiplying against the whole fraction? This ambiguity is alleviated when you write the fraction vertically:
\frac{1}{4}(3+2)
Now it is extremely clear. You could also write:
(1/4)(3+2)
Which is also extremely clear.
To switch between these forms, apply a series of rules.
For instance, switching from improper fractions to mixed fractions, you “simplify” the fraction by extracting whole, divisible parts out of the fraction. As an example:
\begin{aligned} &\frac{12}{5}\\ &= \frac{10}{5}+\frac{2}{5}\\ &= 2+\frac{2}{5}\\ &\boxed{= 2\frac{2}{5}} \end{aligned}
Or, to switch from mixed fractions to improper fractions, you convert the whole part to have the same denominator as the fraction part. As an example:
\begin{aligned} &4\frac{6}{13}\\ &= \frac{13\cdot4}{13}+\frac{6}{13}\\ &= \frac{52}{13}+\frac{6}{13}\\ &= \frac{58}{13} \end{aligned}
Notice that to convert a whole number to a fraction with an arbitrary denominator (say, d), you simply multiply that number by \frac{d}{d}. Since every whole number is technically a fraction (e.g., 4 = \frac{4}{1}), then this because \frac{d\cdot4}{d}.
Decimals are a more straightforward way of representing a number. While they were briefly mentioned in Chapter 1, this will be revisited now that division is understood.
\color{cyan}{\textnormal{Decimals}} are a means of representing any kind of number. You can represent whole numbers with decimals: 4.0; rational numbers: 5.25 and even irrational numbers, like: \pi=3.14159265..., which makes decimals incredibly useful to use.
There’s not much else to say about decimals, except to briefly talk about how they relate to division, and what the positions of each digit represent.
Similar to whole numbers, each digit has a specific place that represents a particular value (a power of 10, more specifically). For instance, 1234 means 1 thousand, 2 hundreds, 3 tens, and 4 ones. Likewise, decimals have similar meanings, albeit in “reverse”. That is, 0.321 represents 1 thousandths, 2 hundredths, and 3 tenths.
There is no “oneths” place.
However, you can follow the pattern into smaller and smaller values. There are: ten-thousandths, hundred-thousandths, millionths, …; although these aren’t used very often.
Now, decimals are often the result of division of some kind. This means that all decimals can be represented by a division operation (and thus, a fraction). HOWEVER, not all decimals represent a rational number, as it may be impossible to represent the number as the ratio (fraction) of 2 whole numbers.
For instance, \pi is the ratio between the circumference of a circle (C) and the circle’s diameter (d) (do not worry about these terms, the idea of a rational number is the important part). That means that \pi=C/d, making it representative by a fraction. However, \pi is irrational because there does not exist 2 whole numbers whose fraction can define it (in fact, the closest is 22/7).
A \color{cyan}{\textnormal{percentage}} is a special way of representing a quantity, where the number is expressed as a fraction of 100. It is denoted with the percent sign: \%.
It literally means “per cent”, which is derived from the Latin per centum, or “by the hundred”.
Percentages are dimensionless (they have no units) and instead express proportions (like I drank 50\% of this juice box!).
Percentages are equal to a fraction where the denominator is zero. That is:
\begin{aligned} 5\% = \frac{5}{100} \\ 10\% = \frac{10}{100} \\ 15\% = \frac{15}{100} \\ \dots \\ 95\% = \frac{95}{100} \\ 100\% = \frac{100}{100} \\ \dots \\ 150\% = \frac{150}{100} \\ \dots \\ \end{aligned}
Notice that percentages can be larger than 100\%, since they are ultimately just a ratio. That also means that percentages can be negative (which is typical for things like percent change or other comparisons.)
Extra Credit: Per Mille
There’s another kind of proportionate value. It is represented by the ‰ symbol. It is called the “per mille” symbol, representing a “per mille” (that is, per thousand).
A per mille represents a similar concept as a percentage, except it the denominator is 1000.
A \color{cyan}{\textnormal{Percent Change}} (aka, \% \text{ Change}) is a means of expressing how a value has changed from one value to another.
Say that you see the price of a 5 lb bag of rice change from \$3 to \$5 in a year. To represent this situation, you may utilize the method of computing percent change:
\% \text{ Change} = \frac{\text{New} - \text{Old}}{\text{Old}}
Therefore, you could compute the percent change for the bag of rice scenario as:
\begin{aligned} &\frac{\$5 - \$3}{\$3} \\ &= \frac{\$2}{\$3} \\ &= \frac{2}{3} \phantom{000000}\text{The units cancel because they are the same!}\\ &\boxed{\approx 66.67\%} \end{aligned}
This means that the price increased by 66.67\%. This is useful because it ensures that changes are comparable, since they are on the same scale (due to the nature of all percentages being ratios of 100).
\color{cyan}{\textnormal{Long Division}} is a method of division where the divisor is placed under a special symbol. The dividend is placed to the left, and the quotient is calculated step by step above. Differences are computed below the divisor. This is a method of doing division by hand.
\begin{aligned} \phantom{00000}274.\bar{3} \\ 45\overline{)12345.0}\\ \phantom{000}-\underline{90\phantom{0}}\phantom{0.0} \\ \phantom{0000}334\phantom{0.0} \\ \phantom{000}-\underline{315\phantom{0}}\phantom{.0} \\ \phantom{00000}195\phantom{.0} \\ \phantom{00000}-\underline{180\phantom{0}}\phantom{.} \\ \phantom{00000}15.0 \\ \phantom{00000}-\underline{13.5} \\ \phantom{00000}1.5 \\ \end{aligned}
Notice that this is a case (among many) where division is not “even”. In other words, the quotient is a decimal. In this case, the “leftover” is known as the remainder.
The \color{cyan}{\textnormal{remainder}} is the number left over from division. Like, 5 / 2 = 2.5. You could also state that 5 / 2 = 2 with a remainder of 1. This remainder can also be used to make a mixed form (number and fraction): 5 / 2 = 2 \frac{1}{2}
Long division should be continued until either enough precision is reached (say, 3 decimal places, perhaps; this depends on your needs) or if the remainder becomes zero. In the case above, you could continue this forever. Notice the repetition of the 15 as 1.5, which is just a factor of 10 less. You can see this pattern will exist forever: 15.0 then 1.5, then 0.15, then 0.015, etc. That is why the long division ends; we know the quotient will end in 3 repeating forever.
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