Fraction (mathematics)

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File:Cake-quarters.jpg
A cake divided into four equal quarters. Each fraction of the cake is represented numerically as 14. It can be seen that two quarters (2 x 14 = 24) is equivalent to half (12) the cake.

In mathematics, a fraction is a way of expressing a quantity based on an amount that is divided into a number of equal-sized parts. For example, each part of a cake split into four equal parts is called a quarter (and represented numerically as 14); two quarters is half the cake, and eight quarters would make two cakes.

Mathematically, a fraction is a quotient of numbers, like 34, or more generally, an element of a quotient field.

In our cake example above, where a quarter is represented numerically as 14, the bottom number, called the denominator, is the total number of equal parts making up the cake as a whole, and the top number, called the numerator, is the number of these parts we have. For example, the fraction 34 represents three quarters. The numerator and denominator may be separated by a slanting line, or may be written above and below a horizontal line.

The numerator and denominator are the "terms" of the fraction. The word "numerator" is related to the word "enumerate," meaning to "tell how many"; thus the numerator tells us how many parts we have in the indicated fraction. To denominate means to "give a name" or "tell what kind"; thus the denominator tells us what kind of parts we have (halves, thirds, fourths, etc.). Note that because it is impossible to divide something into zero equal parts, zero can never be the denominator of a fraction.

The word is also used in related expressions, like continued fraction and algebraic fraction, see Special cases below.

Forms of fractions

Proper and improper fractions

If the numerator and denominator of a fraction are both positive, then the fraction is a proper fraction if the numerator is less than the denominator, but an improper fraction otherwise. If either the numerator or denominator (or both) are negative, their absolute values should be compared to determine whether the fraction is proper or improper.

Mixed numbers

A mixed number is the sum of a whole number and a proper fraction. For instance, you could have two entire cakes and three quarters of another cake. The whole and fractional parts of the number are written right next to each other: 2 + 34 = 234.

An improper fraction can be thought of as another way to write a mixed number; in the "234" example above, imagine that the two entire cakes are each divided into quarters. Each entire cake contributes 44 to the total, so 44 + 44 + 34 = 114 is another way of writing 234.

A mixed number can be converted to an improper fraction in three steps:

  1. Multiply the whole part times the denominator of the fractional part.
  2. Add the numerator of the fractional part to that product.
  3. The resulting sum is the numerator of the new (improper) fraction, and the new denominator is the same as that of the mixed number.

Similarly, an improper fraction can be converted to a mixed number:

  1. Divide the numerator by the denominator.
  2. The quotient (without remainder) becomes the whole part and the remainder becomes the numerator of the fractional part.
  3. The new denominator is the same as that of the original improper fraction.

Equivalent fractions

Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a new fraction that is said to be equivalent to the original fraction.1 The word equivalent means that the two fractions have the same value. For instance, consider the fraction 12. When the numerator and denominator are both multiplied by 2, the result is 24, which has the same value as 12. To see this, imagine cutting the example cake into four pieces; two of the pieces together (24) make up half the cake (12).

We can say, for example, that 13, 26, 39, and 100300 are all equivalent fractions.

Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. We call this reducing the fraction. A fraction in which the numerator and denominator have no factors in common (other than 1) is said to be irreducible or in lowest terms. For instance, 39 is not in lowest terms because both 3 and 9 can be evenly divided by 3. In contrast, 38 is in lowest terms — the only number that's a factor of both 3 and 8 is 1.

Reciprocals and the "invisible denominator"

The reciprocal of a fraction is another fraction with the numerator and denominator swapped. The reciprocal of 37, for instance, is 73.

Because any number divided by 1 results in the same number, it is possible to write any whole number as a fraction by using 1 as the denominator: 17 = 171. (We sometimes call the number 1 the "invisible denominator.") Therefore, we can say that, except for zero, every fraction or whole number has a reciprocal. The reciprocal of 17 would be 117.

Arithmetic with fractions

Fractions, like whole numbers, obey the commutative, associative, and distributive laws, and the rule against division by zero.

Addition and subtraction

Adding fractions

Adding fractions can be a little tricky, since you cannot simply add the numerators and denominators. For example, if we had a cake divided into three pieces, each piece would be 1/3. Then, if we try to add one piece, 1/4, from the cake divided into four pieces, and one piece, 1/3, from the cake divided into three pieces, what would we have? Well, would we have 1/4 + 1/3 = ??? You can see this is NOT equal to 1/7 or 2/7 !!

To add fractions together, they must be changed to equivalent values having the same fractional unit -- the same denominator -- in this case 1/12. How do we do this? By multiplying each fraction by 1. By one? Yes. 1 = 3/3 and 1 = 4/4. Now watch: 1/4 = 1/4 x 1 = 1/4 x 3/3 = 3/12. And 1/3 = 1/3 x 1 = 1/3 x 4/4 = 4/12. So now 1/4 + 1/3 = 3/12 + 4/12 = 7/12 and we have the correct result.

Notice that we only add the numerators together. The denominator does not change, since we are working with the same fractional unit. Another way to see this is: 1/4 + 1/3 = 3/12 + 4/12 = 1/12 x (3 + 4) = 1/12 x 7 = 7/12.

Lets take another example. If you add a half dollar to a quarter, what will you get? You know it's 75 cents, right? When we say 75 cents we have automatically, in our mind, converted each coin into cents (pennies): One half dollar = 50 cents; one quarter = 25 cents; so 1/2 + 1/4 = 50/100 + 25/100 = 75/100 or 75 cents. Of course, we could use a smaller denominator since we know one half dollar equals two quarters. I.e., 1/2 + 1/4 = 2/4 + 1/4 = 3/4. In words, one half plus one quarter equals two quarters plus one quarter equals three quarters, or 75 cents.

So the trick is to find a common fractional unit -- a common denominator -- that will let us simply add the numerators together. Let's take one more example. Find 2/3 + 1/2. We see that the denominators are 3 and 2. We need to find a value that each denominator can be multiplied by to give a common value. Well, it's easy to see that we can multiply 3 by 2, and 2 by 3, to give a common denominator of 6. But remember, you cannot change the value of each fraction, so we must multiply both numerator and denominator by the same number. We now have: 2/3 + 1/2 = 2/2 x 2/3 + 3/3 x 1/2 = 4/6 + 3/6 = 7/6 or 1 + 1/6.

When doing arithmetic with fractions, results should usually be expressed in lowest terms. For instance,

16 + 13 = 16 + 26 = 36 = 12

Note that 36 is not an incorrect answer, because 36 and 12 are equivalent, but the reduced form is preferred, and classroom exercises will nearly always require that final fractional answers to problems be reduced.

Subtracting fractions

The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance, 2/3 - 1/2 = 2/2 × 2/3 - 3/3 × 1/2 = 4/6 - 3/6 = 1/6.

Multiplication and division

Multiplication

By whole numbers

If you consider the cake example above, if you have a quarter of the cake, and you multiple the amount by three, then you end up with three quarters. We can write this numerically as follows:

3\times {1 \over 4}={3 \over 4}

As another example, suppose that five people work for three hours out of a seven hour day (ie. for three seventh of the work day). In total, they will have worked for 15 hours (5 x 3 hours each), or 15 sevenths of a day. Since 7 seventh of a day is a whole day, 14 sevenths is two days, then in total, they will have worked for 2 days and a seventh of day. Numerically:

5\times {3 \over 7}={15 \over 7}=2{1 \over 7}
By fractions

If you consider the cake example above, if you have a quarter of the cake, and you multiple the amount by a third, then you end up with a twelfth of the cake. In other words, a third of a quarter (or a third times a quarter), is a twelfth. Why? Because we are splitting each quarter into three pieces, and four quarters times three makes 12 parts (or twelfths). We can write this numerically as follows:

{1 \over 3}\times {1 \over 4}={1 \over 12}

As another example, suppose that five people do an equal amount work that totals three hours out of a seven hour day. Each person will have done a fifth of the work, so they will have worked for a fifth of three sevenths of a day. Numerically:

{1 \over 5}\times {3 \over 7}={3 \over 35}
General rule

You may have noticed that when we multiply fractions, we simply multiply the two numerators (the top numbers), and multiply the two denominators) (the bottom numbers). For example:

{5 \over 6}\times {7 \over 8}={5\times 7 \over 6\times 8}={35 \over 48}
By mixed numbers

When multiplying mixed numbers, it's best to convert the whole part of the mixed number into a fraction. For example:

3\times 2{3 \over 4}=3\times \left({{8 \over 4}+{3 \over 4}}\right)=3\times {11 \over 4}={33 \over 4}=8{1 \over 4}

In other words, 2{3 \over 4} is the same as \left({{8 \over 4}+{3 \over 4}}\right), making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total). And 33 quarters is 8{1 \over 4} since 8 cakes, each made of quarters, is 32 quarters in total.

Division

To divide by a fraction, simply multiply by the reciprocal of that fraction.

5\div {1 \over 2}=5\times {2 \over 1}=5\times 2=10
{2 \over 3}\div {2 \over 5}={2 \over 3}\times {5 \over 2}={10 \over 6}={5 \over 3}

About 4,000 years ago Egyptians divided with fractions using slightly different methods. Egyptians used the least common multiple technique to divide unit fractions. Examples can be found at

http://egyptianmath.blogspot.com

Special cases

A vulgar fraction (or common fraction) is a rational number written as one integer (the numerator) divided by a non-zero integer (the denominator), for example 4/3 as opposed to 11/3. The line that separates the numerator and the denominator is called the vinculum if it is horizontal, a solidus if it is slanting. A unit fraction is a vulgar fraction with a numerator of 1 (1/7). An Egyptian fraction is the sum of distinct unit fractions (1/3+1/5). A decimal fraction is a vulgar fraction where the denominator is a power of 10 (4/100). A dyadic fraction is a vulgar fraction in which the denominator is a power of two (1/8). A compound fraction is a fraction where the numerator or denominator (or both) contain fractions, {\frac  {2}{3}}{\Bigg /}{\frac  {1}{5}}, these can be simplified to give vulgar fractions.

An expression that has the form of a fraction, but actually represents division by or into an irrational number might be called an "irrational fraction" (an oxymoron). A common example is π/2, the radian measure of a right angle.

Rational numbers are the quotient field of integers. Rational functions are functions evaluated in the form of a fraction, where the numerator and denominator are polynomials. These rational expressions are the quotient field of the polynomials (over some integral domain).

A continued fraction is an expression such as a_{0}+{\frac  {1}{a_{1}+{\frac  {1}{a_{2}+...}}}}, where the ai are integers. This is not an element of a quotient field.

The term partial fraction is used in algebra, when decomposing rational functions. The goal of the method of partial fractions is to write rational functions as sums of other rational functions with denominators of lesser degree.

Pedagogical tools

In Primary Schools, fractions have been demonstrated through Cuisenaire rods.

See also the external links below.

History

See also: Egyptian fractions, history of irrational numbers.

The earliest known use of decimal fractions is ca. 2800 BC as Ancient Indus Valley units of measurement. The Egyptians used Egyptian fractions ca. 1000 BC. The Greeks used unit fractions and later continued fractions and followers of the Greek philosopher Pythagoras, ca. 530 BC, discovered that the square root of two cannot be expressed as a fraction. In 150 BC Jain mathematicians in India wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, operations with fractions.

Footnotes

  • 1.The reason belongs more in the section on multiplying fractions, but in essence, it's this: when I multiply the numerator and denominator of a fraction by 2, for example, I'm really multiplying the entire fraction by 22. But 22 = 1, and multiplying any number by 1 results in the same number.

See also

External links