Remember that to express numbers that are not whole numbers we use

decimal numbers as 75'324 in which every digit has a value which is divided by

ten when we move to the right. So

7 is seventy units

5 is five units

3 is three tenths of a unit

2 two hundredths of the unit

4 is four thousandths of a unit

Types of decimal numbers

As a result of some operations we can get different types of decimal numbers:

Regular numbers: Are decimal numbers with a limited quantity of decimal digits and from them all could be zeros.

For example: 14/5 = 2'8 we find an exact division

Repeating decimals: There is a group of digits that are repeated forever.

Examples: If we divide 4 by 3 we get 1'3333333...

Calculating 13/33= 0'36363636...

The group of repeated decimal digits is called the period on the first case our period is 3 on the second case the period is 36

The number 1'33333... must be written as 1'3 and 0'36363636 as 0'36.

Irrational numbers: They have an unlimited quantity of decimal digits but there is not any period.

Example: Calculating square root of 2 we get 1.414213562... and we don’t find any sequence on the digits we get.

Using this system an angle “a” can be expressed for example a = 43º 43′ 2 ′′ and we need to operate angles expressed in this form.

For time the unit as you know is the hour and is divided into minutes and seconds with the same relationship they have on the angles.

A period of time is expressed as 3 h 5 min 3 s for example.

We need to be able to operate in this system using these two magnitudes.

Addition

We need to add separately degrees or hours minutes and seconds and then convert the seconds into minutes and the minutes into degrees/hours if we get more than 60 subunits.

For example: 35º 23′ 30′′ + 25º 53′ 5 ′′

Adding separately we get 35º 23′ 30′′ + 25º 53′ 58′′ = 60º 76′ 88′′ and as 88′′ = 1′ 28′′ we add 1′ and get 77′ = 1º 1 ′ we add 1º and the solution is 61º 17′ 28′′

A fraction is a number that expresses part of a unit or a part of a quantity.

Fractions are written in the form a/b where a and b are whole numbers, and the number b is not 0.

They can be written also in the form a/b

So if we want to test if 12/20 and 24/20 are equivalent fractions

The first cross-product is the product of the first numerator and the second denominator: 12 × 40 = 480.

The second cross-product is the product of the second numerator and the first denominator: 24 × 20 = 480.

Since the cross-products are the same, the fractions are equivalent.

Simplying Fractions

When numerator and denominator have no common factors the fraction is in the simplest form or in its lowest terms. We also say that the fraction is irreductible.

We know that 4/12 = 2/6 = 1/3

4 and 12 have a common factor (4), so 4/12 can be written as 1/3 (Divide the top and the bottom by 4.)

2 and 6 have a common factor (2), so 2/6 can be written as 1/3 (Divide the top and the bottom by 2.)

However, 1 and 3 have no common factors, so 1/3 is the simplest form of these fractions.

There are two methods of reducing a fraction to the lowest terms.

Method 1:

Divide the numerator and denominator by their HCF.

12/30. The HCF of 12 and 30 is 6 so 12/30=2/5

Method 2:

Divide the numerator and denominator by any common factor. Keep dividing until there are no more common factors.

Comparing and ordering fractions

1. To compare fractions with the same denominator, look at their numerators. The largest fraction is the one with the largest numerator.
2. To compare fractions with different denominators, take the cross product. Compare the cross products.
   1. If the cross-products are equal, the fractions are equivalent.
   2. If the first cross product is larger, the first fraction is larger.
   3. If the second cross product is larger, the second fraction is larger.
3. We can also compare fractions dividing the numerator by the denominator

Adding and subtracting fractions

If the fractions have the same denominator:

The numerator of the sum is found by simply adding the numerators over the denominator. Their difference is the difference of the numerators over the denominator.

We do not add or subtract the denominators!

If the fractions have different denominators

For example for 3/5 + 7/8 follow these steps:

1. Reduce them to a common denominator.

For this problem complete the following steps:

• We find the LCM of the denominators as it is the smallest number that both denominators divide into. For 5 and 8, it is easy to see that 40 is the LCM. We look for equivalent fractions with 40 as common denominator.

In our example 3/5 = ()/40 and 7/8 = ()/40

• We divide every new denominator by the previous one and then we multiply the result by the numerator.

So in 3/5 = 24/40 and in 7/8 = 35/40

• Add the numerators and do not change the denominator.

• Reduce if possible.

Multiplying fractions

When two fractions are multiplied, the result is a fraction with a numerator that is the product of the fraction’s numerators and a denominator that is the product of the fraction’s denominators.

Reduce when possible.