From Förberedande kurs i matematik 1

Theory

Exercises

Calculations with numbers

Calculating with numbers requires you to perform a series of operations. These are the four basic operations of arithmetic. Here are some concepts that are helpful to know in order to understand a mathematical text:

When you add numbers the sum does not depend on the order in which the terms are added together

\displaystyle 3+4+5=3+5+4=5+4+3=12\,\mbox{.}

As regards subtraction the order is important of course.

\displaystyle 5-2=3 \quad \mbox{whereas} \quad 2-5=-3\,\mbox{.}

When we talk about the difference between two numbers we usually mean the difference between the larger and the smaller. Thus, we say the difference between 2 and 5 is 3.

When numbers are multiplied their order is not important.

\displaystyle 3 \cdot 4 \cdot 5=3 \cdot 5 \cdot 4 = 5 \cdot 4 \cdot 3 = 60 \,\mbox{.}

For division the order is of importance.

\displaystyle \frac{6}{3} = 2\quad\mbox{whereas}\quad\frac{3}{6} = 0{,}5 \,\mbox{.}

Hierarchy of arithmetic operations (priority rules)

If several mathematical operations occur in a mathematical expression it is important to have a standard for the order in which the operations are to be carried out. The following rules apply:

• Parentheses ( brackets, "innermost brackets" first)
• Multiplication and division (from left to right)
• Addition and subtraction (from left to right)

"Invisible" parentheses

For division the numerator and the denominator must be calculated separately before the division is carried out. One can therefore say that there are "invisible parentheses" around the numerator and denominator.

This is especially important if calculators are used.

Division

\displaystyle \frac{8+4}{2+4}

must be written as \displaystyle (8 + 4 )/(2 + 4) for a calculator so that the correct answer \displaystyle 2 may be obtained. A common mistake is to write \displaystyle 8 + 4/2 + 4, which the calculator interprets as \displaystyle 8 + 2 + 4 = 14.

Different types of numbers

The numbers we use to describe quantity, size, etc... are generically called the real numbers and can be illustrated by a straight line real-number axis:

The real numbers "fill" the real-number axis, i.e. there are no holes or spaces along the real-number axis. Each point on the real-number axis can be specified by a decimal. The set of real numbers are all the decimals and is denoted by R. The real-number axis also shows the relative magnitude of numbers; a number to the right is always greater than a number to the left. It is standard to classify the real numbers into the following types:

Natural numbers (usually symbolised by the letter N)

The numbers which are used when we calculate “how many”: 0, 1, 2, 3, 4, ...

Integers (Z)

The natural numbers and their negative counterparts: ..., -3, -2, -1, 0, 1, 2, 3, ...

Rational numbers (Q)

All the numbers that can be written as a ratio of whole numbers (fractions). For example:

\displaystyle -\frac{3}{4},\ \frac{3}{2}, \ \frac{37}{128}, \quad\mbox{etc.}

Note that integers are classed as rational numbers because

\displaystyle -1 = \frac{-1}{1},\quad 0 = \frac{0}{1},\quad 1 = \frac{1}{1},\quad 2 = \frac{2}{1},\quad\mbox{etc.}

A rational number can be written in various ways. For example:

\displaystyle 2 = \frac{2}{1}=\frac{4}{2}=\frac{6}{3}=\frac{8}{4} =\frac{100}{50}=\frac{384}{192}\quad\mbox{etc.}

Irrational numbers

The numbers on the real-number axis that can not be written as a fraction are called irrational numbers. Examples of irrational numbers are most roots, for example:

\displaystyle \sqrt{2} and \displaystyle \sqrt{3}, but also numbers such as \displaystyle \pi

Decimal form

All types of real numbers can be written in decimal form, with an arbitrary number of decimal places. Decimal integers written to the right of the decimal point specify the number of tenths, hundredths, thousandths and so on. In the same way as the integers to the left of the decimal point indicate the number of units, tens, hundreds and so on.

A rational number can be written in decimal form by performing the division. Thus \displaystyle \textstyle\frac{3}{4} is the same as "3 divided by 4", i.e. 0.75.

Read about long division on wikipedia.

As can be seen the rational numbers above have a periodic decimal expansion, i.e. the decimal expansion ends up with a finite block of digits that is repeated endlessly. This applies to all rational numbers and distinguishes them from the irrational numbers which do not have a periodic pattern in their decimal expansion. Conversely it is also true that all numbers with a periodic decimal expansion are rational.

Rounding off

Since it is impractical to use long decimal expansions, one often rounds off a number to an appropriate number of decimal places. The standard practise is that the numbers 0, 1, 2, 3 and 4 are rounded down while 5, 6, 7, 8 and 9 are rounded up.

We use the symbol \displaystyle \approx (is approximately equal to) to show that a rounding off has taken place.

Comparing numbers

To indicate the relative size between numbers one uses the symbols > (is greater than), < (is less than) and = (is equal to). The relative size between two numbers can be determined either by giving the numbers in decimal form or by representing rational numbers as fractions with a common denominator.

Exercises