Review of past work  (Page 8/8)

You can add and multiply rational numbers and still get a rational number at the end, which is very useful. If we have 4 integers $a,b,c$ and $d$ , then the rules for adding and multiplying rational numbers are

Notation tip

The statement "4 integers $a,b,c$ and $d$ " can be written formally as $\{a,b,c,d\}\in \mathbb{Z}$ because the $\in$ symbol means in and we say that $a,b,c$ and $d$ are in the set of integers.

Two rational numbers ( $\frac{a}{b}$ and $\frac{c}{d}$ ) represent the same number if $ad=bc$ . It is always best to simplify any rational number, so that the denominator is as small as possible. This can be achieved by dividing both the numerator and the denominator by the same integer. For example, the rational number $1000/10000$ can be divided by 1000 on the top and the bottom, which gives $1/10$ . $\frac{2}{3}$ of a pizza is the same as $\frac{8}{12}$ ( [link] ).

You can also add rational numbers together by finding the lowest common denominator and then adding the numerators. Finding a lowest common denominator means finding the lowest number that both denominators are a factor Some people say divisor instead of factor. of. A factor of a number is an integer which evenly divides that number without leaving a remainder. The following numbers all have a factor of 3

and the following all have factors of 4

The common denominators between 3 and 4 are all the numbers that appear in both of these lists, like 12 and 24. The lowest common denominator of 3 and 4 is the smallest number that has both 3 and 4 as factors, which is 12.

For example, if we wish to add $\frac{3}{4}+\frac{2}{3}$ , we first need to write both fractions so that their denominators are the same by finding the lowest common denominator, which we know is 12. We can do this by multiplying $\frac{3}{4}$ by $\frac{3}{3}$ and $\frac{2}{3}$ by $\frac{4}{4}$ . $\frac{3}{3}$ and $\frac{4}{4}$ are really just complicated ways of writing 1. Multiplying a number by 1 doesn't change the number.

Dividing by a rational number is the same as multiplying by its reciprocal, as long as neither the numerator nor the denominator is zero:

A rational number may be a proper or improper fraction.

Proper fractions have a numerator that is smaller than the denominator. For example,

are proper fractions.

Improper fractions have a numerator that is larger than the denominator. For example,

are improper fractions. Improper fractions can always be written as the sum of an integer and a proper fraction.

Converting rationals into decimal numbers

Converting rationals into decimal numbers is very easy.

If you use a calculator, you can simply divide the numerator by the denominator.

If you do not have a calculator, then you have to use long division.

Since long division was first taught in primary school, it will not be discussed here. If you have trouble with long division, then please ask your friends or your teacher to explain it to you.

Irrational numbers

An irrational number is any real number that is not a rational number. When expressed as decimals, these numbers can never be fully written out as they have an infinite number of decimal places which never fall into a repeating pattern. For example, $\sqrt{2}=1,41421356...$ , $\pi =3,14159265...$ . $\pi$ is a Greek letter and is pronounced “pie”.

Real numbers

1. Identify the number type (rational, irrational, real, integer) of each of the following numbers:
   1. $\frac{c}{d}$ if $c$ is an integer and if $d$ is irrational.
   2. $\frac{3}{2}$
   3. -25
   4. 1,525
   5. $\sqrt{10}$
2. Is the following pair of numbers real and rational or real and irrational? Explain. $\sqrt{4}$ ; $\frac{1}{8}$

Mathematical symbols

The following is a table of the meanings of some mathematical signs and symbols that you should have come across in earlier grades.

So if we write $x>5$ , we say that $x$ is greater than 5 and if we write $x\ge y$ , we mean that $x$ can be greater than or equal to $y$ . Similarly, $<$ means `is less than' and $\le$ means `is less than or equal to'. Instead of saying that $x$ is between 6 and 10, we often write $6<x<10$ . This directly means `six is less than $x$ which in turn is less than ten'.

Mathematical symbols

1. Write the following in symbols:
   1. $x$ is greater than 1
   2. $y$ is less than or equal to $z$
   3. $a$ is greater than or equal to 21
   4. $p$ is greater than or equal to 21 and $p$ is less than or equal to 25

Infinity

Infinity (symbol $\infty$ ) is usually thought of as something like “the largest possible number" or “the furthest possible distance". In mathematics, infinity is often treated as if it were a number, but it is clearly a very different type of “number" than integers or reals.

When talking about recurring decimals and irrational numbers, the term infinite was used to describe never-ending digits.

End of chapter exercises

1. Calculate
   1. $18-6\times 2$
   2. $10+3(2+6)$
   3. $50-10(4-2)+6$
   4. $2\times 9-3(6-1)+1$
   5. $8+24÷4\times 2$
   6. $30-3\times 4+2$
   7. $36÷4(5-2)+6$
   8. $20-4\times 2+3$
   9. $4+6(8+2)-3$
   10. $100-10(2+3)+4$
2. If $p=q+4r$ , then $r=.....$
3. Solve $\frac{x-2}{3}=x-3$