The numbers we know

So far you have learned three kinds of numbers:

• whole numbers: 0, 1, 2, 3, …
• fractions: $\displaystyle \frac{1}{2}$, $\displaystyle \frac{3}{4}$, $\displaystyle 1\frac{1}{2}$, …
• decimals: 0.5, 0.75, 1.75, …

They are all on the same number line:

One point on the line can have several names: $\displaystyle \frac{1}{2}$ and $\displaystyle 0.5$ are the same number, and so are $\displaystyle {\frac{3}{4}}={0.75}$ and $\displaystyle {1\frac{1}{2}}={1.5}$.

There is no whole number between 3 and 4. Between two fractions, though, there are always more fractions:

• Between $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{1}{2}$: we expand both to the denominator 12 and get $\displaystyle \frac{4}{12}$ and $\displaystyle \frac{6}{12}$. Between them is $\displaystyle \frac{5}{12}$.
• Want more? We expand to 24: $\displaystyle \frac{8}{24}$ and $\displaystyle \frac{12}{24}$. Between them are $\displaystyle \frac{9}{24}$, $\displaystyle \frac{10}{24}$ and $\displaystyle \frac{11}{24}$.
• You can go on – to 48, 96, … – and each time there are more numbers between them.

Between two decimals there are always more numbers too: between 0.3 and 0.4 are 0.31, 0.35, 0.38, …, and between 0.3 and 0.31 are 0.301, 0.305, …

So we say that fractions and decimals are dense on the number line: between any two of them there are numbers – as many as you like. Whole numbers are not dense.

Find five numbers between the two numbers, and write them in order from small to large.

Between $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{1}{2}$

Between $\displaystyle 0.2$ and $\displaystyle 0.21$

Each kind of number fits other situations:

• A whole number – when you count: 28 students, 3 buses.
• A fraction – a part of a whole thing or of a group: $\displaystyle \frac{3}{4}$ of the pizza, $\displaystyle \frac{2}{3}$ of the class.
• A decimal – measures: 1.45 m, 2.5 kg, 12.90 ₪.

Put one of the numbers $\displaystyle \frac{3}{5}$, $\displaystyle 0.21$, $\displaystyle 28$ in each sentence.
Explain why you chose it.

• On the second day of the trip, the class walked ___ of the route.
• In three years, Yossi grew by ___ metres.
• ___ students went on the trip.

A thermometer has numbers below zero too. A number with a minus in front of it, like $\displaystyle -2$, is to the left of zero on the number line: $\displaystyle -2$ degrees is two degrees below zero.

On a cold night it was $\displaystyle -2$ degrees on Mount Hermon.
At noon it was $\displaystyle 4$ degrees there.
By how many degrees did the temperature rise?

We walk along the number line from $\displaystyle -2$ to $\displaystyle 4$: two steps to 0, and four more steps to 4 – six steps in all. The temperature rose by 6 degrees, and the segment from $\displaystyle -2$ to $\displaystyle 4$ is 6 long.

At night the temperature in Jerusalem was $\displaystyle -3$ degrees.
The next day at noon it was $\displaystyle 5$ degrees.
By how many degrees did the temperature rise?
Use the number line.

Use the number line.

How long is the segment from $\displaystyle -4$ to $\displaystyle -1$? And the segment from $\displaystyle -4$ to $\displaystyle 2$?

Find five numbers between $\displaystyle -4$ and $\displaystyle -3$.