Integers, place value and directed numbers

Every number topic in this course starts here.

Integers are the whole numbers you count with, the numbers below zero that measure cold, debt and depth, and zero itself between them.

Once you can read a large number digit by digit, say it in words and place any integer on a number line, everything from fractions to standard form has something to stand on.

You need to understand and use integers (positive, negative and zero), understand place value, use directed numbers in practical situations, e.g. temperatures, and order integers.

An integer is a whole number.

It can be positive (1, 2, 3, ...), negative ($-1$, $-2$, $-3$, ...)

or zero.

The numbers $2.5$ and $\frac{1}{2}$ are not integers, because they are not whole.

A negative number is written with a minus sign in front, and you say "negative four" or "minus four" for $-4$.

Zero is an integer and it is neither positive nor negative.

It is the point that separates the two.

The mathematician Brahmagupta, working in India in the 7th century, was one of the first to write down rules for calculating with zero and with negative numbers as numbers in their own right.

The value of a digit depends on its place in the number.

In 3 470 215 the digit 7 is worth seventy thousand, but in 1 207 it is worth only seven.

Each place is worth ten times the place to its right.

• The 3 is worth 3 000 000, the 4 is worth 400 000 and the 7 is worth 70 000.

• The 0 shows there are no thousands. It holds the place so the other digits keep their values. Without it, 3 470 215 would collapse to 347 215, a very different number.

• Large numbers are written in groups of three digits from the right, with a small space between groups, so they are easy to read.

To write a number in words, read it one group of three at a time and say the group's name: "three million, four hundred and seventy thousand, two hundred and fifteen".

A directed number is a number with a sign that shows a direction: up or down, above or below, in credit or in debt.

Temperatures are the everyday example.

A reading of $-6$ °C is 6 degrees below freezing; $5$ °C is 5 degrees above.

On a winter morning like the one below, frost covers the ground because the air is colder than the freezing point of water, 0 °C. The red liquid in the thermometer has sunk low, below the freezing mark, so the reading is a negative number.

A thermometer is a number line stood on its end: up is warmer, down is colder.

The number line below shows the idea.

Positive numbers go to the right of zero and negative numbers to the left.

Moving right means going up: a rise in temperature, a deposit into a bank account, a lift going up.

Moving left means going down.

To find a change, count the steps between the two numbers.

Heights and depths work the same way, with sea level as zero.

In the picture below, the seabird flies at a positive height above the water, the boat floats at sea level (zero), and the diver and the submarine are at negative positions below it.

The submarine is deeper, so its position is the more negative number, just as $-20$ is less than $-5$.

Many students subtract the digits and say $6 - 5 = 1$.

The number line shows why that is wrong.

From $-6$ up to $0$ is 6 degrees, and from $0$ up to $5$ is 5 more, so the total change is $6 + 5 = 11$ degrees.

The jump crosses zero, so the two parts add.

In general, the difference between two temperatures is the larger one take away the smaller one: $5 - (-6) = 5 + 6 = 11$.

Subtracting a negative number is the same as adding the positive number.

The next section shows why.

On the number line, a number is smaller than every number to its right.

So:

• every negative number is smaller than zero, and zero is smaller than every positive number;

• among negative numbers, the one with the bigger digits is the smaller number: $-9 < -4$, because $-9$ is further left (9 degrees below freezing is colder than 4 below).

The symbol $<$ means "is less than" and $>$ means "is greater than".

The pointed end always faces the smaller number.

Directed numbers come back all through the course: coordinates in all four quadrants use negative values, solving equations often gives a negative answer, and a gradient of $-2$ means a line going downhill.

Place value is the base for decimals, rounding, significant figures and standard form in the next unit.

1. Write down the value of the 4 in the number 62 438 Total [1]

2. Write the number 50 307 in words. Total [1]

3. Write the number two million, forty thousand and six in figures. Total [1]

4. Write these temperatures in order. Start with the coldest. $3$ °C $\quad -8$ °C $\quad -1$ °C $\quad 0$ °C $\quad -11$ °C $\quad 6$ °C Total [1]

5. The table shows the temperature at midnight in five towns.

(a) Which town had the lowest temperature? (1)
(b) Work out the difference between the highest temperature and the lowest temperature. (1) By midday, the temperature in Calder had risen by 13 degrees.
(c) Work out the temperature in Calder at midday. (1) Total [3]

6. Kofi says that $-7$ is greater than $-2$ because 7 is greater than 2. Is Kofi correct? Give a reason for your answer. Total [1]

7. In a building the ground floor is floor 0. Floor $-3$ is three floors below the ground floor. A lift starts at floor $-3$. It goes up 9 floors and then down 4 floors. Work out the number of the floor where the lift stops. Total [2]

8. Write down the number that is 1000 more than 49 650 Total [1]

9. Amal's bank account has a balance of $-35$ euros, which means he owes the bank 35 euros. He pays in 120 euros. Work out the new balance of his account. Total [1]

10. A submarine is at $-240$ m, which means 240 m below sea level. It rises by the same number of metres each minute for 5 minutes. It is then at $-65$ m. Work out the number of metres it rises each minute. Total [2]