MTH1W Integers: Location, Direction and Change: Notes and Practice

At 6 a.m. on a January morning in Sudbury the thermometer reads $- 1 4^{\circ} C$ . By 2 p.m. it reads $5^{\circ} C$ . How much warmer did it get? It is tempting to say "9 degrees", because $14 - 5 = 9$ , but the real answer is 19 degrees: the temperature climbed 14 degrees just to reach zero, and then 5 more. Questions like this come up whenever something can sit on either side of a starting point: temperatures above and below freezing, heights above and below sea level, a bank balance that is positive (you have money) or negative (you owe money). The numbers that handle this are the integers, and the tool that answers "how much did it change?" is one short rule: change equals final minus initial.

Integers and the number line

The integers are the whole numbers together with their opposites: $\dots, - 3, - 2, - 1, 0, 1, 2, 3, \dots$ . On a number line, positive integers sit to the right of zero and negative integers sit to the left. Zero is neither positive nor negative; it is the reference point, the place you agree to measure from. In a real situation you have to know what zero stands for: sea level for elevation, the freezing point of water for degrees Celsius, an empty account for money. Two numbers the same distance from zero on opposite sides, such as $- 6$ and $6$ , are opposites. That distance from zero is the absolute value, written $∣ - 6∣ = 6$ . Absolute value tells you "how far", the sign tells you "which side".

Fig 1A number line from -10 to 10 with two moves: from -3 up to +5 (change +8) and from +4 down to -6 (change -10).

Order works the same way everywhere on the line: the number further right is greater. So $- 2 > - 7$ , because $- 2$ is to the right of $- 7$ . This feels backwards at first, since 7 is bigger than 2, but think of temperature: $- 2^{\circ} C$ is warmer than $- 7^{\circ} C$ . Among negative numbers, the one with the larger absolute value is the smaller number.

Situation	Zero means	Positive means	Negative means
Temperature (°C)	water freezes	above freezing	below freezing
Elevation (m)	sea level	above sea level	below sea level
Bank balance (dollars)	nothing in the account	money you have	money you owe
Floors in a building	ground floor	floors above ground	parking levels below ground
Change in a quantity	no change	an increase	a decrease

The sign of an integer depends on the reference point you choose

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Which list is in order from least to greatest?

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A number is 8 units to the left of zero. What is its opposite?

A tall roadside sign shaped like a thermometer reading White River, The Coldest Spot in Canada, 72 below zero, with a station wagon and a camping trailer parked in front. — PhotoThis 1973 sign in White River, Ontario, is a giant thermometer, which is really a vertical number line with values above and below zero. Its 72 below zero is in Fahrenheit, a reminder that you need to know what zero stands for before you compare temperatures.RichardBH from Hamilton, Canada, CC BY 2.0, via Wikimedia Commons

Change = final − initial

To measure a change, always subtract in the same order: the final value minus the initial (starting) value. The answer carries a sign that tells you the direction. A positive change is an increase (warmer, higher, more money); a negative change is a decrease. In the Sudbury example the change is $5 - (- 14)$ . Subtracting a negative number is the same as adding its opposite, so $5 - (- 14) = 5 + 14 = 19$ . The temperature rose 19 degrees, which matches counting on the number line: 14 steps from $- 14$ up to $0$ , then 5 more steps to $5$ .

change = final - initial 5 - (- 14) = 5 + 14 = 19 (- 6) - 4 = - 10

Fig 2A temperature scale for the Sudbury morning: the climb from −14 °C to 5 °C splits into 14 degrees up to 0 and 5 more, a total change of +19 °C.

Read the sign of a change

A change of

+ 8

means "up 8" or "increased by 8". A change of

- 10

means "down 10" or "decreased by 10". The size of the change is its absolute value; the sign is the direction. Swapping the order (initial minus final) gives the right size with the wrong sign.

Adding a positive moves right; adding a negative moves left: $- 3 + 8 = 5$ and $4 + (- 10) = - 6$ .
Subtracting means adding the opposite: $7 - (- 2) = 7 + 2 = 9$ and $- 5 - 3 = - 5 + (- 3) = - 8$ .
Multiplying or dividing two numbers with the same sign gives a positive answer; with different signs, a negative answer: $(- 4) (- 3) = 12$ , $(- 20) \div 5 = - 4$ .

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A bank balance goes from

- 25

dollars (money owed) to

40

dollars. What is the change in the balance, in dollars?

dollars

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The temperature goes from

2^{\circ} C

- 9^{\circ} C

. Which calculation gives the change?

VideoKhan Academy uses the number line to show why subtracting a negative number is the same as adding a positive one, the step you need for a change like 5 − (−14).Watch on YouTube

Signs in fractions, ratios, rates and decimals

A fraction is a division, so a negative sign follows the division sign rules. One negative sign, wherever it sits (in the numerator, in the denominator or in front), makes the whole fraction negative. Two negative signs cancel and make it positive. This matters for rates, which compare a change to the time or distance it took. If a lake's water level changes by $- 6$ cm over 4 days, the rate is $\frac{- 6}{4} = - 1.5$ cm per day: the negative sign says the level is falling. As a decimal the sign simply stays in front, $- 1.5$ .

\frac{- 3}{4} = \frac{3}{- 4} = - \frac{3}{4} = - 0.75 \frac{- 6}{- 8} = \frac{6}{8} = \frac{3}{4} = 0.75

AOne negative sign

$\frac{- 9}{12} = - \frac{3}{4}$
The value is negative
Rate of change: a decrease
Decimal form: $- 0.75$

BTwo negative signs

$\frac{- 9}{- 12} = \frac{3}{4}$
The signs cancel: the value is positive
e.g. a negative change over a negative step
Decimal form: $0.75$

Zero cannot be a denominator

\frac{0}{5} = 0

is fine (zero shared five ways is zero), but

\frac{5}{0}

has no value: no number times 0 gives 5. A rate such as "change per day" only makes sense over a time span that is not zero.

Fig 3A number line from −1 to 1. Four ways to write −3/4 (one negative sign) point to the same spot left of 0; −6/−8 = 6/8 = 3/4 (two negative signs) points to the spot the same distance right of 0.

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Which of these fractions is positive?

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A pond's water level changes by −9 cm over 6 days. What is the rate of change, in centimetres per day?

cm per day

Integers: Location, Direction and Change

Key things to remember

Integers and the number line

Change = final − initial

Signs in fractions, ratios, rates and decimals

Worked examples

Where marks get lost

Flashcards 8

Practice