This document discusses integers and operations involving positive and negative numbers. It begins by explaining positive and negative numbers in real-world contexts like temperature, money, and depth. It then defines integers as the set of whole numbers along with their negatives. The document goes on to cover absolute value, comparing integers on a number line, and the rules for adding, subtracting, multiplying, and dividing integers based on the signs of the numbers. It concludes with a brief discussion of exponents, roots, and the order of operations when combining different types of calculations involving integers.

1. Lesson 4 INTEGERS
LESSON 4 INTEGERS
1. POSITIVE AND NEGATIVE NUMBERS
We can use negative numbers in different situations:
·Positions:
A submarine which is sailing 700 m below sea level: – 700 m
The second floor of a subterranean garage: – 2
·Money:
He is €400 overdrawn: – 400
·Temperatures:
The temperature in Siberia has risen 25ºC below zero: -25ºC
Negative numbers are less than zero.
Use a number line to order negative numbers. For instance, it is easy to see that -2 is a
higher number than -5 because it is further to the right on the number line.
Negative numbers are always written with a ‘-’ sign in front of them and they are
counted from zero to the left. We read them as “negative” or “minus”, i.e., -5 is read as
“negative five” or “minus five”.
2. THE SET OF INTEGERS
If we include the negative numbers with the whole numbers, we have a new set of
numbers that are called integers.
Integers: {…, -3, -2, -1, 0, 1, 2, 3, …}
The Integers include zero, the counting numbers, and the negative of the counting
numbers, to make a list of numbers that stretch in either direction infinitely.
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2. Lesson 4 INTEGERS
Absolute Value means only how far a number is from zero:
“6″ is 6 away from zero, and “-6″ is also 6 away from zero. So the absolute value of 6 is
6, and the absolute value of -6 is also 6.
So in practice “absolute value” means to remove any negative sign in front of a number,
and to think of all numbers as positive (or zero).
To show that you want the absolute value of something, you put “|” marks either side
(they are called “bars”), like these examples:
|-5| = 5 |7| = 7
Opposite. For every positive integer, there’s a negative integer an equal distance from
the origin. Two integers that lie the same distance from the origin in opposite directions
are called opposites. For example, “negative 5″ is the opposite of “positive 5.”
Comparing numbers. Writing numbers down on a Number Line makes it easy to tell
which numbers are bigger or smaller.
• Between a positive number and a negative number, the positive number is always
larger:
5 > -2 1 > -3 2 > -100
• Between two positive numbers, the larger number is the one with the largest
absolute value:
7>2 10 > 4 300 > 200
• Between two negative numbers, the larger number is the one with the smallest
absolute value:
-7 < -2 -10 < -4 -300 < -200
Example: John owes $3, Virginia owes $5 but Alex doesn’t owe anything, in fact he has
$3 in his pocket. Place these people on the number line to find who is poorest and who
is richest.
Having money in your pocket is positive. But owing money is negative. So John has “-3″,
Virginia “-5″ and Alex “+3″.
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3. Lesson 4 INTEGERS
Now it is easy to see that Virginia is poorer than John (-5 is less than -3) and John is
poorer than Alex (-3 is smaller than 3), and Alex is, of course, the richest!
3. ADDING AND SUBTRACTING INTEGERS
Using the Number Line
When we add, we move right on the number line.
When we subtract, we move left on the number line
Example:
To evaluate -3+2 , we start at -3 and move 2 places to the right
The answer is -1.
To evaluate -3-2 , we start at -3 and move 2 places to the left
The answer is -5.
More examples.1
Using Rules
Rule 1 :
The sum of two or more positive integers is a positive integer.
The sum of two or more negative integers is a negative integer.
Example:
(+7) + (+ 4) = + 11
(-8) + (– 9) = – 17
Rule 2 :
To find the sum of a positive and a negative integer:
Subtract the two numbers (ignore the signs) and then keep the sign of the larger integer.
(– 9) + (+ 7) = – 2
(– 3) + (+ 5) = + 2
1 http://www.mathsisfun.com/numbers/number-line-using.html
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4. Lesson 4 INTEGERS
4. ADDITIONS AND SUBTRACTIONS WITH BRACKETS
Remember: If a number has no sign it usually means that it is a positive number.
Example: 5 is really +5
Adding positive numbers is just simple addition. Example: 2 + 3 = 5 is really saying
“Positive 2 plus Positive 3 equals Positive 5″
You could write it as (+2) + (+3) = (+5)
Subtracting positive numbers is just simple subtraction.Example: 6 – 3 = 3 is really
saying
“Positive 6 minus Positive 3 equals Positive 3″
You could write it as (+6) – (+3) = (+3).
Subtracting a negative is the
same as adding.
Example: 6-(-3) = 6+3 = 9.
Subtracting a positive
or
Adding a negative
is
subtraction
15 - (+3) = 12
15 + (-3) = 12
Try playing Casey Runner2, you need to know the rules of positive and negative to
succeed!
5. MULTIPLICATION AND DIVISION
To multiply two or more integers we have to multiply the signs and then the absolute
value of the numbers. The rule for multiplying and dividing the signs is very similar to
the rule for adding and subtracting. When the signs are different the answer is negative
and when the signs are the same the answer is positive:
2 http://www.mathsisfun.com/numbers/casey-runner.html
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5. Lesson 4 INTEGERS
Now try a Test Bite3
6. POWERS AND ROOTS
The sign of a power depends on the base and the exponent.
If the base is positive, the power will be positive, as you learnt in lesson 2.
If the base is negative,
and the exponent is even, the power will be positive:
(–3)4 = (–3)(–3)(–3)(–3)
= (9)(9)
= 81
and the exponent is odd, the power will be negative:
(–3)5 = (–3)(–3)(–3)(–3)(–3)
= (9)(9)(–3)
= –243
Be careful about grouping. To avoid confusion, use brackets () in cases like this:
With () : (-2)2 = (-2) × (-2) = 4
Without () : -22 = -(22) = – (2 × 2) = -4
The rules for square roots are as follows:
a positive integer has two square roots, one is positive and the other one is negative (its
opposite):
a negative integer has no square roots.
Combined operations.
When expressions have more than one operation, we have to follow rules for the order
of operations. These are the same rules as for natural numbers:
Rule 1: First perform any calculations inside the brackets.
Rule 2: Do powers and roots.
Rule 3: Next perform all multiplications and divisions, working from left to right.
Rule 4: Lastly, perform all additions and subtractions.
To finish, try this exercises.4
3 http://www.bbc.co.uk/apps/ifl/schools/gcsebitesize/maths/quizengine?quiz=negativenumbers&templateStyle=maths
4 http://www.regentsprep.org/Regents/math/ALGEBRA/AOP3/Smixed.htm
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6. Lesson 4 INTEGERS
GLOSSARY
Look for the following words in this dictionary5 and copy them in your notebook.
Sign
Absolute value
Opposite
Even
Odd
5 http://www.amathsdictionaryforkids.com/dictionary.html
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