Basic Characteristics of Addition and Multiplication of Real Numbers.
Everyone knows 2 < 5. Now multiply both sides by −1. Does −2 < −5 hold? No: it becomes −2 > −5. We obey some rules of addition and multiplication without thinking, and in one of them the inequality sign flips.
1. Six rules for adding and multiplying
If a and b are real numbers, then both a + b and ab are real numbers.
Before the laws get their names, test them. Drag a, b and c to anything you like, and both sides of each law work themselves out as you go.
law sandboxdrag a, b, c
Do the two sides agree?
Swapping the order changes nothing: a + b = b + a and ab = ba. With three numbers, which pair you join first does not matter either: (a + b) + c = a + (b + c) and (ab)c = a(bc).
Adding 0 changes nothing and multiplying by 1 changes nothing: a + 0 = 0 + a = a and a·1 = 1·a = a. The book says only two real numbers, 0 and 1, do this, and 0 ≠ 1.
Every a has an opposite for addition: a + (−a) = 0. If a ≠ 0, it has one for multiplication too: a·(1/a) = 1.
Addition and multiplication meet in this rule: a(b + c) = ab + ac. Six equations are below; tap one to see which rule it shows.
name itCompare the two sides
Match the property
equation
no.
3 + 5 = 5 + 3
2
(2 × 3) × 4 = 2 × (3 × 4)
3
7 + 0 = 7
4
5 + (−5) = 0
5
4 × 1/4 = 1
5
2(3 + 4) = 2·3 + 2·4
6
Commutative. Reverse the order and the sum is the same: a + b = b + a (characteristic 2).
Associative. Which two you multiply first does not change the result: (ab)c = a(bc) (characteristic 3).
Identity, 0. Adding 0 leaves the number as it was: a + 0 = a (characteristic 4).
Additive inverse. a and −a add up to 0: a + (−a) = 0 (characteristic 5).
Multiplicative inverse. For a ≠ 0, a·(1/a) = 1 (characteristic 5).
Distributive. The multiplier splits over both parts of the sum: a(b + c) = ab + ac (characteristic 6).
All six seen.
2. Three rules for inequalities
For any two real numbers a and b, one of three things is always true: a < b, or a = b, or a > b.
Adding the same number to both sides keeps the inequality: if a < b, then a + c < b + c. So does multiplying by a positive number: if a < b and c > 0, then ac < bc.
But if c < 0, then ac > bc. Multiplying by a negative number reverses the inequality. Take 2 < 5 and push c below zero.
handleTry a negative c
Multiply an inequality
The line is drawn by hand, not to scale.
3. Check yourself
Three questions, one at a time.
check yourselfpick one answer
Question 1
Under which condition does a·(1/a) = 1 hold?
It also holds for negative a, e.g. (−2)(−1/2) = 1; the only condition is a ≠ 0.
True for a = 1, but also for every other nonzero a; the condition is a ≠ 0.
It also holds for positive a, e.g. 2 × 1/2 = 1; the condition is a ≠ 0.
Right. Book: if a ≠ 0 then a·(1/a) = 1; there is no dividing by zero.
Now the second.
check yourselfpick one answer
Question 2
If a, b, c are real numbers and a < b, which of the following is correct?
With c > 0 the sign stays: ac < bc. The sign flips only when c < 0.
Right. By the book, if a < b and c < 0 then ac > bc: multiplying by a negative reverses the sign.
Dividing by a negative c also flips the sign, giving a/c > b/c; the correct one is ac > bc with c < 0.
Dividing by a positive c keeps the sign: a/c < b/c. The correct answer is ac > bc with c < 0.
And the last.
check yourselfpick one answer
Question 3
Writing 3(4 + 5) by the rule a(b + c) = ab + ac gives which?
Right. From a(b + c) = ab + ac, 3(4 + 5) = 3 × 4 + 3 × 5 = 27.
The 3 multiplies only the 4, not the 5: this gives 17, not 27.
This swaps the roles of + and ×, giving 23; the correct form is 3 × 4 + 3 × 5.
This is 7 × 8 = 56, not the distributive rule; the correct one is 3 × 4 + 3 × 5 = 27.
The rules of this page, on a few cards.
cut from the notetap to turn over
Seven cards
QuestionIf a and b are real numbers, what kind of numbers are a + b and ab?
AnswerBoth are real numbers
QuestionFor which two real numbers do a + 0 = a and a·1 = a hold?
Answer0 and 1 (with 0 ≠ 1)
QuestionWhat condition on a is needed for a·(1/a) = 1?
Answera ≠ 0
Questiona(b + c) = …
Answerab + ac
QuestionWhich three relations can hold between two real numbers a and b, one of them always?
Answera < b or a = b or a > b
QuestionIf a < b and c > 0, how are ac and bc related?
Answerac < bc
QuestionIf a < b and c < 0, how are ac and bc related?
Answerac > bc (the sign reverses)
Source: Mathematics, Classes 9-10 (NCTB), Chapter 1 “Real Numbers”, the “Basic Characteristics of Addition and Multiplication of Real Numbers” part.