General Mathematics · Class 9-10 · Chapter 1 · Real numbers
How addition and multiplication behave.
Nothing new is taught in this section; what gets done is naming what you have been using without thinking. The naming is not wasted, because in algebra and inequalities these names have to be written beside every step. And the last rule on the list genuinely earns its keep, because it is the one most often broken.
1. Nine properties
Let a, b and c be any real numbers. Closure says that a + b and ab are both real; adding or multiplying never takes you outside the reals. Commutativity says swapping the order changes nothing, and associativity says moving the brackets changes nothing either. These are two different statements, and very easy to confuse.
Then the identities: a + 0 = a and a × 1 = a. The book says there are only two such real numbers, 0 and 1. Then the inverses: a + (−a) = 0, and if a is not zero, a × 1/a = 1. Notice that last condition, because zero has no multiplicative inverse; this is exactly where the q ≠ 0 in the definition of a rational number comes from. And distributivity: a(b + c) = ab + ac.
One thing is worth adding: every one of these holds for both kinds of real number, rational and irrational alike. √2 + (−√2) = 0 is every bit as true as 5 + (−5) = 0.
name and ruletap to turn over
Nine properties, nine cards
nameClosure
rulea + b and ab are both real. √2 + 3 is real, and so is √2 × 3.
nameCommutative, addition
rulea + b = b + a. For instance 7 + 4 = 4 + 7 = 11.
nameCommutative, multiplication
ruleab = ba. For instance 7 × 4 = 4 × 7 = 28.
nameAssociative
rule(a + b) + c = a + (b + c). For instance (2 + 3) + 5 = 2 + (3 + 5) = 10.
nameAdditive identity
rulea + 0 = a. For instance -9 + 0 = -9. The identity is 0.
nameMultiplicative identity
rulea × 1 = a. For instance √5 × 1 = √5. The identity is 1.
nameAdditive inverse
rulea + (−a) = 0. For instance √2 + (−√2) = 0; it works for irrationals too.
nameMultiplicative inverse
ruleIf a ≠ 0 then a × 1/a = 1. For instance 4 × 1/4 = 1. Zero has none.
nameDistributive
rulea(b + c) = ab + ac. For instance 3(4 + 5) = 3·4 + 3·5 = 27.
There is one reason commutativity and associativity get confused: in both, things move about. The difference is what moves. Under commutativity two terms swap places; under associativity the terms stay exactly where they were and only the brackets move. Sorting the eight equations below into three bins makes the difference show itself.
sort thempick a chip, then a bin
Which property is at work?
Pick up an equation, then drop it in a bin. What actually moved between the two sides is what names it.
Right. Next one.
Not there. Did terms swap places, did brackets move, or did an outside number spread inside?
All in place. How to tell them apart: terms swapping is commutative, brackets moving is associative, and an outside factor landing on each inside term is distributive.
check yourself
One question
7(10 + 2) = 70 + 14. Which property is this?
Under commutativity two terms swap places, as in 5 + 3 = 3 + 5. Nothing swapped here.
Under associativity the brackets move while the terms stay put. Here the outside 7 has moved inside, which is a different matter.
Right. The outside 7 has landed on both terms inside the bracket: a(b + c) = ab + ac. 7 × 10 = 70 and 7 × 2 = 14.
The identity properties are about 0 and 1, as in a + 0 = a. Neither appears here.
2. Three rules about order
The remaining three are about comparison. Trichotomy says that for any two reals a and b, exactly one of a < b, a = b and a > b holds: never two, and never none. The second says that if a < b, adding the same number to both sides leaves the inequality intact, whatever c is.
The third is the one that matters, and the one most often broken. If a < b and the multiplier is positive, then ac < bc and the sign stays as it was. But if the multiplier is negative, then ac > bc: the inequality turns over. And if the multiplier is zero, both sides become zero and the inequality disappears altogether. Make c negative below and watch.
calculatormake c negative
When does the sign turn over?
With the rule in hand, test it on a question.
check yourself
Another question
Given -2 < 5, which of these is correct?
The sign has been left as it was, but the multiplier -3 is negative. Work it out: 6 and -15, and 6 is not the smaller.
Right. (-2)(-3) = 6 and (5)(-3) = -15, and 6 > -15. The multiplier is negative, so the inequality turned over.
They would be equal only if the multiplier were zero. -3 is not, so the two sides stay apart.
Both are real numbers, and trichotomy says any two reals can always be compared.
And the last question is about the one exception, which comes back through the whole chapter.
check yourself
Last question
Which number has no multiplicative inverse?
The inverse of 1 is 1 itself, since 1 × 1 = 1.
The inverse of -1 is -1 itself, since (-1) × (-1) = 1.
Right. A multiplicative inverse is something that multiplies to give 1. But 0 times anything is 0, never 1. This is why the definition of a rational number carries the condition q ≠ 0.