The families sit inside one another.
Every natural number is whole, every whole number is an integer, every integer is rational, and every rational number is real. Nothing is both rational and irrational — those two divide the reals in half.
- Home
- Grade 7-12 Math Review
- The Real Number System
The Real Number System
Every number belongs to several families at once. The properties are the rules those families obey.
Classify the number
To which of these does −4 belong?
−4 is an integer; it is also rational, because it can be written as −41; and every rational number is real. So it belongs to all three. B is wrong because whole numbers start at 0 and go up — negatives are not whole numbers. C is impossible: rational and irrational are opposites, and no number is both.
Rational or irrational: can it be written as a fraction?
Rational = ab with a and b integers and b ≠ 0.
| Number | As a fraction | Verdict |
|---|---|---|
| 0.75 | 34 | rational |
| 0.333… | 13 | rational — it repeats |
| −6 | −61 | rational |
| √9 | 3, so 31 | rational — it simplifies |
| √7 | cannot be | irrational |
| π | cannot be | irrational |
A root sign does not make a number irrational
√9 is 3, an ordinary integer. Simplify first, then judge. And 227 is a rational approximation of π, not π itself — π never repeats and never ends.
The five properties, named by what moved.
| Property | What changed | Example |
|---|---|---|
| Commutative | the order | 7 + 3 = 3 + 7 |
| Associative | the grouping | (2 + 5) + 8 = 2 + (5 + 8) |
| Distributive | spread across a sum | 4(10 + 3) = 40 + 12 |
| Identity | nothing — 0 or 1 added or multiplied | 9 + 0 = 9, 9 × 1 = 9 |
| Inverse | undone to 0 or 1 | 9 + (−9) = 0, 9 × 19 = 1 |
order moved → commutative
brackets moved → associative
both → look again
The quickest test: if the numbers are in a different order, it is commutative; if the same numbers sit in different brackets, it is associative.
These properties are mental-arithmetic tools.
They are not vocabulary to memorise — they let you rearrange a calculation into an easier one.
25 × 17 × 4
awkward left to right
= 25 × 4 × 17
commutative
= 100 × 17 = 1,700
associative, then trivial
6 × 98
no calculator
= 6(100 − 2)
distributive
= 600 − 12 = 588
in your head
A repeating decimal is a fraction in disguise.
This is why every repeating decimal is rational. Write 0.272727… as a fraction in lowest terms.
x = 0.272727…
two digits repeat
100x = 27.272727…
shift by one full block
99x = 27
subtract: the tails cancel
x = 2799 = 311
divide top and bottom by 9
Check: 3 ÷ 11 = 0.2727… ✓ One repeating digit gives a denominator of 9 (0.555… = 59); two give 99.
27100 is a different number
Writing 0.272727… as 27100 gives 0.27, which stops. The repeating part needs 99 underneath, not 100, so the answer is 2799 = 311.
Placing irrational numbers on the number line.
Trap a square root between the two perfect squares around it.
Between which two consecutive integers does √45 lie, and which tenth is it closest to?
36 < 45 < 49
the perfect squares either side
6 < √45 < 7
take the square root of each
6.7 × 6.7 = 44.89, 6.8 × 6.8 = 46.24
test the tenths near 7
√45 ≈ 6.7
44.89 is far closer to 45
Order −1.6, −√3 and −53 from least to greatest.
−1.6, −1.732…, −1.666…
every number as a decimal
−√3, −53, −1.6
the most negative comes first
Negatives flip the order
1.6 < 53 < √3, so it is tempting to write −1.6, −53, −√3. With the minus signs the biggest size is the smallest number, so the order turns round: −√3, −53, −1.6.
Variations: a root that simplifies, a sum, and a product.
Still "simplify first, then judge"; these are the cases that fool a quick glance.
Name every family −√363 belongs to.
−√363 = −63 = −2
simplify first
integer, rational, real
not whole, not natural: it is negative
A root sign, a fraction bar and a minus sign all at once, and it is still just −2.
Is 5 + √2 rational or irrational?
5 + 1.41421356…
√2 never ends or repeats
6.41421356…
adding 5 changes only the whole part
Rational + irrational is always irrational: the endless tail cannot be cancelled by a fraction.
Is √3 × √12 rational or irrational?
√3 × √12 = √36
multiply under one root
= 6
rational
Two irrational numbers can multiply to a rational one, so "irrational × irrational" has no fixed answer.
The two inverses.
Additive inverse gets you to 0. Multiplicative inverse gets you to 1.
7
additive −7, multiplicative 17
7 + (−7) = 0, 7 × 17 = 1
−25
additive 25, multiplicative −52
flip it, keep the sign
0
additive 0, multiplicative none
nothing times 0 gives 1
Zero is the one number with no multiplicative inverse, which is the same fact as "you cannot divide by zero", said in different words.
On the exam
Expect "which of these is irrational", "which property is shown" and "between which two integers" items. The fastest route is to simplify every root and fraction first, then judge; the 10-second check for a property is to ask what moved, and for an estimate, to square your answer.
Now you try
Practice problem
Name the property
5 × (8 + 2) = 5 × 8 + 5 × 2
The 5 has been spread across a sum inside brackets, which is the distributive property. Commutative would mean the order changed and nothing else; associative would mean the brackets moved but the operation stayed the same throughout. Here a multiplication turned into two multiplications and an addition, which only distribution does.
Hint: ask what the brackets did. If something outside them got multiplied into everything inside, it is distribution.