Notes · Unit 1
Real Numbers
Eleven short slides on the number system, number line, properties, inequalities, radicals and exponents.
Press Next (or the right arrow key) to move on. On most slides, each press reveals one more idea.
Number systemsNumber linePropertiesInequalitiesRadicalsExponents
1 · The number family
Each set sits inside the next
R Real numbers every point on the number line
Q Rational numbers p/q with q ≠ 0, such as ½ and −¾
Z Integers … −2, −1, 0, 1, 2 …
W Whole numbers 0, 1, 2, 3 …
N Natural numbers 1, 2, 3 …
Q′ Irrational numbers such as √2 and π. Together with Q they make up R.
No number is both rational and irrational, so Q ∩ Q′ = ∅.
2 · Rational or irrational?
The decimal tells you
Rational
Ends, or repeats in a block
3/8 = 0.3751/3 = 0.333…√16 = 4
Irrational
Never ends, never repeats
√2 = 1.41421…π = 3.14159…0.1010010001…
Watch out: √16 looks like a root, but it equals 4, so it is rational. A root is irrational only when it does not come out as a fraction.
3 · The number line
Every real number has a place
- Positive numbers lie to the right of 0. Negative numbers lie to the left.
- Fractions like ½ sit between integers. The teal dots are rational.
- Irrational numbers such as √2 ≈ 1.414 also have exact places. Rational and irrational points together fill the whole line.
4 · Properties of real numbers
Rules that always hold
Commutative
a + b = b + aab = ba
Associative
(a+b)+c = a+(b+c)(ab)c = a(bc)
Identity
a + 0 = aa × 1 = a
Inverse
a + (−a) = 0a × 1/a = 1 (a ≠ 0)
Distributive
a(b + c) = ab + ac5(x + 2) = 5x + 10
Why 0 has no multiplicative inverse: a multiplicative inverse must give product 1, but 0 × anything is 0.
5 · Order and inequalities
When does the sign flip?
Add the same number to both sides
sign stays: 3 < 5 ⇒ 5 < 7 (added 2)
sign stays: 3 < 5 ⇒ 5 < 7 (added 2)
Multiply by a positive number
sign stays: 3 < 5 ⇒ 6 < 10 (times 2)
sign stays: 3 < 5 ⇒ 6 < 10 (times 2)
Multiply by a negative number
sign flips: 2 < 3 ⇒ −2 > −3
sign flips: 2 < 3 ⇒ −2 > −3
Take reciprocals of positive numbers
sign flips: 2 < 5 ⇒ ½ > ⅕
sign flips: 2 < 5 ⇒ ½ > ⅕
On a number line, x > −4 uses a hollow circle at −4 because −4 is not included. x ≥ −4 uses a filled circle.
6 · Radicals
The nth root
If an = b, then a = n√b = b1/n
√ is the radical signn is the indexb is the radicand
Product rule (same index only)
n√a · n√b = n√(ab)
√3 × √12 = √36 = 6
n√a · n√b = n√(ab)
√3 × √12 = √36 = 6
Reducing the index (x > 0)
8√(x4) = x4/8 = x1/2 = √x
8√(x4) = x4/8 = x1/2 = √x
7 · Exponents
Laws of exponents
am · an = am+n add powers when multiplying
am ÷ an = am−n subtract powers when dividing
(am)n = amn multiply powers
a−n = 1/an negative power means reciprocal (a ≠ 0)
Worked example
(8/27)−2/3 = (27/8)2/3 = (3√(27/8))2 = (3/2)2 = 9/4
(8/27)−2/3 = (27/8)2/3 = (3√(27/8))2 = (3/2)2 = 9/4
8 · Common mistakes
Five traps to avoid
Cancelling a zero. ac = bc gives a = b only if c ≠ 0. For example 3 × 0 = 5 × 0, but 3 ≠ 5.
Reordering subtraction. 5 − 3 ≠ 3 − 5, so subtraction is not commutative.
Assuming irrational + irrational is irrational. √2 + (−√2) = 0, which is rational.
Merging different indices. √2 · 3√3 cannot use the product rule.
Forgetting to flip. Multiplying 2 < 3 by −1 gives −2 > −3, not −2 < −3.
9 · Quick check
Try these, then tap to check
10 · Your turn
Practise Unit 1
Three timed MCQ tests with an explanation after every answer.
Practice questions are independently written and are not official FBISE papers.