NumberFoundation + Higher★ Core topic
Integers, Place Value
& Ordering Numbers.
An integer is a whole number — positive, negative or zero. Each digit's value depends on its position. To order, start at the largest place value. To add/subtract negatives, picture the number line.
I · Key definitions
Integer
A whole number. Positive, negative, or zero. No fractions, no decimals.
e.g. −7, 0, 42
Place value
The value a digit holds based on its position.
in 374, the 7 = 70
Digit
A single numeric symbol: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
374 has 3 digits
Ascending
Smallest to largest. Climbing up.
−5 < 0 < 3 < 8
Descending
Largest to smallest. Going down.
8 > 3 > 0 > −5
Negative
A number less than zero, written with a minus sign.
e.g. −3, −17, −100
II · The place value chart
Millions
H.Thous.
T.Thous.
Thous.
Hundreds
Tens
Ones
Tenths
Hund'ths
Thous'ths
Reads: Two million, four hundred and nine thousand, three hundred and seventy-five point eight six two. The green 3 = 300; the gold 8 = 0.8.
III · The number line
← SMALLER
LARGER →
−5
−4
−3
−2
−1
1
2
3
4
5
Further left = smaller. So −5 is smaller than −1. Any negative is smaller than any positive. Zero sits in the middle — neither positive nor negative.
IV · Comparing · symbols & method
The symbols
a < b a is less than b
a > b a is greater than b
a ≤ b less than or equal
a ≥ b greater than or equal
the alligator's mouth always opens toward the bigger number
Method · ordering
1. Count digits — more = bigger
2. Same length? Compare leftmost digit
3. If equal, compare next digit right
2,345 > 987 (4 digits > 3) · 524 > 498 (5 > 4)
Ordering decimals
1. Line up decimal points · 2. Pad with zeros · 3. Compare left to right
0.5, 0.45, 0.503 → pad → 0.45 < 0.5 < 0.503
V · Negative numbers · the sign rules
Same signs → positive. Different signs → negative. Same rules for division (÷).
VI · Adding & subtracting negatives
Two signs next to each other
+ + = + · − − = +
+ − = − · − + = −
5 − (−3) = 5 + 3 = 8 · 5 + (−3) = 5 − 3 = 2
Number-line picture
Add → move right · Subtract → move left
−2 + 5 → start at −2, move 5 right → 3
3 − 7 → start at 3, move 7 left → −4
VII · Absolute value · |x|
Absolute value = distance from zero on the number line — always positive or zero. Strip off the sign.
VIII · Rounding rules
The rounding rule · digit to the right
0–4 → round down
5–9 → round up
Round to the nearest 10, 100, 1000
348 → 350 (10)
348 → 300 (100)
4,730 → 5,000 (1000)
Decimal places (d.p.) & significant figures (s.f.)
6.2837 → 6.28 (2 d.p.)
6.2837 → 6.3 (2 s.f.)
0.00462 → 0.0046 (2 s.f.)
IX · Order of operations · BIDMAS
Top to bottom. Brackets first. Indices next. Then divide and multiply together, left to right. Finally add and subtract together, left to right.
X · Worked examples
Order these ascending
3 marks
Arrange in ascending order: −4, 203, −17, 30, 0
i.Separate negatives from positives. Negatives: −4, −17. Positives (and zero): 0, 30, 203.
ii.For negatives: further from zero = smaller. So −17 < −4.
iii.For positives: by digit count. 0 < 30 < 203.
−17 < −4 < 0 < 30 < 203
Calculate with negatives & BIDMAS
2 marks
Work out: −3 + 4 × (−2)²
i.Indices first: (−2)² = (−2)(−2) = 4 (same signs → positive)
ii.Expression becomes: −3 + 4 × 4
iii.Multiply before add: 4 × 4 = 16 → −3 + 16
iv.Add: −3 + 16 = 13
−3 + 4 × (−2)² = 13
XI · Common mistakes & examiner tips
Common mistakes
Forgetting that −1 > −5. More negative = smaller, not bigger.
Mixing up d.p. vs s.f. 0.00621 to 2 s.f. = 0.0062, not 0.01.
Dropping the sign after multiplying. (−3)(−2) = +6, not −6.
Doing add before multiply. BIDMAS always wins.
Ignoring the decimal point when comparing: 0.5 > 0.45, not the other way.
Examiner tips
Sketch a number line in the margin — wins easy marks on negatives.
Rewrite ++, −−, +−, −+ in one step before calculating.
Show your BIDMAS working line by line — method marks are easy here.
Estimate before calculating. If the answer looks wildly off, a sign slipped.
Never round early. Keep full accuracy until the final step.