2.1 Place Value & Whole Numbers

Place value questions appear in every ICAS Mathematics paper, from the Introductory paper (whole numbers to 100) right through to Paper J. They are rarely labelled "place value" — instead they hide inside questions about ordering, rounding, skip counting and digit puzzles. Because personal calculators are not allowed, ICAS rewards students who understand the structure of numbers well enough to work quickly and accurately on paper.

What Place Value Means

Our number system is a base-ten system: each place is worth ten times the place to its right. Moving one place to the left multiplies a digit's value by ten; moving one place to the right divides it by ten.

Digit Value vs Number Value

This is the single most tested idea. The digit is just the symbol; its value is the digit multiplied by its place. In 473 286, the digit 7 sits in the ten-thousands place, so its value is 7 × 10 000 = 70 000 — not 7, not 7 000.

Worked example: What is the value of the digit 4 in 2 406 715?

1. Locate the 4: it is in the hundred-thousands place.
2. Multiply: 4 × 100 000 = 400 000.

ICAS loves to offer the distractor "4 000" here, catching students who count places from the left instead of from the right. Always count places starting at the ones.

Reading and Writing Large Numbers

Read a large number in groups of three digits, working left to right: millions, then thousands, then the last group.

• 3 405 028 is read "three million, four hundred and five thousand, twenty-eight".
• Notice the zero in 3 405 028: it is a placeholder holding the ten-thousands place. You do not say it, but you must write it.

Writing from words: "six hundred and two thousand and forty" = 602 040. Say it back to yourself: six hundred and two thousand (602 000) plus forty (40). The trap answer 620 400 swaps the groups — check each group of three separately.

For younger papers (Introductory and Paper A) the same ideas apply to numbers to 100: know that 47 is 4 tens and 7 ones, that 70 is bigger than 17, and how to count on in twos, fives and tens.

Skip Counting and Number Patterns

Skip counting means counting forward (or backward) in equal steps. ICAS phrases these as "continue the pattern" or "what is the missing number?"

Worked example: 1 250, 1 500, 1 750, ___, 2 250.

• Find the step: 1 500 − 1 250 = 250.
• Check with the next pair: 1 750 − 1 500 = 250. Consistent.
• Missing number: 1 750 + 250 = 2 000. Confirm: 2 000 + 250 = 2 250. ✓

Always find the step from two consecutive terms, then verify it against a third term before answering.

Ordering and Comparing Whole Numbers

To compare numbers, first count the digits — more digits means a bigger number. If the digit counts are equal, compare from the left, one place at a time, until the digits differ.

Worked example: Order from smallest to largest: 408 097, 480 097, 408 970, 408 790.

1. All have six digits. Compare hundred-thousands: 480 097 has 8, the others have 0 — so 480 097 is the largest.
2. The remaining three all start 408 ___. Compare the hundreds digit: 0, 9, 7. So 408 097 < 408 790 < 408 970.
3. Final order: 408 097, 408 790, 408 970, 480 097.

Rounding Whole Numbers

Rounding to a place means finding the nearest multiple of that place.

• Look at the digit one place to the right of the target place.
• If it is 5, 6, 7, 8 or 9, round the target digit up.
• If it is 0, 1, 2, 3 or 4, leave the target digit unchanged.
• Every digit to the right of the target place becomes zero.

Worked examples with 46 753:

Note the last row: rounding 46 753 to the nearest 10 000 pushes past 49 999 up to 50 000 — a classic ICAS trap. Rounding can change a whole group of digits, not just one.

Rounding is also your best estimation tool. Before any calculation in a later section, round each number to a friendly value and predict the size of the answer; it catches most slips instantly.

Classic ICAS Traps

• Placeholder zeros: 4 080 − 4 008 = 72, not 0. Keep every zero in its place.
• Digit rearrangement: "Using the digits 3, 0, 7 and 2 once each, make the largest possible number" → 7 320 (largest digits in the highest places). The smallest is 2 037 — zero cannot lead a number.
• "Ten times larger" questions: making a number ten times larger appends a zero (460 → 4 600); it does not add ten.
• Nearest vs next: "the number 100 more than 9 950" is 10 050 — the hundreds column overflows into the thousands.