The Number Properties That Make Mental Maths Easy

Natural numbers are the counting numbers 1, 2, 3, 4 and so on. Whole numbers are the natural numbers together with 0. That single extra element is the entire difference: every natural number is a whole number, but 0 is a whole number that is not natural.

This page covers everything in the ICSE Class 6 Mathematics chapter on natural and whole numbers: successors and predecessors, the properties that let you reorder a calculation safely, and how rounding and estimation let you check an answer without doing the full sum.

The successor of a number is the number that comes immediately after it, found by adding 1. The predecessor is the number immediately before it, found by subtracting 1.

The successor of 499 is 500, and its predecessor is 498. Every whole number has a successor, so the list of whole numbers never ends.

But predecessors run out. 0 has no predecessor among the whole numbers, because $0-1$ is not a whole number. This is exactly why whole numbers have a smallest member (0) but no largest one.

For example, house numbers on a street behave like successors — each plot is one more than the last — but there is no house numbered one less than the first.

A point students often miss: the smallest natural number is 1, while the smallest whole number is 0. Questions deliberately test that distinction.

Four properties do the heavy lifting, and each one is a licence to rewrite a sum into an easier shape.

Closure: adding or multiplying two whole numbers always gives a whole number. $7+5=12$ and $7\times 5=35$ are both whole numbers.

Commutative: order does not matter. $a+b=b+a$ and $a\times b=b\times a$.

Associative: grouping does not matter. $(a+b)+c=a+(b+c)$.

Distributive: multiplication spreads across a sum, $a\times (b+c)=a\times b+a\times c$.

There are also two special numbers. 0 is the additive identity, since $a+0=a$, and 1 is the multiplicative identity, since $a\times 1=a$.

For example, to find $25\times 17\times 4$, commutativity and associativity let you reorder it as $(25\times 4)\times 17=100\times 17=1700$, which needs no written working at all.

One boundary case matters: subtraction and division are not commutative or associative. $9-4$ is not $4-9$, so these shortcuts apply to addition and multiplication only.

Rounding replaces a number with a nearby, simpler one. Find the digit to the right of the place you are rounding to: if it is 5 or more, round up; if it is less than 5, round down, and replace everything after the rounding place with zeros.

Round 4,678 to the nearest ten: the digit after the tens place is 8, so round up to 4,680.

Round the same number to the nearest hundred: the digit after the hundreds place is 7, so round up to 4,700.

Round it to the nearest thousand: the digit after the thousands place is 6, so round up to 5,000.

Notice that the rounded answer depends entirely on which place you were asked for — the same number becomes 4680, 4700 or 5000. So always read the question's place before rounding.

In real life, a shopkeeper quoting "about Rs 5,000" for goods worth Rs 4,678 has rounded to the nearest thousand.

Estimation means rounding each number first, then calculating, to get an answer that is quickly obtained and close enough to judge by.

Estimate $612+387$. Rounded to the nearest hundred these become $600+400=1000$. The exact answer is 999, so the estimate was excellent.

Estimate $48\times 21$. Rounded to the nearest ten, $50\times 20=1000$. The exact answer is 1008.

Estimate $5,940÷29$. Rounding gives $6000÷30=200$, and the exact answer is close to that.

The real use of estimation is catching blunders. If you work out $48\times 21$ on paper and get 108, the estimate of about 1000 tells you instantly that a digit went missing — without needing to redo the multiplication.

Estimates are deliberately approximate, so never present one as the exact answer unless the question asks you to estimate.