Natural Numbers

Worked example

1. 1

   $\sqrt{25}=5$ — Work out the value first: 5 × 5 = 25, so √25 = 5. It is whole and positive, so it is natural.

2. 2

   $\frac{18}{6}=3$ — A fraction line means divide. 18 ÷ 6 = 3 with nothing left over, so this is natural too.

3. 3

   $(-2{)}^2=(-2)\times (-2)$ — Squaring means multiply the number by itself. Keep the −2 in brackets so the minus is not lost.

4. 4

   $(-2)\times (-2)=4$ — Minus times minus is plus, so the answer is 4. The minus has gone, so this is natural.

5. 5

   $\frac{10}{4}=2.5$ — 10 ÷ 4 = 2.5. It has a decimal part, so it fails the whole-number test: not natural.

Worked example

1. 1

   $3.2<n<8.7$ — The question: n is bigger than 3.2 and smaller than 8.7. Neither end is a whole number.

2. 2

   $4$ — The first whole number above 3.2 is 4. 3 is too small, because 3 < 3.2.

3. 3

   $8$ — The last whole number below 8.7 is 8. 9 is too big, because 9 > 8.7.

4. 4

   $4,5,6,7,8$ — Write out every whole number from 4 to 8, so none is missed.

5. 5

   $\text{count}=5$ — Count the list. There are 5 natural numbers.

Worked example

1. 1

   $12,13,14,\dots ,40$ — The question. Both ends count, but writing 29 numbers out is slow and easy to get wrong.

2. 2

   $40-12=28$ — Last minus first gives 28. This counts the gaps between the numbers, not the numbers.

3. 3

   $28+1=29$ — There is always one more number than gaps, so add 1. There are 29 numbers.

Worked example

1. 1

   $n^2<30$ — The question: which natural numbers have a square less than 30? Start at 1, the first natural number.

2. 2

   $1^2=1,2^2=4,3^2=9$ — Square each one in turn. All are less than 30, so 1, 2 and 3 fit.

3. 3

   $4^2=16,5^2=25$ — Still less than 30, so 4 and 5 fit too.

4. 4

   $6^2=36$ — 36 is more than 30, so 6 fails. Bigger numbers give bigger squares, so you can stop here.

5. 5

   $n=1,2,3,4,5$ — The answer: five natural numbers.

Worked example

1. 1

   $5+8=13$ — Add: the total of two counting numbers is bigger than both, so adding is always natural.

2. 2

   $5\times 8=40$ — Times: this is 8 + 8 + 8 + 8 + 8, just repeated adding, so it is always natural too.

3. 3

   $8-5=3$ — Take away, bigger minus smaller: 3 is natural.

4. 4

   $5-8=-3$ — Now the other order. Taking a bigger number away goes below zero, and −3 is not natural.

5. 5

   $40÷8=5$ — Divide: 40 ÷ 8 = 5 exactly, so this time the answer is natural.

6. 6

   $8÷5=1.6$ — But 8 ÷ 5 leaves a decimal. 1.6 is not whole, so dividing only sometimes works.

Worked example

1. 1

   $305040$ — The question. Count three digits in from the right. That splits it into 305 and 040.

2. 2

   $305040$ — The left group is the thousands. Read it, then say thousand: three hundred and five thousand.

3. 3

   $305040$ — The right group is read on its own. 040 is forty: the 0 at the front only holds a place, so you say nothing for it.

4. 4

   $305\text{ thousand and }40$ — Join the parts: three hundred and five thousand and forty. Say “and” before a last part under 100.

Worked example

1. 1

   $2\text{ million}$ — Millions come first, so write 2. Two groups of three digits must follow it.

2. 2

   $2070$ — Seventy thousand fills the thousands group. Seventy is 70, so write 070: the 0 in front keeps three digits.

3. 3

   $2070006$ — Six fills the last group. Write 006, again three digits.

4. 4

   $2070006$ — The answer. Check: a number from one million up to ten million has 7 digits.