0580 · 1.1Types of number

Place value and number families

0:00Twelve counters can form several equal groups, but thirteen counters cannot, unless each group contains just one counter. That small difference reveals an important property of numbers. Numbers have different features, and recognising those features helps us solve problems. In this video, we will connect number families, factors, multiples and reciprocals, rather than memorising separate lists. We need two starting ideas about multiplication and division. Multiplication can describe equal groups, such as three groups of four making twelve.

0:29Division tells us whether a quantity can be shared into equal groups without a remainder. We will use those ideas to explain why some numbers divide others exactly. Reading a number correctly depends on place value, not simply its length. Grouping digits into blocks of three helps us recognise thousands, millions and billions. Six billion is written as six followed by nine zeros. Ten thousand and seven contains ten thousands and seven units, with zeros holding the empty places.

0:55For an exam-style conversion, write six billion in figures and write the displayed number in words. Six billion becomes six thousand million, giving the digits shown on screen. The second number separates into 10000 + 7. Its name is therefore ten thousand and seven, not ten thousand and seventy. Natural numbers are the counting numbers, starting with one, two and three, and continuing without end. Integers include positive whole numbers, zero and negative whole numbers.

1:22Negative three is an integer, but three fifths is not. Zero is neither positive nor negative, although it belongs to the integers. Some mathematical sources include zero among the natural numbers, so you may encounter that convention elsewhere. Here, we use natural numbers for counting from one. More importantly, number descriptions can overlap. Sixteen is a natural number and an integer, and it also has other properties we will discover.

Factors, primes and powers

1:46A factor of a positive integer divides it exactly, leaving no remainder. Three is a factor of twelve because 12 ÷ 3 = 4. The matching multiplication statement is 3 × 4 = 12. Factors therefore come in pairs, which helps us find them systematically. For twelve, the factor pairs are one and twelve, two and six, and three and four. Reading both numbers from each pair gives all six positive factors.

2:12A prime number is an integer greater than one with exactly two positive factors. Those factors are one and the number itself. Thirteen is prime because only one and thirteen divide it exactly. One is not prime because it has only one positive factor, not two. Two is the only even prime, since every larger even number also has two as a factor. A positive integer greater than one that is not prime is called composite.

2:35A square number is obtained by multiplying an integer by itself. 4 × 4 gives sixteen, which can be arranged as a square of dots. A cube number comes from multiplying an integer by itself and then by itself again. 2 × 2 × 2 gives eight, representing two layers of a two by two arrangement. The positive square numbers begin one, four, nine, sixteen and twenty-five. The positive cube numbers begin one, eight, twenty-seven and sixty-four.

3:03Learn these lists, because spotting them quickly saves time without a calculator. A common mistake is to confuse squaring with doubling, so 5² is twenty-five, not ten.

Prime factors, HCF and LCM

3:13Prime factors are factors that are themselves prime numbers. Expressing a number as a product of its prime factors means breaking it into multiplication using only primes. These primes act like building blocks for the original number. We stop splitting only when every remaining factor is prime. Express seventy-two as a product of its prime factors. First, 72 = 2 × 36. Splitting thirty-six gives 2 × 2 × 18.

3:40Splitting eighteen into 2 × 9 gives three twos multiplied by nine. Finally, splitting nine leaves 2 × 2 × 2 × 3 × 3. Every remaining factor is now prime, so the decomposition is complete. We can write the result as 2³ × 3². 2³ = 8, and 3² = 9. 8 × 9 returns seventy-two, which checks our answer. Common factors divide both numbers exactly. For example, the common factors of twelve and eighteen are one, two, three and six.

4:16The highest common factor is the greatest positive integer that divides both. Prime factorisation reveals the building blocks available in both numbers. To make a common factor, we can use only the prime factors they share. Find the highest common factor of seventy-two and ninety. We already know that 72 = 2³ × 3². 90 = 2 × 45. Splitting forty-five gives 2 × 3 × 15. Splitting fifteen gives 2 × 3 × 3 × 5.

4:49Ninety therefore equals 2 × 3² × 5. Both numbers contain at least one factor of two and two factors of three. Their highest common factor is 2 × 3², which becomes 2 × 9. This gives eighteen, with no additional shared prime factors available. A multiple is obtained by multiplying a number by an integer. The positive multiples of six begin six, twelve, eighteen and twenty-four. Common multiples appear in both numbers’ lists of multiples.

5:19The lowest common multiple is the smallest positive number in both lists, so zero is not our answer. Find the lowest common multiple of seventy-two and ninety using their prime factors. A multiple of seventy-two needs at least three factors of two and two factors of three. A multiple of ninety also needs a factor of five. The smallest collection satisfying both requirements is 2³ × 3² × 5. This becomes 8 × 9 × 5.

5:47Multiplying eight by nine gives seventy-two, leaving 72 × 5. The result is three hundred and sixty. Dividing by seventy-two gives five, while dividing by ninety gives four, confirming that both divide it exactly. The distinction explains a common exam mistake. For the highest common factor, use the smallest number of copies of each shared prime. For the lowest common multiple, use the largest number of copies needed by either number.

6:13Multiplying the original numbers can produce a common multiple, but not necessarily the lowest one.

Rational numbers and reciprocals

6:18A rational number can be written as one integer divided by another integer, with a non-zero denominator. Three fifths is rational, and negative four is rational because it equals −4/1. Terminating decimals are rational because they can be written as fractions with denominators such as ten or one hundred. 0.75 = 75/100. Recurring decimals are also rational, even though their digits continue forever. One third gives zero point three recurring, with the same digit repeating.

6:47An irrational number cannot be written as a ratio of two integers. Its decimal expansion neither terminates nor eventually repeats, so a long decimal alone does not establish irrationality. √2 and pi are examples of irrational numbers. However, √9 is three, which is rational. A square root symbol does not automatically make a number irrational. A calculator displays only finitely many digits, so its screen cannot prove whether the full decimal eventually repeats.

7:15The reciprocal of a non-zero number is one divided by that number. Multiplying a number by its reciprocal gives one. For a fraction, swapping the numerator and denominator works because everything then cancels in the product. The reciprocal of five is one fifth, while the reciprocal of three fifths is five thirds. Find the reciprocal of −3/4. 1 ÷ −3/4 gives −4/3. Checking the product gives 12/12, which equals one.

7:41The negative sign remains because two negative factors produce a positive product. Zero has no reciprocal because division by zero is undefined. These identification and factorisation methods must work without a calculator on Paper Two. Paper Four permits a calculator, but clear working still supports your answers. Questions may ask you to identify number types, express prime factors, or find a highest common factor or lowest common multiple. Our answers here are exact, so rounding to three significant figures is unnecessary.

8:11Number families describe properties, and one number can have several descriptions. Factors divide exactly, while multiples contain complete copies of a number. Prime factors explain the difference between the greatest shared divisor and the smallest shared multiple. Rational numbers are ratios of integers, irrational numbers are not, and reciprocals multiply with their original numbers to give one.