C1.1, C1.6; E1.1, E1.6 - Place value and whole-number calculations

C1.1, C1.6; E1.1, E1.6 - Place value and whole-number calculations

A zero can change a number enormously without adding anything of its own. You will read and write large whole numbers, explain the value of each digit, and use the four operations with checks that can catch a misplaced digit.

Place value in large numbers

How can the same digit mean 77 in one number and 7 000 0007\,000\,000 in another? Its position gives it its value: seven loose objects and seven groups of a million are very different amounts.

Our place-value system uses ten digits, from 00 to 99. Moving one place left makes the value represented by a digit ten times as large. Ten ones make a ten; ten tens make a hundred; ten hundreds make a thousand. This pattern continues however large the number becomes.

Place value

The place value of a digit is the value of the position it occupies. Its contribution to the number is the digit multiplied by that place value. For example, the 77 in 47 20547\,205 contributes 7×1 000=7 0007\times1\,000=7\,000.

Read a large number in groups of three digits from the right. Spaces separate the groups without changing the value. Each group has hundreds, tens and ones of its named unit.

BillionsMillionsThousandsUnits
33407407020020605605

The table represents 3 407 020 6053\,407\,020\,605. A million is 1 000 0001\,000\,000; a billion is 1 000 000 0001\,000\,000\,000, which is a thousand million. The group 020020 represents twenty thousand. Its first zero keeps the hundreds-of-thousands position empty.

Worked example 1: a digit in the millions group

Question. Write down the value of the digit 77 in 3 407 020 6053\,407\,020\,605.

Plan. Locate the group containing 77, then identify its position within that group.

407 million=400 million+7 millionsplit the group7×1 000 000=7 000 000seven millions\begin{aligned} 407\text{ million}&=400\text{ million}+7\text{ million} && \text{split the group}\\ 7\times1\,000\,000&=7\,000\,000 && \text{seven millions} \end{aligned}

Answer. 7 000 0007\,000\,000.

Check. Removing those seven millions gives 3 400 020 6053\,400\,020\,605. Only the millions digit has changed.

Move the 77 one place left within the millions group. Only its position changes, so its value becomes ten times as large.

Worked example 2: the same digit one place left

Question. Write down the value of the digit 77 in 3 470 020 6053\,470\,020\,605.

Plan. Read the millions group as hundreds, tens and ones of a million.

470 million=400 million+70 millionsplit the group7×10 000 000=70 000 000seven ten-millions\begin{aligned} 470\text{ million}&=400\text{ million}+70\text{ million} && \text{split the group}\\ 7\times10\,000\,000&=70\,000\,000 && \text{seven ten-millions} \end{aligned}

Answer. 70 000 00070\,000\,000.

Check. 70 000 000÷10=7 000 00070\,000\,000\div10=7\,000\,000, matching the value one place to the right in Example 1.

Zero contributes nothing in its own position, but it holds other digits in their positions. Removing the zero from 205205 gives 2525, a different number.

Watch out

The digit and its value are different answers. In 52 80452\,804, the digit in the thousands place is 22; its value is 2 0002\,000.

Numbers in words and figures

Group names let you read a long number without counting every digit separately. Start with the leftmost group, say its number and its group name, then move right. Skip a group containing only zeros when speaking, but keep all three places when writing it in figures.

For example, 8 000 0148\,000\,014 is “eight million and fourteen”. There are no thousands to say. In figures, the empty thousands group must still be 000000, and fourteen must occupy the final group as 014014.

Worked example 3: reading a billion-sized number

Question. Write 2 063 004 0182\,063\,004\,018 in words.

Plan. Name each nonzero group, working from billions to units.

2∣063∣004∣018⟶2 billion, 63 million, 4 thousand, 18read each group2 063 004 018=2 000 000 000+63 000 000+4 000+18check the group values\begin{aligned} 2\mid063\mid004\mid018 &\longrightarrow 2\text{ billion},\ 63\text{ million},\ 4\text{ thousand},\ 18 && \text{read each group}\\ 2\,063\,004\,018&=2\,000\,000\,000+63\,000\,000+4\,000+18 && \text{check the group values} \end{aligned}

Answer. Two billion, sixty-three million, four thousand and eighteen.

Check. The four spoken group values add back to the original number. The zeros have kept 6363, 44 and 1818 in the correct groups.

Now reverse the direction: the wording is the same, and you must rebuild the figure groups. This is a useful independent way to check a conversion.

Worked example 4: rebuilding the figure groups

Question. Write “two billion, sixty-three million, four thousand and eighteen” in figures.

Plan. Reserve three places for each group after the first, filling unused places with zeros.

billions∣millions∣thousands∣units⟶2∣063∣004∣018keep all group places2 000 000 000+63 000 000+4 000+18=2 063 004 018combine the values\begin{aligned} \text{billions}\mid\text{millions}\mid\text{thousands}\mid\text{units} &\longrightarrow2\mid063\mid004\mid018 && \text{keep all group places}\\ 2\,000\,000\,000+63\,000\,000+4\,000+18&=2\,063\,004\,018 && \text{combine the values} \end{aligned}

Answer. 2 063 004 0182\,063\,004\,018.

Check. Read it back aloud: two billion, sixty-three million, four thousand and eighteen. Each group matches the question.

You may see commas used as group separators in ordinary writing. Here spaces separate thousands, while a decimal point is reserved for a part of a whole number.

Watch out

“Five million and nine” is 5 000 0095\,000\,009, not 5 0095\,009. Keep the empty thousands group as well as the two zeros before the 99.

Whole-number calculations and inverse checks

Whole-number arithmetic works by combining or separating equal place-value units. Addition combines amounts, subtraction finds a remainder or gap, multiplication combines equal groups, and division finds the number or size of equal groups.

In column addition and subtraction, line up units with units, tens with tens, and so on. Ten units can be exchanged for one ten; one hundred can be exchanged for ten tens. These exchanges change the way an amount is written, without changing the amount itself.

Worked example 5: combining two amounts

Question. A store has 3 7863\,786 notebooks and receives another 2 4572\,457. Work out the new total.

Plan. Add matching places, exchanging whenever a column reaches ten.

6+7=13write 3 units, carry 1 ten8+5+1=14write 4 tens, carry 1 hundred7+4+1=12write 2 hundreds, carry 1 thousand3+2+1=6combine thousands3 786+2 457=6 243read the resulting places\begin{aligned} 6+7&=13 && \text{write 3 units, carry 1 ten}\\ 8+5+1&=14 && \text{write 4 tens, carry 1 hundred}\\ 7+4+1&=12 && \text{write 2 hundreds, carry 1 thousand}\\ 3+2+1&=6 && \text{combine thousands}\\ 3\,786+2\,457&=6\,243 && \text{read the resulting places} \end{aligned}

Answer. 6 2436\,243 notebooks.

Check. Split the added amount: 3 786+2 000+400+50+7=6 2433\,786+2\,000+400+50+7=6\,243. This reaches the same total without the column carries.

Reverse the operation with the same three amounts. Addition and subtraction are inverse operations: one undoes the other.

Worked example 6: undoing the addition

Question. A store has 6 2436\,243 notebooks and sends out 2 4572\,457. Work out how many remain.

Plan. Subtract matching places, exchanging from the next place when necessary.

6 243=5 000+1 100+130+13exchange across the columns13−7=6units13−5=8tens11−4=7hundreds5−2=3thousands6 243−2 457=3 786combine the remaining places\begin{aligned} 6\,243&=5\,000+1\,100+130+13 && \text{exchange across the columns}\\ 13-7&=6 && \text{units}\\ 13-5&=8 && \text{tens}\\ 11-4&=7 && \text{hundreds}\\ 5-2&=3 && \text{thousands}\\ 6\,243-2\,457&=3\,786 && \text{combine the remaining places} \end{aligned}

Answer. 3 7863\,786 notebooks.

Check. 3 786+2 457=6 2433\,786+2\,457=6\,243, so the remainder and the amount removed reconstruct the original stock.

Multiplication can also be split by place value. For 2424 equal groups, calculate 2020 groups and 44 groups, then add. In division, keep track of how much has been shared: any remainder must be smaller than the nonzero divisor. Division by zero is undefined.

Worked example 7: multiplying equal groups

Question. Each box holds 3636 pencils. Work out the number of pencils in 2424 full boxes.

Plan. Split 2424 into 20+420+4, then combine the two products.

36×20=720twenty full boxes36×4=144four more boxes36×24=720+144=864combine all twenty-four boxes\begin{aligned} 36\times20&=720 && \text{twenty full boxes}\\ 36\times4&=144 && \text{four more boxes}\\ 36\times24&=720+144=864 && \text{combine all twenty-four boxes} \end{aligned}

Answer. 864864 pencils.

Check. Halve 2424 and double 3636: 72×12=720+144=86472\times12=720+144=864. Regrouping gives the same total.

Reverse this multiplication to find a missing group count. The quantities stay the same; the unknown changes from total pencils to number of boxes.

Worked example 8: division as missing groups

Question. 864864 pencils are packed into boxes of 3636. Work out the number of full boxes.

Plan. Remove convenient multiples of 3636, counting the boxes used.

864−20×36=144fill twenty boxes144÷36=4fill four more boxes864÷36=20+4=24count all the boxes\begin{aligned} 864-20\times36&=144 && \text{fill twenty boxes}\\ 144\div36&=4 && \text{fill four more boxes}\\ 864\div36&=20+4=24 && \text{count all the boxes} \end{aligned}

Answer. 2424 boxes, with no pencils left over.

Check. 24×36=86424\times36=864, which reconstructs the total. For a division with a remainder, check divisor times quotient plus remainder instead.

An inverse check asks a different question from the original calculation, so it can reveal an error you might repeat by simply doing the same calculation again. When a division has a remainder, its meaning matters: leftover objects may need an extra container even though the quotient counts only the full ones.

Watch out

In 36×2436\times24, the 22 means 2020, so that partial product is 720720, not 7272. Losing this place value makes a plausible-looking answer much too small.

Quick Recap

  • A digit's position determines its value; each place to the left is ten times as large.
  • Read large numbers in three-digit groups. A billion is a thousand million.
  • Keep placeholder zeros when converting words into figures.
  • Add and subtract matching places; split multiplication into useful groups and check division by multiplication.

The opening 77 can mean 77 or 7 000 0007\,000\,000 because its position tells you what size units it counts.