From tally marks to place value

Imagine you are a shepherd $30{,}000$ years ago and you want to remember how many sheep you have. You scratch one mark on a bone for each sheep. After $50$ sheep your bone has $50$ marks, and counting them takes nearly as long as counting the sheep themselves. There must be a better way , and there is.

Concept

Tally marks were the first writing of numbers. You group them in fives , $||||\!\!\!/$ , so you can count by fives instead of by ones. This is a small improvement but still painful for large amounts.

Egyptian numerals (about $5000$ years ago) used different symbols for $1$, $10$, $100$, $1000$ and so on. The number $342$ would be three "$100$" symbols, four "$10$" symbols, and two "$1$" symbols , a lot of writing for one number.

Roman numerals (still seen on clock faces and chapter titles) used $\text{I, V, X, L, C, D, M}$. The number $1989$ is $\text{MCMLXXXIX}$. Try multiplying $\text{XLVII} \times \text{XIX}$ on a piece of paper , it is nearly impossible without converting.

The breakthrough was place value. In our system, the same digit means different things depending on where it stands. In $342$, the $3$ means $300$, the $4$ means $40$, and the $2$ means $2$. We are saying:

$342 = 3 \times 100 + 4 \times 10 + 2 \times 1 = 3 \times 10^2 + 4 \times 10^1 + 2 \times 10^0.$

The same ten digits , $0$ through $9$ , can write every whole number, no matter how big, just by adjusting positions. This is the decimal (base-$10$) system. It is so good that we now use it everywhere from postal codes to bank balances.

A few hidden gifts of place value:

• Adding and subtracting is reduced to a column-by-column algorithm.
• Multiplying and dividing become long but mechanical.
• Reading any whole number is fast , "$1{,}24{,}305$" is one lakh, twenty-four thousand, three hundred five.

Place value also extends past the decimal point. After the dot, digits represent tenths, hundredths, thousandths:

$0.327 = \tfrac{3}{10} + \tfrac{2}{100} + \tfrac{7}{1000}.$

This is the very same idea , each step right divides the place value by $10$.

There are bases other than $10$. Computers use base $2$ (binary), with only digits $0$ and $1$. The number $13$ in binary is $1101$, meaning $1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 8 + 4 + 1 = 13$. Base $60$ is why we still measure time in $60$-minute hours , a leftover from ancient Babylon.

Worked examples

Example 1. Write $5046$ using place value notation.

• $5046 = 5 \times 10^3 + 0 \times 10^2 + 4 \times 10^1 + 6 \times 10^0$
• The $0$ in the hundreds place is essential , without it, the number would read as $546$.

Example 2. What does the $7$ stand for in $470{,}320$?

• The $7$ is in the ten-thousands place.
• It means $7 \times 10{,}000 = 70{,}000$.

Example 3. Convert Roman $\text{MCDXLIV}$ to our system.

• $M = 1000$, $CD = 400$, $XL = 40$, $IV = 4$.
• Total: $1000 + 400 + 40 + 4 = 1444$.

Example 4. Convert $1101_{2}$ (binary) to base $10$.

• $1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 4 + 0 + 1 = 13$.

Try it yourself

1. Write $6{,}049$ in expanded place-value form.
2. What does each $4$ mean in $4{,}494$?
3. Convert $\text{LXXVIII}$ to base $10$.
4. Convert $389$ to Roman numerals.
5. Convert $10110_{2}$ (binary) to base $10$.
6. Write $42$ in binary.
7. Without place value, addition would be painful. Add $\text{XXVII} + \text{XLV}$ (you may convert first).
8. Why does the number $402$ specifically need the digit $0$?

Activity

Bead counters. Cut out three columns labelled "hundreds", "tens", "units". Use beads or beans to fill them according to any number ($265$ = two hundreds, six tens, five units). Now move beads from one column to another. Each move of ten beads from one column to one bead of the next demonstrates "regrouping". This is exactly what happens in mental arithmetic but made visible.