Place Value

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For integers

We probably mostly understand place value for integers.

$$4321 = 4\times1000 + 3\times100 + 2\times10 + 1$$

Multiplying by 10

An understanding of place value, together with the distributive property, is the key to understanding why we mulitply by 10 by just slapping a zero on the end.

\begin{align} 10\times321 &= 10\times(3\times100 + 2\times10 + 1) \\ &= 3\times1000 + 2\times100 + 10 = 3210 \end{align}

For decimals

Place value for decimals is very similar but we use powers of $1/10$ to the right of the decimal point.

$$1.23 = 1 + 2\times\frac{1}{10} + 3\times\frac{1}{100}$$

Multiplying by 10

The right way to multiply by 10 is now to slide the decimal point to the right.

\begin{align} 10\times1.23 &= 10\times\left(1 + 2\times\frac{1}{10} + 3\times\frac{1}{100}\right) \\ &= 10\times1 + 10\times\frac{2}{10} + 10\times\frac{3}{100} \\ &= 10 + 2 + \frac{3}{10} = 12.3 \end{align}

Dividing by 10

We can divide by 10 by sliding the decimal point to the left.

\begin{align} \frac{1.23}{10} &= \frac{1}{10}\times\left(1 + 2\times\frac{1}{10} + 3\times\frac{1}{100}\right) \\ &= \frac{1}{10}\times1 + \frac{1}{10}\times\frac{2}{10} + \frac{1}{10}\times\frac{3}{100} \\ &= \frac{1}{10} + \frac{2}{100} + \frac{3}{1000} = 0.123 \end{align}

Computing tips

  • $10\%$: Move the decimal over one spot to the left
    • Example: $(0.1)\times\$22.50 = \$2.25$
  • $20\%$: Move the decimal over and double
    • Example: $(0.2)\times\$22.50 = \$4.50$
  • $5\%$: Move the decimal over and cut in half
    • Example: $(0.05)\times\$22.50 = \frac{1}{2}\times\$2.25 = \$1.13$
  • $15\%$: Move the decimal over one spot to the left, cut in half, and add those two together
    • Example: $(0.15)\times\$22.50 = \$2.25 + \$1.13 = \$3.38$

The $20\%$ trick follows exactly from associativity:

$$ \begin{align} 0.2 \times \$22.50 &= (2\times0.1) \times \$22.50 \\ &= 2\times(0.1\times\$22.50) \\ &= 2\times\$2.25 = \$4.50 \end{align} $$

The $15\%$ trick follows exactly from distributivity:

$$ \begin{align} 0.15 \times \$22.50 &= (0.1+0.05) \times \$22.50 \\ &= 0.1\times\$22.50 + 0.05\times\$22.50 \\ &= \$2.25 + \$1.13 = \$3.38 \end{align} $$

Other multiplication tricks

I ran a Google search for "multiplication tricks" and found this page.

All the tricks there can be explained using the algebraic properties of multiplication.

$\times\,4$

Trick: Double, then double again.

Example: $4\times9$

  • First double: $9\to18$
  • Then, double again: $18\to36$

Explanation: $4\times n = (2\times2)\times n = 2\times(2\times n)$

$\times\,5$

Trick: Multiply by 10, then cut in half.

Example: $5\times9$

  • Multiply by $10$: $9\to90$
  • Then, cut in half: $90\to45$

Explanation: $$5\times n = \left(\frac{1}{2}\times10\right)\times n = \frac{1}{2}\times(10\times n)$$

$\times\,9$

Trick: Multiply by 10, then subtract the number itself.

Example: $9\times13$

  • Multiply by $10$: $13\to130$
  • Then, subtract $13$: $130\to117$

Explanation: $9\times n = (10-1)\times n = 10\times n - n$