Category Archives: Fractions

October 4, 2025 · 8:36 am

Converting base 10 numbers to base 2

Converting integers to base 2 is reasonably easy.

For example, what is 82 in base 2?

Think about powers of 2

$n$	$2^n$
$0$	$1$ (‘ones’)
$1$	$2$ (‘tens’)
$2$	$4$ (‘hundreds)
$3$	$8$ (‘thousands’)

Make $82$ the sum of powers of $2$ .

$82=64+16+2=1\times 2^6+0\times 2^5+ 1\times 2^4+0\times 2^3+0\times 2^2+1 \times 2^1+0\times 2^0=1010010$

We follow the same approach for real numbers

$n$	$2^n$
$-3$	$\frac{1}{8}=0.125$
$-2$	$\frac{1}{4}=0.25$
$-1$	$\frac{1}{2}=0.5$
$0$	$1$ (‘ones’)
$1$	$2$ (‘tens’)
$2$	$4$ (‘hundreds)
$3$	$8$ (‘thousands’)

Convert $0.765625$ to base $2$

$0.765625\times 2=1.53125$ the first number is 1

$0.53125\times 2=1.0625$ the second number is 1

$0.0625\times 2=0.125$ the third number is 0

$0.125\times 2=0.25$ the fourth number is 0

$0.25\times 2=0.5$ the fifth number is 0

$0.5\times 2=1$ the sixth number is 1 and we have finished

$0.765625=0.110001_2$

What about something like $2.\overline{4}$ ?

The non-decimal part $2=10$

$0.\overline{4}\times 2=0.\overline{8}$ first number is zero

$0\overline{8}\times 2=1.\overline{7}$ second number is 1

$0.\overline{7}\times 2=1.\overline{5}$ third number is 1

$0.\overline{5}\times 2=1.\overline{1}$ fourth number is 1

$0.\overline{1}\times 2=0.\overline{2}$ fifth number is 0

$0.\overline{2}\times 2=0.\overline{4}$ sixth number is 0

We are back to where we started, so $2.\overline{4}=10.0111000111000..._2=10.\overline{0.11100}_2$

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Filed under Arithmetic, Decimals, Fractions, Number Bases

Tagged as base 2, binary, number bases

July 1, 2025 · 10:35 am

Synthetic Division (for factorising and/or solving polynomials)

I use synthetic division to factorise polynomials with a degree greater than 2. For example, $f(x)=x^3+2x^2-5x-6$

It works best with monic polynomials but can be adapted to non-monic ones (see example below).

The only problem is that you need to find a root to start.

Try the factors of $-6$ i.e. $(-1, 1, 2, -2, 3, -3, 6, -6)$

$f(-1)=(-1)^3+2(-1)^2-5(-1)-6=0$

Hence, $x=-1$ is a root and $(x+1)$ is a factor of the polynomial.

Set up as follows

Bring the first number down

Multiply by the root and place under the second co-efficient

Add down

Repeat the process

The numbers at the bottom $(1, 1, -6)$ are the coefficients of the polynomial factor.

We now know $x^3+2x^2-5x-6=(x+1)(x^2+x-6)$ .

We can factorise the quadratic in the usual way.

$x^2+x-6=(x+3)(x-2)$

Hence $x^3+2x^2-5x-6=(x+1)(x+3)(x-2)$ .

Let’s try a non-monic example

Factorise $6x^4+39x^3+91x^2+89x+30$

I know $-2$ is a root. Otherwise I would try the factors of 30.

Use synthetic division

Because this was non-monic we need to divide our new co-efficients $(6, 27, 37, 15)$ by 6 (the co-efficient of the $x^4$ term)

$x^3+\frac{9}{2}x^2+\frac{37}{6}+\frac{5}{2}$

We now need to go again. I know that $\frac{-3}{2}$ is a root and $(2x+3)$ is a factor.

Our quadratic factor is $x^2+3x+5/3$ , which is $3x^2+9x+5$ .

The quadratic factor doesn’t have integer factors so,

$6x^4+39x^3+91x^2+89x+30=(x+2)(2x+3)(3x^2+9x+5)$

I think this is much quicker than polynomial long division.

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Filed under Algebra, Factorising, Factorising, Fractions, Polynomials, Quadratic, Simplifying fractions, Solving Equations

Tagged as non monic synthetic division, Synthetic division

June 19, 2025 · 7:18 am

Converting recurring (non-terminating) decimals to fractions

The easiest approach is to jump right in with some examples.

Example 1

Convert $0.\overline{5}$ to a fraction.

Let $x=0.\overline{5}$

(1) $\begin{equation*}x=0.\overline{5}\end{equation*}$

(2) $\begin{equation*}10x=5.\overline{5}\end{equation*}$

Subtract equation $1$ from equation $2$

$\begin{equation*}9x=5\end{equation}$

Hence $x=\frac{5}{9}$ so $0.\overline{5}=\frac{5}{9}$

Example 2

Convert $0.\overline{12}$ to a fraction.

Let $x=0.\overline{12}$

(3) $\begin{equation*}x=0.\overline{12}\end{equation*}$

(4) $\begin{equation*}100x=12.\overline{12}\end{equation*}$

Subtract equation $3$ from equation $4$ .

$\begin{equation*}99x=12\end{equation}$

$\begin{equation*}x=\frac{12}{99}=\frac{4}{33}\end{equation}$

Example 3

Convert $0.1\overline{23}$ to a fraction

Let $x=0.1\overline{23}$

(5) $\begin{equation*}x=0.1\overline{23}\end{equation*}$

(6) $\begin{equation*}10x=1.\overline{23}\end{equation*}$

(7) $\begin{equation*}1000x=123.\overline{23}\end{equation*}$

Subtract equation $6$ from equation $7$

$\begin{equation*}990x=122\end{equation}$

$\begin{equation*}x=\frac{122}{990}=\frac{61}{495}\end{equation}$