Edexcel GCSE Mathematics

## Place value and ordering Each digit in a number is worth ten times the digit to its right, so in $4\,062.35$ the 6 is worth 60, the 3 is worth $\frac{3}{10}$ and the 5 is worth $\frac{5}{100}$. A zero holds a place open without adding anything. Place value is what lets you multiply and divide by 10, 100 and 1000 by shifting digits, and what lets you compare decimals: $0.09$ is less than $0.4$ because $0.4$ has 4 tenths and $0.09$ has none. To order decimals, write them with the same number of decimal places (pad with zeros) and compare digit by digit from the left. To order fractions, give them a common denominator or convert them to decimals. On a number line the smaller number is always to the left, so $-7 < -2$, even though 7 is bigger than 2. The inequality symbols are $<$ (less than), $>$ (greater than), $\le$ (less than or equal to) and $\ge$ (greater than or equal to), and $\ne$ means not equal to. The open end of the symbol faces the bigger number. For example $-3 \le n < 2$ means $n$ can be $-3$ but must be smaller than $2$, so the integers that fit are $-3, -2, -1, 0, 1$. ## Negative numbers Adding a negative number is the same as subtracting the positive, and subtracting a negative is the same as adding the positive. $$ \begin{aligned} 5 + (-3) &= 2 \\ 5 - (-3) &= 8 \\ -5 - 3 &= -8 \end{aligned} $$ When multiplying or dividing, two numbers with the same sign give a positive answer and two with different signs give a negative one. So $(-4) \times (-6) = 24$…