Number Properties & Simplification

At a glance 12 formulas

1. 1Order of operationsBODMAS: brackets first, then 'of', then division and multiplication from left to right, then addition and subtraction, also from left to right ($10 - 4 + 2 = 8$).
2. 2Odd and evenOdd + odd and even + even are even; odd + even is odd. A product is odd only when every factor is odd.
3. 3Primes1 is not prime and 2 is the only even prime. Every prime above 3 has the form $6k \pm 1$. To test $N$, try only the primes up to $\sqrt{N}$.
4. 4Perfect squaresA perfect square ends in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8, and it has an odd number of factors.
5. 5Recurring decimalsPure: repeating block over that many 9s, $0.\overline{36} = \tfrac{36}{99}$. Mixed: all digits minus the non-repeating part, over 9s then 0s, $0.1\overline{6} = \tfrac{15}{90}$.
6. 6Digits and place value$\overline{ab} = 10a + b$$\overline{ab} - \overline{ba} = 9(a - b),\ \ \overline{ab} + \overline{ba} = 11(a + b)$
7. 7Base b$(pqr)_b = p\,b^2 + q\,b + r$$\text{every digit is less than } b$
8. 8Finding the baseIn the units column, if the digits add up to $s$ but the digit written is $d$ with a carry of 1, the base is $s - d$.
9. 9Counting one digitFrom 1 to 99 each digit 1 to 9 appears 20 times: 10 times as units, 10 times as tens. Count place by place: units, tens, hundreds.
10. 10Digits to number pages$\text{digits for pages } 1 \text{ to } N = 3N - 108$$\text{when } N \text{ has three digits}$
11. 11Skipping the squares$\text{non-squares in } 1 \text{ to } N = N - \lfloor \sqrt{N} \rfloor$
12. 12Telescoping$\dfrac{1}{n(n+1)} = \dfrac{1}{n} - \dfrac{1}{n+1}$$\dfrac{1}{n(n+2)} = \dfrac{1}{2}\left(\dfrac{1}{n} - \dfrac{1}{n+2}\right)$