N1 - Numbers and Arithmetic

Numbers are rounded to make them easier to handle...

• Numbers are rounded up if the 'next' digit is 5 or more, e.g. 

     8374 rounded to the nearest 10 = 8370

• When rounding to a given number of significant figures, start counting at the first non-zero digit, e.g.

    16,487,593 rounded to 2 SF (significant figures) = 16,000,000

• Round measurements to a realistic degree of accuracy, e.g.

    If you are using a meter rules, give your answer to the nearest half centimeter.

If the next digit is 5 or more, round up.

For SF, start counting at the non-zero digit.

If the height of a tree is given as five meters, correct to the nearest meter...

It could be anywhere between 4.5 meters and 5.5 meters...

4.5m = lower bound

5.5m = upper bound

• The upper bound is not actually included in the range of possible values...
• The lower bound is the lowest measurement a rounded value could be...
• The upper bound is the highest measurement a rounded value could be...

E.G. 

• The upper and lower bounds of 5.8m = 5.85m/5.75m 
• The upper and lower bounds of 35mm, correct to the nearest 5mm = 37.5mm/32.5mm

Multiplying doesn't always make numbers bigger, and dividing doesn't always make numbers smaller...

• Multipying by numbers less  than 1 = makes positive numbers smaller
• Dividing by numbers less than 1 = makes positive numbers bigger
• Multiplying/Dividing by 1 = no change
• Multiplying by numbers more than 1 = makes positive numbers bigger
• Dividing by numbers more than 1 = makes positive numbers smaller 

E.G. 8 x 0.5 = 4 

• Dividing by a number is the same as multiplying by it's reciprocal... 

9 ÷ 4 = 9 x ¼ = 2¼ 

• Multiplying  by a number is the same as dividing by it's reciprocal...

2 x 4 = 2 ÷ ¼ = 8

You can use approximations to one SF to make estimates...

You need to be careful when estimating with powers: 

• 1.3 is quite close to 1, but 1.3³ is not close to 1³!

E.G. (563 + 1.58) ÷ (327 - 4.72) = 600 ÷ 300 = 200

200 = estimate 

• You can use standard form to make estimations with very large or small numbers 

Estimating involves rounding  any numbers that are hard to work with!

You can add or subtract fractions if they have the same denominator...

• You can make two demoninators the same by multiplying them together, but whatever you do to the denominator, you have to do to the numerator, too.

½ + ¼ = 4/8 + 2/8 = 6/8 = 3/4 

• To multiply fractions, multiply the numerators together, and multiple the demoninators together.

¼  x ½ = 1/8 

• Dividing by any number is the same as multiplying by its multiplicative inverse (e.g. 4/1 is the multiplicative inverse for 1/4). 

½ ÷ ¼  = ½ x 4/1 = 4/2 = 2/1 = 2

By dividing the numerator by the denominator, you can convert a fraction to a decimal!

Terminating or Recurring?

Terminating = the denominator has only got the factors 2 and 5 (e.g. 8= 2x2x2) 

Recurring = the denominator has got factors other than 2 and/or 5 (e.g. 6 = 2x3) 

To convert a terminating decimal to a fraction, write the decimal as the numerator (e.g. 0.385 as 385) and the denominator as a power of 10. 

0.385 = 385/1000 = 77/200

To convert a recurring decimal, do the following... 

x = 0.36(recurring) > 100x = 36.36(recurring) > 99x = 36 > x = 36/99 = 4/11

Converting between Fractions, Percentages and Decimals...

• To convert a decimal to a percentage, multiply it by 100%. 

0.65 = 0.65 x 100% = 65%

• To convert a percentage to a decimal, divide it by 100.

18.3% = 18.3 ÷ 100 = 0.183

• To convert a fraction to a percentage, divide the numerator by the denominator and then multiply it by 100%.

5/8 = 5÷8 = 0.625 x 100% = 62.5%