Numbers Large and Small

Unitary Rates Describe Proportions A unitary rate is a way of describing a proportional relationship 400 grains of rice = 7.0g 800 grains = 14.0g 200 grains = 3.5g 100 grains = 1.75g 1600 grains = 28.0g Two quantities related in this way are said to be proportional. When the number of grains gets smaller, the corresponding mass gets smaller

If any number of grains of rice is divided by the corresponding mass, the result is the same: 400 = 57.142857 7.0 So, no matter how many grains of rice, the number of grains divided by their mass should be 57.142857 Any two quantities that are proportional produce a constant number when one of the quantities is divided by the other: A/B = k A and B represent the two quantities, and k represents the constant number produced when one quantity is divided by the other. In science, the units are included so that the constant has a sensible meaning:   400 grains = 57.142857 grains 7.0 g 1g

1grain You could also divide the mass by the number of grains. 7.0g = 0.0175 g 400 grains 1 grain The two proportionality constants are unitary rates: grains per gram, and grams per grain We can use this unitary rate to calculate the mass of any number of grains of rice. What is the mass of 1 000 000 grains of rice? ? grams = 1 000 000 grains x 0.0175g = 17 500 g 1grain ? kilograms = 17 500 g x 1kg = 17.5 kg 1000 g

Scientific Notation Most large numbers in science have a long string of zeros at the end, and the small ones have a long string of zeros at the beginning. Scientific Notation writes the number as a product of some number between one and ten multiplied by some power of ten. 32 represents an exponent number. 2 is called the exponent; 3 is called the base The exponent indicates the number of times the base would be multiplied by itself : 33 = 3x3x3 17 500 = 1.75 x 104 0.000175 = 1.75 x 10-4

Multiplying and Dividing Exponential Numbers (10 x 10 x 10 x 10) x (10 x 10 x 10) = 104 x 103 = 107 The product of two powers of ten is obtained by adding the exponents.