N1, N2 - Integers, decimals and written arithmetic

Ordering numbers and comparison symbols

Which is warmer: $-0.5\,\mathrm{°C}$ or $-0.57\,\mathrm{°C}$? The digits alone can tempt you into the wrong answer. Their positions on a number line settle it.

An integer is a whole number, including zero and negative whole numbers. A decimal can lie between integers: $-2.4$ is between $-3$ and $-2$. Numbers increase to the right. Thus $-8 < -3$, even though $-8$ is farther from zero. Distance from zero and numerical order are different ideas.

[DIAGRAM: asset_name: Signed numbers on a number line; asset_slug: 1ma1_m01_signed_order; caption: Numbers farther right are greater, including negative decimals.; description: A number line from minus 4 to 2 marks minus 3.2, minus 1.5, zero and 0.8 in increasing order, with an arrow showing that values increase to the right.]

Use these symbols to express the comparison precisely.

The open end of an inequality faces the larger value. Reversing the numbers reverses the sign: $-4 < -1$ and $-1 > -4$ say the same thing. To compare positive decimals, align decimal points and use trailing zeros to make the columns visible.

Now change only the signs. The positive magnitudes have the same order, but the negative values lie to the left of zero.

A longer decimal need not be larger: there is even no greatest decimal below $1$. Given any number $d < 1$, its midpoint with $1$ is larger than $d$ but still below $1$.

Try the same comparison with a different final digit in one value.

The value of a digit

A digit's position tells you its contribution to the number. Each move one column left makes that contribution ten times as large; each move right divides it by ten.

In $4{,}072{,}305$, the $4$ contributes four million, the $7$ contributes seventy thousand and the $3$ contributes three hundred. Zeros hold empty places. Removing the zero between $4$ and $7$ would change the value. Commas group digits for reading; they do not change the number.

The first column after the decimal point is always tenths. In $0.00062$, the $6$ is in the ten-thousandths column, so its value is $0.0006$.

Scaling changes the values of all the digits together.

Change only multiplication to division. The digits move in the opposite direction.

The same reasoning gives $62 \div 1000 = 0.062$: six tens become six hundredths and two units become two thousandths.

Use the same division with one changed digit.

Written operations and negative integers

Written methods keep units aligned and record exchanges, so you can handle large calculations one small step at a time.

For addition, align units under units. In $478 + 267$, $8 + 7 = 15$: write $5$ units and carry $1$ ten. Then $7 + 6 + 1 = 14$: write $4$ tens and carry $1$ hundred. Finally $4 + 2 + 1 = 7$, giving $745$.

For $602 - 278$, exchange a hundred for ten tens, then one ten for ten units. This preserves the value of $602$:

The inverse check is $324 + 278 = 602$. For signed addition, moving right adds a positive value and moving left adds a negative value. Thus $-7 + 12 = 5$ and $-7 + (-12) = -19$. Subtracting a negative reverses that leftward move: $-7 - (-12) = -7 + 12 = 5$.

For multiplication or division, matching signs give a positive answer and different signs give a negative answer. Work out the magnitude first.

Change only the sign of the first factor. The written multiplication is unchanged.

Long division uses the same place values. To find $936 \div 24$, $24$ goes into $93$ three times. Write $3$ in the tens column and subtract $72$, leaving $21$ tens. Bring down $6$ to make $216$. Nine lots of $24$ give $216$, so the quotient is $39$. Check $39 \times 24 = 936$. Therefore $(-936) \div (-24) = 39$ too.

With a positive divisor, a remainder must be at least zero and less than the divisor. You can continue into decimal places when appropriate. Division by zero is undefined: for example, no number multiplied by zero gives $5$.

Change the magnitude of the negative factor and keep the multiplication method.

Calculating with decimals

Decimal calculations use integer methods once you keep track of each digit's place value.

For addition and subtraction, align decimal points. Write $12.7 + 0.86$ as $12.70 + 0.86 = 13.56$. Similarly, $5.2 - 1.87 = 5.20 - 1.87 = 3.33$. The added zeros allow exchanges in the hundredths column. Adding less than one to $12.7$ cannot produce a result near $21$.

For multiplication, first use integers, then restore the scale. For division, scale both numbers by the same power of ten until the divisor is an integer. These rules are not on the formula sheet; learn them through place value.

Change only the sign of $2.4$. The scale and positive magnitude stay the same.

For division, $6.84 \div 0.3 = 68.4 \div 3 = 22.8$. Multiplying both numbers by $10$ preserves their ratio. Multiplying just one would change it. Check $22.8 \times 0.3 = 6.84$; changing the dividend to $-6.84$ gives quotient $-22.8$.

When the divisor is already whole, keep the quotient's decimal point aligned with the dividend's. For $7.35 \div 5$, divide $7$ units to get $1$ unit with $2$ left. The remainder and $3$ tenths make $23$ tenths: divide to get $4$ tenths with $3$ tenths left. Together with $5$ hundredths this is $35$ hundredths, giving $7$ hundredths. The result is $1.47$, and $1.47 \times 5 = 7.35$.

The next product changes only the magnitude of the negative factor in the second example.

Quick Recap

• Compare place values first; negative values farther left are smaller.
• The six comparison symbols distinguish equality, inequality and whether equality is allowed.
• Written arithmetic aligns like places and records exchanges or partial products.
• Decimal multiplication restores both scales; decimal division scales both inputs together.
• Inverse operations and rough size checks help catch sign and place-value errors.

The warmer temperature is $-0.5\,\mathrm{°C}$, because $-0.5 > -0.57$.